CFA Level I Quantitative Methods formulas

Time value of money, returns, probability, statistics and hypothesis tests: the formulas behind most of the calculator work in the course.

258 formulas from the Quantitative Methods lessons (11–14% of the exam), 159 to memorize. Each says how deeply to learn it and opens to a quick worked example with the keystrokes for the TI BA II Plus and the HP 12C.

Memorize only Advanced (hide foundations) Hide what I know

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

Module 1 Returns of Financial Assets

Required return: risk-free rate, inflation and risk premia

Required return build-up Use it

r_i \approx r_{rf} + \text{infl} + \sum_{j=1}^{n} f_{ij}\, rp_j

Quick example

Real risk-free 1.0%, inflation 2.5%, market premium 5.0% (exposure 1.1), size premium 2.0% (exposure 0.4), value premium 1.5% (exposure 0.3). 1.0 + 2.5 + 5.5 + 0.8 + 0.45 = 10.25%.

On your calculator

1+2.5+(1.1×5)+(0.4×2)+(0.3×1.5)= 10.25

1.1ENTER5× 5.5
0.4ENTER2×+ 6.3
0.3ENTER1.5×+ 6.75
3.5+ 10.25, adding the 1.0 and 2.5

ENTER enter (on the HP, puts the number on the stack)

Real risk-free rate (exact) Memorize

r_{rf} = \dfrac{1 + r_f}{1 + \text{infl}} - 1

Quick example

A bill yields 6.0% and inflation is 2.5%. 1.06 ÷ 1.025 − 1 = 3.41%, against 3.50% from subtraction.

On your calculator

1.06÷1.025−1= 0.0341, so 3.41%

1.06ENTER1.025÷1− 0.0341

ENTER enter (on the HP, puts the number on the stack)

Excess return Memorize

\dot{r} = \dfrac{1 + r_i}{1 + r_f} - 1 \approx r_i - r_f

Quick example

Shares return 8.4% while a bill returns 2.0%. 1.084 ÷ 1.02 − 1 = 6.27% exactly, against 6.40% by subtraction.

On your calculator

1.084÷1.02−1= 0.0627

1.084ENTER1.02÷1− 0.0627

ENTER enter (on the HP, puts the number on the stack)

Holding period return

Holding period return Memorize

r = \frac{P_1 - P_0 + Inc}{P_0} = r_{price} + r_{distribution}

Quick example

Buy at 40.00, receive 0.80 of dividends, sell at 43.00. (43.00 − 40.00 + 0.80) ÷ 40.00 = 9.5%.

On your calculator

43−40+0.8=÷40= 0.095

43ENTER40−0.8+40÷ 0.095

ENTER enter (on the HP, puts the number on the stack)

Price return Memorize

r_{price} = \frac{P_1 - P_0}{P_0}

Quick example

Bought at 60.00, sold at 63.00. 3.00 ÷ 60.00 = 5.0%.

Capital distribution return Memorize

r_{distribution} = \frac{Inc}{P_0}

Quick example

Dividends of 1.20 a year on a share bought at 48.00. 1.20 ÷ 48.00 = 2.5%, the dividend yield.

Multi-period holding period return Memorize

r = (1 + r_1)(1 + r_2)\cdots(1 + r_T) - 1

Quick example

Returns of 8%, −5% and 12%. 1.08 × 0.95 × 1.12 − 1 = 14.91%.

On your calculator

1.08×0.95×1.12−1= 0.1491

1.08ENTER0.95×1.12×1− 0.1491

ENTER enter (on the HP, puts the number on the stack)

Arithmetic and geometric mean returns

Arithmetic mean return Memorize

\bar{r} = \frac{1}{T}\sum_{t=1}^{T} r_t

Quick example

+60%, −40% and +10%. (60 − 40 + 10) ÷ 3 = 10% a year.

Geometric mean return Memorize

\bar{r}_G = \left[\prod_{t=1}^{T}(1 + r_t)\right]^{1/T} - 1

Quick example

+60%, −40% and +10%. (1.60 × 0.60 × 1.10)^(1/3) − 1 = 1.056^(1/3) − 1 = 1.83% a year.

On your calculator

1.6×0.6×1.1= growth of 1 over three years, 1.056
y^x31/x= the cube root, 1.0183
−1= 0.0183

y^x power · 1/x reciprocal

1.6ENTER0.6×1.1× 1.056
31/xy^x the cube root, 1.0183
1− 0.0183

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Annualized and continuously compounded returns

Annualized return Memorize

r_{annualized} = (1 + r_{period})^{c} - 1

Quick example

2.5% over 3 months, so c = 4. 1.025⁴ − 1 = 10.38%.

On your calculator

1.025y^x4=−1= 0.1038

y^x power

1.025ENTER4y^x1− 0.1038

ENTER enter (on the HP, puts the number on the stack) · y^x power

Continuous growth Use it

FV = PV \times e^{\tilde{r}\,t}

Quick example

GBP 1,000 at 6% for one year: 1,000 × e^0.06 = 1,061.84.

On your calculator

0.062NDe^x×1000= 1,061.84

2ND second function (the shifted label above a key) · e^x e to the power x

0.06ge^x1000× 1,061.84

g blue shift key · e^x e to the power x

Continuously compounded return Memorize

\tilde{r}_{t,t+1} = \ln\left(\frac{P_{t+1}}{P_t}\right) = \ln(1 + r_{t,t+1})

Quick example

A price rises from 38 to 43. ln(43 ÷ 38) = ln 1.1316 = 12.36%.

On your calculator

43÷38=LN 0.1236

LN natural logarithm

43ENTER38÷gLN 0.1236

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · LN natural logarithm

Gross, net, after-tax, real and leveraged returns

From total to net Use it

r_{gross} = r_{total} - \text{transaction costs};\ r_{net} = r_{gross} - \text{management fees} - \text{administrative expenses}

Quick example

Total 8.0%, transaction costs 0.2%, management fee 0.6%, administrative expenses 0.1%. Gross 8.0 − 0.2 = 7.8%; net 7.8 − 0.6 − 0.1 = 7.1%.

On your calculator

8−0.2= 7.8, gross
−0.6−0.1= 7.1, net

8ENTER0.2− 7.8, gross
0.6−0.1− 7.1, net

ENTER enter (on the HP, puts the number on the stack)

After-tax nominal return Memorize

r_{net-tax} = r_{price}(1 - tax_{capital\ gains}) + r_{distribution}(1 - tax_{distributions})

Quick example

Price return 6%, distribution return 3%, gains taxed at 25%, distributions at 40%. 6 × 0.75 + 3 × 0.60 = 4.5 + 1.8 = 6.3%.

On your calculator

6×0.75=STO1 4.5
3×0.6+RCL1= 6.3

STO store · RCL recall a stored number

6ENTER0.75× 4.5
3ENTER0.6×+ 6.3

ENTER enter (on the HP, puts the number on the stack)

Real return (exact) Memorize

r_{real} = \frac{1 + r_i}{1 + \text{infl}} - 1

Quick example

Nominal 6.5%, inflation 2.5%. 1.065 ÷ 1.025 − 1 = 3.90%, against 4.00% from subtraction.

On your calculator

1.065÷1.025−1= 0.0390

1.065ENTER1.025÷1− 0.0390

ENTER enter (on the HP, puts the number on the stack)

Leveraged return Memorize

r_{LP} = r_P + \frac{V_D}{V_E}\left(r_P - r_d\right)

Quick example

Equity 600,000, borrowed 400,000 at 5%, portfolio return 9%. 9% + (400 ÷ 600)(9% − 5%) = 11.67%.

On your calculator

9−5=×400÷600+9= in percent: 11.67

9ENTER5−400×600÷9+ 11.67

ENTER enter (on the HP, puts the number on the stack)

Call option return Use it

r_{call} = \frac{\max(0,\ P_T - K) - \text{premium}}{\text{premium}}

Quick example

Strike 30, premium 2, share ends at 35. (5 − 2) ÷ 2 = 150%. The break-even is 32, and at or below 30 the call loses the whole premium.

On your calculator

35−30−2=÷2= 1.50, so 150%

35ENTER30−2−2÷ 1.50

ENTER enter (on the HP, puts the number on the stack)

Module 2 Types of Financial Returns

Types of returns: assets, instruments and indicators

Total return: price plus distribution Memorize

r = \frac{P_1 - P_0 + \text{Inc}}{P_0} = \frac{P_1 - P_0}{P_0} + \frac{\text{Inc}}{P_0}

Quick example

Buy at 60.00, receive a 1.50 dividend, sell at 63.00. (63.00 − 60.00 + 1.50) ÷ 60.00 = 7.5%: price return 5.0% plus distribution return 2.5%.

On your calculator

63−60+1.5=÷60= 0.075

63ENTER60−1.5+60÷ 0.075

ENTER enter (on the HP, puts the number on the stack)

Government bill: return over its life Memorize

r_{period} = \frac{\text{Face}}{\text{Price}} - 1

Quick example

A 182-day bill with face value 10,000 is priced at 9,780. 10,000 ÷ 9,780 − 1 = 2.2495% over its life.

On your calculator

10000÷9780−1= 0.0225

10000ENTER9780÷1− 0.0225

ENTER enter (on the HP, puts the number on the stack)

Government bill: annualized return Memorize

r_{annual} = (1 + r_{period})^{365/\text{days}} - 1

Quick example

The same 182-day bill. 1.022495 raised to 365 ÷ 182, minus 1, is 4.56%. Doubling the period return would give 4.50%.

On your calculator

10000÷9780=y^x(365÷182)=−1= 0.0456

y^x power

10000ENTER9780÷365ENTER182÷y^x1− 0.0456

ENTER enter (on the HP, puts the number on the stack) · y^x power

Currency return Use it

r_{currency} = \frac{S_1}{S_0} - 1

Quick example

The home-currency price of one unit of the foreign currency rises from 0.9200 to 0.9568. 0.9568 ÷ 0.9200 − 1 = +4.00%.

On your calculator

0.9568÷0.92−1= 0.04

0.9568ENTER0.92÷1− 0.04

ENTER enter (on the HP, puts the number on the stack)

Foreign asset in your home currency Memorize

r_{home} = (1 + r_{local})(1 + r_{currency}) - 1

Quick example

Local return 5.0%, and the foreign currency falls 3.0% against yours. 1.05 × 0.97 − 1 = 1.85%.

On your calculator

1.05×0.97−1= 0.0185

1.05ENTER0.97×1− 0.0185

ENTER enter (on the HP, puts the number on the stack)

Geometric mean return Memorize

r_G = \left[(1 + r_1)(1 + r_2)\cdots(1 + r_n)\right]^{1/n} - 1

Quick example

Returns of +30% and −20%. The arithmetic mean is 5.00%. (1.30 × 0.80)^(1/2) − 1 = 1.98%.

On your calculator

1.3×0.8=y^x0.5=−1= 0.0198

y^x power

1.3ENTER0.8×0.5y^x1− 0.0198

ENTER enter (on the HP, puts the number on the stack) · y^x power

Long-run asset class returns and their components

Relative outperformance (ratio form) Memorize

r_{rel} = \frac{1 + r_A}{1 + r_B} - 1

Quick example

Equities earn 10.4% a year and bonds 7.1%. 1.104 ÷ 1.071 − 1 = 3.08% a year. The simple difference, 3.30%, overstates it.

On your calculator

1.104÷1.071−1= 0.0308

1.104ENTER1.071÷1− 0.0308

ENTER enter (on the HP, puts the number on the stack)

Macro and micro risk and the risk-return trade-off

Range of annual returns Use it

\text{Range} = r_{best} - r_{worst}

Quick example

An asset's best year is +18% and its worst is −9%. The range is 18 − (−9) = 27 percentage points.

On your calculator

18−9+|-= 27

+|- change sign

18ENTER9CHS− 27

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Market risk premium (ratio form, exact) Use it

\text{MRP} = \frac{1 + E[r_m]}{1 + r_f} - 1

Quick example

Expected market return 6.5%, risk-free rate 2.0%. 1.065 ÷ 1.02 − 1 = 4.41%.

On your calculator

1.065÷1.02−1= 0.0441

1.065ENTER1.02÷1− 0.0441

ENTER enter (on the HP, puts the number on the stack)

Market risk premium (difference form, approximate) Use it

\text{MRP} \approx E[r_m] - r_f

Quick example

The same inputs: 6.5% − 2.0% = 4.50%. It is 0.09 percentage points above the exact 4.41%.

On your calculator

6.5−2= 4.5

6.5ENTER2− 4.5

ENTER enter (on the HP, puts the number on the stack)

Portfolio return Memorize

r_{portfolio} = w_1 r_1 + w_2 r_2 + \cdots + w_N r_N = \sum_{i=1}^{N} w_i r_i

Quick example

50% in equities at 8.8%, 35% in bonds at 5.4%, 15% in bills at 2.2%. 4.40 + 1.89 + 0.33 = 6.62%.

On your calculator

(0.5×8.8)+(0.35×5.4)+(0.15×2.2)= 6.62

0.5ENTER8.8×0.35ENTER5.4×+0.15ENTER2.2×+ 6.62

ENTER enter (on the HP, puts the number on the stack)

Module 3 Benchmarking Returns

Money-weighted vs time-weighted rate of return

Money-weighted rate of return Use it

0 = \sum_{t=0}^{T} \frac{CF_t}{(1 + r_{MW})^t}

Quick example

Flows of −80 at t = 0, −100 at t = 1 and +216 at t = 2. The internal rate of return of those flows, and so the money-weighted rate of return, is 13.30% a year.

On your calculator

CF2NDCLR WORK open the cash flow worksheet and clear it
80+|-ENTER↓ CF0 = −80
100+|-ENTER↓ C01 = −100, and F01 stays at 1
↓216ENTER C02 = 216
IRRCPT 13.30

CF cash flow worksheet · 2ND second function (the shifted label above a key) · CLR WORK clear the worksheet · +|- change sign · ENTER enter (on the HP, puts the number on the stack) · IRR internal rate of return · CPT compute

fCLEAR REG clear the cash flow registers
80CHSgCF0 the first flow
100CHSgCFj t = 1
216gCFj t = 2
fIRR 13.30

f gold shift key · CLEAR REG clear all registers · CHS change sign · g blue shift key · CF0 cash flow at time zero · CFj cash flow j · IRR internal rate of return

Time-weighted rate of return Memorize

r_{TW} = (1 + r_1)(1 + r_2)\cdots(1 + r_T) - 1

Quick example

Quarterly returns of 5%, −5%, 5% and 4% between cash flows. 1.05 × 0.95 × 1.05 × 1.04 − 1 = 8.93% for the year.

On your calculator

1.05×0.95×1.05×1.04−1= 0.0893

1.05ENTER0.95×1.05×1.04×1− 0.0893

ENTER enter (on the HP, puts the number on the stack)

Time-weighted rate of return over several years (annualized) Memorize

r_{TW} = \left[(1 + r_1)(1 + r_2)\cdots(1 + r_T)\right]^{1/T} - 1

Quick example

Year 1: 80 to 100 (+25%). Year 2: 100 to 108 (+8%). (1.25 × 1.08)^(1/2) − 1 = 16.19% a year.

On your calculator

1.25×1.08= year 1 × year 2 growth, 1.35
√x−1= two years, so the square root: 0.1619

√x square root

1.25ENTER1.08× 1.35
g√x1− 0.1619

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Annualizing a sub-period money-weighted rate of return Use it

r_{MW,\,annual} = (1 + r_{MW})^{k} - 1

Quick example

A quarterly internal rate of return of 2.0371% annualizes to 1.020371⁴ − 1 = 8.40%, not 4 × 2.0371% = 8.15%.

On your calculator

1.020371y^x4−1= 0.0840

y^x power

1.020371ENTER4y^x1− 0.0840

ENTER enter (on the HP, puts the number on the stack) · y^x power

Index value, price return and total return

Index value Use it

V_t = \frac{\text{aggregate reference value}_t}{D}

Quick example

Constituents worth 10,800 in total and a divisor of 10.8 give an index of 1,000. When they are worth 11,360 the index is 11,360 ÷ 10.8 = 1,051.85.

On your calculator

11360÷10.8= 1051.85

11360ENTER10.8÷ 1051.85

ENTER enter (on the HP, puts the number on the stack)

Index price return Memorize

r_{price} = \frac{V_1 - V_0}{V_0}

Quick example

A price return index moves from 2,000 to 2,060. (2,060 − 2,000) ÷ 2,000 = 3.00%.

On your calculator

2060−2000=÷2000= 0.03

2060ENTER2000−2000÷ 0.03

ENTER enter (on the HP, puts the number on the stack)

Index total return Memorize

r_{total} = \frac{V_1 - V_0 + Inc}{V_0}

Quick example

Index 2,000 to 2,060 with 24 points of income (capital distributions). (2,060 − 2,000 + 24) ÷ 2,000 = 4.20%.

On your calculator

2060−2000+24=÷2000= 0.042

2060ENTER2000−24+2000÷ 0.042

ENTER enter (on the HP, puts the number on the stack)

Index return from constituent returns Memorize

r_{index} = \sum_i w_i r_i

Quick example

Weights 50%, 30% and 20%. Price returns of 6%, 2% and −4% give 2.8%. Adding dividend yields of 2%, 1% and 3% makes total returns of 8%, 3% and −1%, so the index total return is 0.50 × 8 + 0.30 × 3 + 0.20 × (−1) = 4.7%.

On your calculator

(0.5×8)+(0.3×3)−(0.2×1)= 4.7

0.5ENTER8×0.3ENTER3×+0.2ENTER1×− 4.7

ENTER enter (on the HP, puts the number on the stack)

Price-weighted and equal-weighted indexes

Price-weighted index value Memorize

V_{PRI} = \frac{\sum_i P_i}{D}

Quick example

Prices 30, 60 and 90 with a divisor of 0.18 (set for a base of 1,000). 180 ÷ 0.18 = 1,000.

On your calculator

30+60+90=÷0.18= 1000

30ENTER60+90+0.18÷ 1000

ENTER enter (on the HP, puts the number on the stack)

Price weight and price-weighted return Memorize

w_i = \frac{P_i}{\sum_j P_j}, \qquad r = \frac{\sum P_1 - \sum P_0}{\sum P_0}

Quick example

Prices go from 30, 60, 90 to 33, 57, 99. The sum rises from 180 to 189, so the return is 9 ÷ 180 = 5.00%. The 90 share carries 90 ÷ 180 = 50% of the weight.

On your calculator

189÷180−1= 0.05

189ENTER180÷1− 0.05

ENTER enter (on the HP, puts the number on the stack)

Divisor after a split Memorize

D_{new} = \frac{\sum P_{after}}{V_{before}}

Quick example

Prices 40, 60 and 100 with a divisor of 2.00 give an index of 100. The 100 share splits 2-for-1, so the sum falls to 150. New divisor = 150 ÷ 100 = 1.50.

On your calculator

40+60+50=÷100= 1.5

40ENTER60+50+100÷ 1.5

ENTER enter (on the HP, puts the number on the stack)

Equal-weighted index value Use it

V_{EQI} = \frac{\sum_i n_i P_i}{D}, \quad n_1 P_1 = n_2 P_2 = \cdots = n_N P_N

Quick example

Hold 90 X at 80, 288 Y at 25 and 160 Z at 45 (7,200 in each). 21,600 ÷ 21.6 = 1,000.

On your calculator

(90×80)+(288×25)+(160×45)=÷21.6= 1000

90ENTER80×288ENTER25×+160ENTER45×+21.6÷ 1000

ENTER enter (on the HP, puts the number on the stack)

Equal weight and equal-weighted return Memorize

w_i = \frac{1}{N}, \qquad r = \frac{1}{N}\sum_i r_i \ \text{(since the last rebalancing)}

Quick example

Four shares return 8%, −2%, 5% and 1%. (8 − 2 + 5 + 1) ÷ 4 = 3.00%.

On your calculator

8+2+|-+5+1=÷4= 3

+|- change sign

8ENTER2CHS+5+1+4÷ 3

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Market-cap, float-adjusted and fundamental weighting

Market-cap weight Memorize

w_i = \frac{Q_i P_i}{\sum_j Q_j P_j}

Quick example

Share A: 20 million shares at 35. Share B: 15 million shares at 28. Values 700 and 420, total 1,120. A carries 700 ÷ 1,120 = 62.5%.

On your calculator

(20×35)+(15×28)= 1120
700÷1120= 0.625

20ENTER35×15ENTER28×+ 1120
700ENTER1120÷ 0.625

ENTER enter (on the HP, puts the number on the stack)

Market-cap-weighted index value Memorize

V_{MCI} = \frac{\sum_i Q_i P_i}{D}

Quick example

Total market value 10,800 at the start, divisor 10.8: index 1,000. Market value 11,360 at the end: 11,360 ÷ 10.8 = 1,051.85.

On your calculator

11360÷10.8= 1051.85

11360ENTER10.8÷ 1051.85

ENTER enter (on the HP, puts the number on the stack)

Market-cap-weighted index return Memorize

r = \frac{\sum_i Q_i P_{i,1} + Div}{\sum_i Q_i P_{i,0}} - 1

Quick example

Total market value of the constituents goes from 1,200 to 1,230, with 15 of dividends (Div, the total paid by the constituents). (1,230 + 15) ÷ 1,200 − 1 = 3.75%. Without dividends the price return is 2.50%.

On your calculator

1230+15=÷1200−1= 0.0375

1230ENTER15+1200÷1− 0.0375

ENTER enter (on the HP, puts the number on the stack)

Float-adjusted market-cap weight Memorize

w_i = \frac{f_i Q_i P_i}{\sum_j f_j Q_j P_j}

Quick example

A holder owns 40% of a company with 100 million shares at 30. With float 60%, the index counts 0.60 × 100 × 30 = 1,800 of the 3,000 market value.

On your calculator

0.6×100×30= 1800

0.6ENTER100×30× 1800

ENTER enter (on the HP, puts the number on the stack)

Contribution to index return Use it

\text{contribution}_i = w_i \times r_i, \qquad r_{index} = \sum_i w_i r_i

Quick example

Weight 34.09%, return −4%: contribution 0.3409 × (−4%) = −1.36 points.

On your calculator

0.3409×4+|-= −1.36

+|- change sign

0.3409ENTER4CHS× −1.36

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Fundamental weight Memorize

w_i = \frac{F_i}{\sum_j F_j}

Quick example

Earnings of 60, 30 and 10 give weights of 60%, 30% and 10%, whatever the share prices are.

Rebalancing, reconstitution and what indexes are used for

Shares after rebalancing an equal-weighted index Use it

n_i^{new} = \frac{V_{aggregate} \div N}{P_i}

Quick example

Holdings worth 23,472 across 3 shares; Y trades at 24. 23,472 ÷ 3 ÷ 24 = 326 shares.

On your calculator

23472÷3=÷24= 326

23472ENTER3÷24÷ 326

ENTER enter (on the HP, puts the number on the stack)

Module 4 Time Value of Money in Finance

Pricing bonds: zero-coupon, coupon and perpetual

Future value Memorize

FV_t = PV_0 \times (1 + r)^{t}

Quick example

EUR 2,000 invested for 4 years at 5% a year, compounded annually, grows to 2,000 × 1.05⁴ = EUR 2,431.01.

On your calculator

2NDCLR TVM with P/Y set to 1
4N5I/Y2000+|-PV
CPTFV 2,431.01

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · +|- change sign · PV present value · CPT compute · FV future value

fCLEAR FIN
4n5i2000CHSPV
FV 2,431.01

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · CHS change sign · PV present value · FV future value

Present value Memorize

PV_0 = FV_t \times (1 + r)^{-t}

Quick example

EUR 2,439.78 due in 4 years, discounted at 5% compounded quarterly (1.25% for 16 quarters). 2,439.78 ÷ 1.0125¹⁶ = EUR 2,000.00.

On your calculator

2NDCLR TVM
16N1.25I/Y2439.78FV
CPTPV −2,000.00

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · FV future value · CPT compute · PV present value

fCLEAR FIN
16n1.25i2439.78FV
PV −2,000.00

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · FV future value · PV present value

Compounding m times a year Use it

FV_t = PV_0 \times \left(1 + \dfrac{r}{m}\right)^{m \times t}

Quick example

EUR 2,000 for 4 years at 5% a year compounded quarterly (m = 4): 2,000 × 1.0125¹⁶ = EUR 2,439.78. It really earns 1.0125⁴ − 1 = 5.09% a year.

On your calculator

2NDCLR TVM
16N1.25I/Y2000+|-PV
CPTFV 2,439.78

fCLEAR FIN
16n1.25i2000CHSPV
FV 2,439.78

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · CHS change sign · PV present value · FV future value

Price of a zero-coupon bond Memorize

PV = \dfrac{FV}{(1 + r)^{t}}

Quick example

Face value 100, matures in 8 years, yield 4.5%. 100 ÷ 1.045⁸ = 70.32.

On your calculator

2NDCLR TVM with P/Y set to 1
8N4.5I/Y100FV
CPTPV −70.32

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · FV future value · CPT compute · PV present value

fCLEAR FIN
8n4.5i100FV
PV −70.32

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · FV future value · PV present value

Price of a zero-coupon bond (continuous compounding) Know the idea

PV = FV \times e^{-rt}

Quick example

Face value 100, 8 years, continuously compounded yield 4.5%. 100 × e^(−0.36) = 69.77.

On your calculator

0.36+|-2NDe^x×100= 69.77

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.36CHSge^x100× 69.77

CHS change sign · g blue shift key · e^x e to the power x

Price of a coupon bond Memorize

PV = \sum_{k=1}^{N} \dfrac{PMT}{(1 + r)^{k}} + \dfrac{FV}{(1 + r)^{N}}

Quick example

5 years, 3.5% annual coupon, face value 100, yield 5%. N = 5, I/Y = 5, PMT = 3.5, FV = 100 gives 93.51, a discount.

On your calculator

2NDCLR TVM
5N5I/Y3.5PMT100FV
CPTPV −93.51

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · FV future value · CPT compute · PV present value

fCLEAR FIN
5n5i3.5PMT100FV
PV −93.51

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · FV future value · PV present value

Floating-rate coupon Use it

PMT = FV \times \dfrac{\text{reference rate} + \text{margin}}{\text{payments a year}}

Quick example

Face value USD 1,000,000, paid quarterly, reference rate 3.40%, margin 85 bp. 1,000,000 × 4.25% ÷ 4 = USD 10,625.

On your calculator

3.4+0.85=÷100÷4×1000000= 10,625

3.4ENTER0.85+100÷4÷1000000× 10,625

ENTER enter (on the HP, puts the number on the stack)

Price of a perpetual bond Memorize

PV = \dfrac{PMT}{r}

Quick example

Pays 4.25 a year forever, yield 5%. 4.25 ÷ 0.05 = 85.00.

On your calculator

4.25÷0.05= 85

4.25ENTER0.05÷ 85

ENTER enter (on the HP, puts the number on the stack)

Annuities and level-payment loans

Present value of an ordinary annuity Memorize

PV = PMT \times \dfrac{1 - (1 + r)^{-t}}{r}

Quick example

EUR 2,000 at the end of each year for 6 years, discounted at 5%. 2,000 × (1 − 1.05⁻⁶) ÷ 0.05 = EUR 10,151.38.

On your calculator

2NDCLR TVM with P/Y set to 1
6N5I/Y2000PMT
CPTPV −10,151.38

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PMT payment · CPT compute · PV present value

fCLEAR FIN
6n5i2000PMT
PV −10,151.38

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PMT payment · PV present value

Level payment on a loan Memorize

PMT = \dfrac{r \times PV}{1 - (1 + r)^{-t}}

Quick example

Borrow 350,000 over 30 years at 4.2% a year, paid monthly. r = 0.35% a month, t = 360. PMT = 1,711.56 a month.

On your calculator

2NDCLR TVM
360N0.35I/Y350000PV
CPTPMT −1,711.56

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · I/Y interest rate per year (per period when P/Y is 1) · PV present value · CPT compute · PMT payment

fCLEAR FIN
360n0.35i350000PV
PMT −1,711.56

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · i interest rate per period · PV present value · PMT payment

Interest in a payment Use it

\text{Interest} = r \times \text{opening balance}, \quad \text{Principal} = PMT - \text{Interest}

Quick example

Balance 250,000 at 0.4% a month, payment 1,432.49. Interest 1,000.00, principal 432.49.

On your calculator

250000×0.004= 1,000.00 interest
+|-+1432.49= 432.49 principal

+|- change sign

250000ENTER0.004× 1,000.00 interest
CHS1432.49+ 432.49 principal

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Valuing a stock from its dividends

Value with no growth Memorize

V_0 = \dfrac{D}{r}

Quick example

A preferred share pays 2.70 a year and investors require 9%. 2.70 ÷ 0.09 = 30.00.

On your calculator

2.7÷0.09= 30

2.7ENTER0.09÷ 30

ENTER enter (on the HP, puts the number on the stack)

Gordon growth model Memorize

V_0 = \dfrac{D_1}{r - g}, \quad D_1 = D_0\,(1 + g)

Quick example

Last dividend 2.80, growth 2% forever, required return 7%. D₁ = 2.856, so 2.856 ÷ 0.05 = 57.12.

On your calculator

2.8×1.02÷(0.07−0.02)= 57.12

2.8ENTER1.02× D₁
0.07ENTER0.02− r − g
÷ 57.12

ENTER enter (on the HP, puts the number on the stack)

Two-stage dividend discount model Use it

V_0 = \sum_{t=1}^{n} \dfrac{D_0 (1 + g_s)^{t}}{(1 + r)^{t}} + \dfrac{V_n}{(1 + r)^{n}}, \quad V_n = \dfrac{D_{n+1}}{r - g_l}

Quick example

D₀ 1.20, growth g_s = 15% for 2 years then g_l = 3.5% forever, r = 9.5%. D₁ 1.38, D₂ 1.587, V₂ = 1.587 × 1.035 ÷ 0.06 = 27.3758. Value = 1.38 ÷ 1.095 + (1.587 + 27.3758) ÷ 1.095² = 25.42.

On your calculator

CF2NDCLR WORK
0ENTER↓1.38ENTER↓↓28.96275ENTER
NPV9.5ENTER↓CPT 25.42

CF cash flow worksheet · 2ND second function (the shifted label above a key) · CLR WORK clear the worksheet · ENTER enter (on the HP, puts the number on the stack) · NPV net present value · CPT compute

fCLEAR REG
0gCF01.38gCFj28.96275gCFj9.5i
fNPV 25.42

f gold shift key · CLEAR REG clear all registers · g blue shift key · CF0 cash flow at time zero · CFj cash flow j · i interest rate per period · NPV net present value

Implied return and implied growth

Implied return of a zero-coupon bond Memorize

r = \left(\dfrac{FV}{PV}\right)^{1/t} - 1

Quick example

A 5-year zero is priced at 82.50 per 100. (100 ÷ 82.50)^(1/5) − 1 = 3.92% a year.

On your calculator

2NDCLR TVM with P/Y set to 1
5N82.5+|-PV100FV
CPTI/Y 3.92

2ND second function (the shifted label above a key) · CLR TVM clear the time value of money keys · N number of periods · +|- change sign · PV present value · FV future value · CPT compute · I/Y interest rate per year (per period when P/Y is 1)

fCLEAR FIN
5n82.5CHSPV100FV
i 3.92

f gold shift key · CLEAR FIN clear the financial registers · n number of periods · CHS change sign · PV present value · FV future value · i interest rate per period

Implied growth rate Memorize

g = r - \dfrac{D_1}{P_0}

Quick example

Forward P/E 18, payout ratio 45%, required return 8%: D₁ ÷ P₀ = 0.45 ÷ 18 = 2.5%, so g = 8% − 2.5% = 5.5%.

On your calculator

0.08−(0.45÷18)= 0.055

0.08ENTER0.45ENTER18÷− 0.055

ENTER enter (on the HP, puts the number on the stack)

Implied growth from the last dividend Use it

g = \dfrac{r\,P_0 - D_0}{P_0 + D_0}

Quick example

Price 36, last dividend 1.20, required return 9%. (0.09 × 36 − 1.20) ÷ (36 + 1.20) = 5.48%.

On your calculator

(0.09×36−1.2)÷(36+1.2)= 0.0548

0.09ENTER36×1.2−36ENTER1.2+÷ 0.0548

ENTER enter (on the HP, puts the number on the stack)

Forward P/E from the Gordon model Use it

\dfrac{P_0}{E_1} = \dfrac{D_1 / E_1}{r - g}

Quick example

Payout ratio 40%, required return 9%, growth 5%. 0.40 ÷ 0.04 = 10.0×.

On your calculator

0.4÷(0.09−0.05)= 10

0.4ENTER0.09ENTER0.05−÷ 10

ENTER enter (on the HP, puts the number on the stack)

Trailing P/E from the Gordon model Use it

\dfrac{P_0}{E_0} = \dfrac{(D_0 / E_0)(1 + g)}{r - g}

Quick example

Payout 40%, growth 5%, required return 9%: 0.40 × 1.05 ÷ 0.04 = 10.5×.

On your calculator

0.4×1.05÷(0.09−0.05)= 10.5

0.4ENTER1.05×0.09ENTER0.05−÷ 10.5

ENTER enter (on the HP, puts the number on the stack)

Cash flow additivity and implied forward rates

Cash flow additivity Use it

PV(A + B) = PV(A) + PV(B)

Quick example

A 2-year 5% coupon bond equals a 1-year zero paying 5 plus a 2-year zero paying 105. At spot rates 3.0% and 3.8%: 5 ÷ 1.03 + 105 ÷ 1.038² = 102.31.

On your calculator

5÷1.03=STO1 4.8544
105÷(1.038y^x2)=+RCL1= 102.31

STO store · y^x power · RCL recall a stored number

5ENTER1.03÷105ENTER1.038ENTER2y^x÷+ 102.31

ENTER enter (on the HP, puts the number on the stack) · y^x power

Implied one-year forward rate Memorize

(1 + S_2)^2 = (1 + S_1)(1 + F_{1,2}) \;\Rightarrow\; F_{1,2} = \dfrac{(1 + S_2)^2}{1 + S_1} - 1

Quick example

S₁ = 3.0%, S₂ = 3.8%. 1.038² ÷ 1.03 − 1 = 4.61%.

On your calculator

1.038y^x2=÷1.03−1= 0.0461

y^x power

1.038ENTER2y^x1.03÷1− 0.0461

ENTER enter (on the HP, puts the number on the stack) · y^x power

Implied forward rate, general form Memorize

(1 + S_B)^{B} = (1 + S_A)^{A}\,(1 + F_{A,B})^{B-A}

Quick example

S₁ = 3.0%, S₃ = 4.2%. The 2-year rate from year 1 to year 3, F₁,₃: (1.042³ ÷ 1.03)^(1/2) − 1 = 4.81% a year.

On your calculator

1.042y^x3=÷1.03=y^x0.5=−1= 0.0481

y^x power

1.042ENTER3y^x1.03÷0.5y^x1− 0.0481

ENTER enter (on the HP, puts the number on the stack) · y^x power

Forward rate from zero-coupon prices Use it

F_{1,2} = \dfrac{P_1}{P_2} - 1

Quick example

1-year zero 97.09, 2-year zero 92.81 per 100. 97.09 ÷ 92.81 − 1 = 4.61%.

On your calculator

97.09÷92.81−1= 0.0461

97.09ENTER92.81÷1− 0.0461

ENTER enter (on the HP, puts the number on the stack)

Forward exchange rates and option values by no arbitrage

Forward exchange rate (continuous compounding) Memorize

F = S\,e^{(r_P - r_B)\,T}

Quick example

INR 83.00 per USD, continuously compounded rates INR 6.5%, USD 4.5%, one year. 83.00 × e^0.02 = 84.68.

On your calculator

0.065−0.045=2NDe^x×83= 84.68

2ND second function (the shifted label above a key) · e^x e to the power x

0.065ENTER0.045−ge^x83× 84.68

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Forward exchange rate (annual compounding) Use it

F = S \times \dfrac{(1 + r_P)^{T}}{(1 + r_B)^{T}}

Quick example

CHF 0.9400 per EUR, one-year rates CHF 1.5%, EUR 3.0% (annual compounding). 0.9400 × 1.015 ÷ 1.03 = 0.9263.

On your calculator

0.94×1.015÷1.03= 0.9263

0.94ENTER1.015×1.03÷ 0.9263

ENTER enter (on the HP, puts the number on the stack)

Forward exchange rate (simple rates) Know the idea

F = S \times \dfrac{1 + r_P\,T}{1 + r_B\,T}

Quick example

USD 1.0800 per EUR, simple rates of 4% a year (USD) and 2% a year (EUR), six months. 1.08 × 1.02 ÷ 1.01 = 1.0907.

On your calculator

1.08×1.02÷1.01= 1.0907

1.08ENTER1.02×1.01÷ 1.0907

ENTER enter (on the HP, puts the number on the stack)

Hedge ratio Memorize

h = \dfrac{c^{+} - c^{-}}{S^{+} - S^{-}}

Quick example

Share 80, then 100 or 64. Call, exercise price 90, pays 10 or 0. h = 10 ÷ 36 = 0.2778 shares per call.

On your calculator

10÷36= 0.2778

10ENTER36÷ 0.2778

ENTER enter (on the HP, puts the number on the stack)

Call value by replication Use it

c_0 = h\,S_0 - \dfrac{h\,S^{-} - c^{-}}{1 + r}

Quick example

Same tree, risk-free rate 4%. 0.2778 × 80 − (0.2778 × 64 − 0) ÷ 1.04 = 22.22 − 17.09 = 5.13.

On your calculator

10÷36=STO1 0.2778
RCL1×64÷1.04=STO2 17.09
RCL1×80−RCL2= 5.13

STO store · RCL recall a stored number

10ENTER36÷STO1 0.2778
64×1.04÷STO2 17.09
RCL180×RCL2− 5.13

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Put value by replication Use it

p_0 = \dfrac{h\,S^{+} + p^{+}}{1 + r} - h\,S_0, \quad h = \dfrac{p^{-} - p^{+}}{S^{+} - S^{-}}

Quick example

Share 80, then 100 or 64. Put, exercise price 90, pays 0 or 26, r = 4%. h = 26 ÷ 36 = 0.7222; p₀ = (0.7222 × 100 + 0) ÷ 1.04 − 0.7222 × 80 = 11.67.

On your calculator

26÷36=STO1 0.7222
RCL1×100÷1.04=STO2 69.44
RCL1×80=STO3 57.78
RCL2−RCL3= 11.67

STO store · RCL recall a stored number

26ENTER36÷STO1 0.7222
100×1.04÷STO2 69.44
RCL2RCL180×− 11.67

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Module 5 Statistical Characteristics of Asset Returns

Mean, median, mode and what outliers do to them

Mean Memorize

\mu = \frac{\sum_{i=1}^{N} X_i}{N}, \qquad \bar{X} = \frac{\sum_{i=1}^{n} X_i}{n}

Quick example

Four quarterly returns of 3%, −1%, 4% and 2%. (3 − 1 + 4 + 2) ÷ 4 = 2.00%.

On your calculator

3+1+|-+4+2=÷4= 2

+|- change sign

3ENTER1CHS+4+2+4÷ 2

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Median position Memorize

n \text{ odd: position } \frac{n+1}{2}; \quad n \text{ even: average positions } \frac{n}{2} \text{ and } \frac{n}{2}+1

Quick example

Six sorted returns 1.1, 1.9, 2.4, 2.8, 3.5, 6.0. Average the 3rd and 4th: (2.4 + 2.8) ÷ 2 = 2.60%.

On your calculator

2.4+2.8=÷2= 2.6

2.4ENTER2.8+2÷ 2.6

ENTER enter (on the HP, puts the number on the stack)

Quartiles, percentiles and box plots

Percentile position, (n + 1) method Use it

L_y = (n + 1)\,\frac{y}{100}

Quick example

19 sorted returns, 80th percentile, (n + 1) method. L = (19 + 1) × 0.80 = 16, so the 16th value exactly.

On your calculator

19+1=×0.8= 16

19ENTER1+0.8× 16

ENTER enter (on the HP, puts the number on the stack)

Interpolated percentile Use it

X_y = X_{\text{lower}} + (L - \text{lower})\,(X_{\text{upper}} - X_{\text{lower}})

Quick example

L = 12.6 sits between the 12th value, 3.1, and the 13th, 3.6. 3.1 + 0.6 × (3.6 − 3.1) = 3.40%.

On your calculator

3.6−3.1=×0.6+3.1= 3.4

3.6ENTER3.1−0.6×3.1+ 3.4

ENTER enter (on the HP, puts the number on the stack)

Interquartile range Memorize

IQR = Q_3 - Q_1

Quick example

Q1 = −0.8% and Q3 = 2.2%. IQR = 2.2 − (−0.8) = 3.0 percentage points.

On your calculator

2.2−0.8+|-= 3

+|- change sign

2.2ENTER0.8CHS− 3

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Outlier fences Memorize

F_U = Q_3 + 1.5\,IQR, \qquad F_L = Q_1 - 1.5\,IQR

Quick example

Q1 = −0.6% and Q3 = 2.4%, so IQR = 3.0. Upper fence 2.4 + 1.5 × 3.0 = 6.90%; lower fence −0.6 − 4.5 = −5.10%.

On your calculator

2.4−0.6+|-=×1.5+2.4= 6.9

+|- change sign

2.4ENTER0.6CHS−1.5×2.4+ 6.9

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Range, mean absolute deviation, variance and standard deviation

Range Use it

\text{Range} = X_{\max} - X_{\min}

Quick example

Returns of 4%, −2%, 9% and 1%. Range = 9 − (−2) = 11 percentage points.

On your calculator

9−2+|-= 11

+|- change sign

9ENTER2CHS− 11

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Mean absolute deviation Use it

MAD = \frac{\sum_{i=1}^{n} |X_i - \bar{X}|}{n}

Quick example

Returns of 7%, −1%, 4% and 6% have mean 4. Distances 3, 5, 0, 2 add to 10, and 10 ÷ 4 = 2.5%.

On your calculator

3+5+0+2=÷4= 2.5

3ENTER5+0+2+4÷ 2.5

ENTER enter (on the HP, puts the number on the stack)

Median absolute deviation Use it

\text{Median absolute deviation} = \text{Median}\left(\,|X_i - \text{Median}(X)|\,\right)

Quick example

Returns 1%, 2%, 4%, 9%, 30%. Median 4; distances 3, 2, 0, 5, 26; sorted 0, 2, 3, 5, 26; median 3%. The 30% barely matters.

Population variance Memorize

\sigma^2 = \frac{\sum_{i=1}^{N} (X_i - \mu)^2}{N}

Quick example

A fund family has exactly four funds, returning 7%, −1%, 4% and 6%. μ = 4; squared deviations 9, 25, 0, 4 = 38; σ² = 38 ÷ 4 = 9.5, σ = 3.08%.

On your calculator

38÷4= 9.5
√x 3.0822

√x square root

38ENTER4÷ 9.5
g√x 3.0822

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Population standard deviation Use it

\sigma = \sqrt{\sigma^2}

Quick example

Population variance 9.5 (%²). √9.5 = 3.08%.

On your calculator

9.5√x 3.0822

√x square root

9.5g√x 3.0822

g blue shift key · √x square root

Sample variance Memorize

s^2 = \frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n - 1}

Quick example

Squared deviations of 4 returns add to 38. 38 ÷ (4 − 1) = 12.67 (%²).

On your calculator

38÷(4−1)= 12.67

38ENTER4ENTER1−÷ 12.67

ENTER enter (on the HP, puts the number on the stack)

Sample standard deviation Memorize

s = \sqrt{\frac{\sum_{i=1}^{n} (X_i - \bar{X})^2}{n - 1}}

Quick example

Variance 12.67 (%²). √12.67 = 3.56%.

On your calculator

12.67√x 3.56

√x square root

12.67g√x 3.56

g blue shift key · √x square root

Sample variance from a population variance Use it

s^2 = \sigma^2 \times \frac{n}{n - 1}

Quick example

σ² = 0.0090 from 30 returns. s² = 0.0090 × 30 ÷ 29 = 0.009310; s = 9.65%.

On your calculator

0.009×30÷29=√x 0.0965

√x square root

0.009ENTER30×29÷g√x 0.0965

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Semi-deviation, downside deviation and the coefficient of variation

Semi-variance Know the idea

\sigma^2_{Semi} = \sum_{X_i < \bar{X}} \frac{(X_i - \bar{X})^2}{n}

Quick example

Returns 1%, 3%, 5%, 7% (mean 4). Below the mean: 1 and 3, squared distances 9 + 1 = 10. Divided by n = 4 as printed: 2.5 (%²). Most questions divide by n − 1 = 3 instead: 3.33.

On your calculator

10÷4= 2.5
10÷3= 3.3333

10ENTER4÷ 2.5
10ENTER3÷ 3.3333

ENTER enter (on the HP, puts the number on the stack)

Target semi-deviation (target downside deviation) Memorize

s_{Target} = \sqrt{\sum_{\text{all } X_i \le B} \frac{(X_i - B)^2}{n - 1}}

Quick example

Returns 1%, −2%, 4%, −5%, 3% and a 0% target. Shortfalls −2 and −5: (4 + 25) ÷ (5 − 1) = 7.25, and √7.25 = 2.69%.

On your calculator

2x²+5x²= 29
÷4=√x 2.69

x² square · √x square root

2ENTER×5ENTER×+ 29
4÷g√x 2.69

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Coefficient of variation Memorize

CV = \frac{s}{\bar{X}}

Quick example

Mean 8%, standard deviation 10%. CV = 10 ÷ 8 = 1.25 units of risk per unit of return.

On your calculator

10÷8= 1.25

10ENTER8÷ 1.25

ENTER enter (on the HP, puts the number on the stack)

Skewness and kurtosis

Sample skewness Know the idea

\text{Skew} = \frac{1}{n}\sum_{i=1}^{n}\left(\frac{X_i - \bar{X}}{s}\right)^{3}

Quick example

Average cubed deviation −0.9 (in %³) and s = 1.2%. −0.9 ÷ 1.2³ = −0.52: negative skew.

On your calculator

1.2y^x3=1/x×0.9+|-= -0.5208

y^x power · 1/x reciprocal · +|- change sign

0.9CHSENTER1.2ENTER3y^x÷ -0.5208

CHS change sign · ENTER enter (on the HP, puts the number on the stack) · y^x power

Excess kurtosis Memorize

K_E = K - 3

Quick example

Kurtosis of 2.7. Excess kurtosis = 2.7 − 3 = −0.3, so thin-tailed (platykurtic).

On your calculator

2.7−3= -0.3

2.7ENTER3− -0.3

ENTER enter (on the HP, puts the number on the stack)

Sample excess kurtosis (100 or more observations) Know the idea

\text{Kurt}_E = \frac{1}{n}\sum_{i=1}^{n}\left(\frac{X_i - \bar{X}}{s}\right)^{4} - 3

Quick example

Average fourth-power deviation 9.9 (in %⁴) and s = 1.2%. 9.9 ÷ 1.2⁴ − 3 = 4.77 − 3 = 1.77: fat-tailed.

On your calculator

1.2y^x4=1/x×9.9−3= 1.7743

y^x power · 1/x reciprocal

9.9ENTER1.2ENTER4y^x÷3− 1.7743

ENTER enter (on the HP, puts the number on the stack) · y^x power

Covariance and correlation

Covariance Memorize

\sigma_{X,Y} = \frac{\sum_{i=1}^{n} (X_i - \mu_X)(Y_i - \mu_Y)}{n}

Quick example

Cross-products of the deviations add to 10 over n = 4 pairs. 10 ÷ 4 = 2.50 (%²).

On your calculator

10÷4= 2.5

10ENTER4÷ 2.5

ENTER enter (on the HP, puts the number on the stack)

Sample covariance Use it

s_{X,Y} = \frac{\sum_{i=1}^{n} (X_i - \bar{X})(Y_i - \bar{Y})}{n - 1}

Quick example

Cross-products of the deviations add to 10 over n = 4 pairs. 10 ÷ (4 − 1) = 3.33 (%²).

On your calculator

10÷(4−1)= 3.33

10ENTER4ENTER1−÷ 3.33

ENTER enter (on the HP, puts the number on the stack)

Correlation coefficient Memorize

\rho_{X,Y} = \frac{\sigma_{X,Y}}{\sigma_X\,\sigma_Y}

Quick example

Covariance 0.0045, standard deviations 15% and 10%. ρ = 0.0045 ÷ (0.15 × 0.10) = 0.30.

On your calculator

0.0045÷(0.15×0.1)= 0.3

0.0045ENTER0.15ENTER0.1×÷ 0.3

ENTER enter (on the HP, puts the number on the stack)

Scaling volatility over time: data frequency and the root-of-time law

Variance over T periods Memorize

\sigma^2_T \approx T \times \sigma^2_1

Quick example

Daily variance 0.0002. Over 22 trading days: 0.0002 × 22 = 0.0044.

On your calculator

0.0002×22= 0.0044

0.0002ENTER22× 0.0044

ENTER enter (on the HP, puts the number on the stack)

Root-of-time law Memorize

\sigma_T \approx \sqrt{T} \times \sigma_1

Quick example

Weekly standard deviation 2.5%. Annual: 2.5 × √52 = 18.03%. Multiplying by 52 (130%) is the trap.

On your calculator

52√x×2.5= 18.0278

√x square root

52g√x2.5× 18.0278

g blue shift key · √x square root

From a long horizon to a short one Use it

\sigma_1 \approx \dfrac{\sigma_T}{\sqrt{T}}

Quick example

Annual standard deviation 24%. Daily: 24 ÷ √252 = 1.51%.

On your calculator

24÷252√x= 1.5119

√x square root

24ENTER252g√x÷ 1.5119

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Module 6 Statistical Distributions

Random variables and distribution functions

Cumulative distribution function Memorize

F(x) = P(X \le x)

Quick example

A cumulative table reads 0.09, 0.25 and 0.69 at x = 0, 1 and 2. P(X ≤ 1) = F(1) = 0.25.

Probability of a range, continuous variable Memorize

P(a \le X \le b) = \int_a^b f(x)\,dx = F(b) - F(a)

Quick example

F(x) = x² on the range 0 to 1. P(0.2 ≤ X ≤ 0.5) = 0.25 − 0.04 = 0.21.

On your calculator

0.5x²−0.2x²= 0.21

x² square

0.5ENTER×0.2ENTER×− 0.21

ENTER enter (on the HP, puts the number on the stack)

Probability of a range, discrete variable Use it

P(a \le X \le b) = \sum_{x=a}^{b} P(X = x)

Quick example

Mass of 0.12, 0.38 and 0.28 at x = 1, 2 and 3. P(1 ≤ X ≤ 3) = 0.78.

On your calculator

0.12+0.38+0.28= 0.78

0.12ENTER0.38+0.28+ 0.78

ENTER enter (on the HP, puts the number on the stack)

Probabilities add to 1 Memorize

\sum_x P(X = x) = 1, \qquad \int_{-\infty}^{\infty} f(x)\,dx = 1

Quick example

Four outcomes have probabilities 0.22, 0.41, 0.19 and p. p = 1 − 0.82 = 0.18.

On your calculator

1−0.22−0.41−0.19= 0.18

1ENTER0.22−0.41−0.19− 0.18

ENTER enter (on the HP, puts the number on the stack)

Density as the slope of the CDF Know the idea

f(x) = \dfrac{dF(x)}{dx}

Quick example

F(x) = x² gives f(x) = 2x, so the density at x = 0.5 is 1.0.

Unconditional expected value, variance and standard deviation

Expected value Memorize

E[X] = P(x_1)\,x_1 + P(x_2)\,x_2 + \cdots + P(x_n)\,x_n = \sum_{i=1}^{n} P(x_i)\,x_i

Quick example

A return is 11% with probability 0.4 and 5% with probability 0.6. E = 0.4 × 11 + 0.6 × 5 = 7.4%.

On your calculator

0.4×11+(0.6×5)= 7.4

0.4ENTER11×0.6ENTER5×+ 7.4

ENTER enter (on the HP, puts the number on the stack)

Variance of a forecast Memorize

\mathrm{Var}(X) = \sigma_X^2 = \sum_{i=1}^{n} P(x_i)\,[x_i - E[X]]^2

Quick example

Same forecast, E[X] = 7.4. 0.4 × (11 − 7.4)² + 0.6 × (5 − 7.4)² = 5.184 + 3.456 = 8.64 (%²).

On your calculator

(11−7.4)x²×0.4+((5−7.4)x²×0.6)= 8.64

x² square

11ENTER7.4−ENTER×0.4×5ENTER7.4−ENTER×0.6×+ 8.64

ENTER enter (on the HP, puts the number on the stack)

Variance shortcut Memorize

\mathrm{Var}(X) = E\big[(X - E[X])^2\big] = E[X^2] - (E[X])^2

Quick example

Returns 11% (0.4) and 5% (0.6). E[X] = 7.4, E[X²] = 0.4 × 121 + 0.6 × 25 = 63.4. Var = 63.4 − 54.76 = 8.64 (%²).

On your calculator

0.4×11x²+(0.6×5x²)=−7.4x²= 8.64

x² square

11ENTER×0.4×5ENTER×0.6×+7.4ENTER×− 8.64

ENTER enter (on the HP, puts the number on the stack)

Standard deviation of a forecast Memorize

\sigma_X = \sqrt{\mathrm{Var}(X)}

Quick example

Variance 8.64 (%²). √8.64 = 2.94%.

On your calculator

8.64√x 2.94

√x square root

8.64g√x 2.94

g blue shift key · √x square root

Distance from the mean in standard deviations Use it

\dfrac{x - E[X]}{\sigma_X}

Quick example

E[X] = 7.4%, σ = 2.94%. An outcome of 2%: (2 − 7.4) ÷ 2.94 = −1.84.

On your calculator

2−7.4=÷2.94= -1.8367

2ENTER7.4−2.94÷ -1.8367

ENTER enter (on the HP, puts the number on the stack)

Discrete distributions: uniform, binomial and Poisson

Discrete uniform mean and variance Memorize

\mu = \dfrac{a+b}{2}, \quad \sigma^2 = \dfrac{(b-a+1)^2 - 1}{12}

Quick example

Whole numbers 0 to 4, all equally likely: mean 2, variance (25 − 1) ÷ 12 = 2.

Binomial probability Use it

P(X = x) = \binom{n}{x} p^x (1-p)^{n-x}

Quick example

n = 5, p = 0.3: P(X = 2) = 10 × 0.09 × 0.343 = 0.3087.

On your calculator

52NDnCr2=×(0.3y^x2)×(0.7y^x3)= 0.3087

2ND second function (the shifted label above a key) · y^x power

5gn!2gn!÷3gn!÷0.3ENTER2y^x×0.7ENTER3y^x× 0.3087

g blue shift key · ENTER enter (on the HP, puts the number on the stack) · y^x power

Binomial mean and variance Memorize

\mu = np, \quad \sigma^2 = np(1-p)

Quick example

n = 36, p = 0.25: mean 9, variance 36 × 0.25 × 0.75 = 6.75.

Binomial price node Use it

S = S_0\,u^{i}\,d^{j}

Quick example

S₀ = 50, u = 1.2, d = 0.9, two ups and one down: 50 × 1.44 × 0.9 = 64.80.

On your calculator

50×1.2x²×0.9= 64.8

x² square

50ENTER1.2ENTER××0.9× 64.8

ENTER enter (on the HP, puts the number on the stack)

Normal approximation to the binomial Know the idea

X \approx N\big(np,\; np(1-p)\big) \text{ if } np > 5 \text{ and } n(1-p) > 5

Quick example

n = 100, p = 0.1: np = 10 and n(1 − p) = 90 are both above 5, so X is about N(10, 9).

Poisson probability Use it

P(k) = \dfrac{(\lambda t)^k e^{-\lambda t}}{k!}

Quick example

λ = 2 a day, one day: P(1) = 2 × e^(−2) = 0.2707.

On your calculator

2×2+|-2NDe^x= 0.2707

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

2ENTER2CHSge^x× 0.2707

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · g blue shift key · e^x e to the power x

Poisson mean and variance Memorize

\mu = \sigma^2 = \lambda \quad \text{(one unit of time)}

Quick example

5 defaults a year on average: variance 5, standard deviation 2.24.

Probability of default by time t Memorize

P(\text{default by } t) = 1 - e^{-\lambda t}, \quad E[t_d] = \dfrac{1}{\lambda}

Quick example

λ = 1.5% a year, 6 years: 1 − e^(−0.09) = 8.61%. Expected time to default 1 ÷ 0.015 = 66.7 years.

On your calculator

0.015×6=+|-2NDe^x+|-+1= 0.0861

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.015ENTER6×CHSge^xCHS1+ 0.0861

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · g blue shift key · e^x e to the power x

Continuous distributions: uniform, normal and logistic

Continuous uniform mean and variance Memorize

\mu = \dfrac{a+b}{2}, \quad \sigma^2 = \dfrac{(b-a)^2}{12}

Quick example

Uniform on [0, 12]: mean 6, variance 144 ÷ 12 = 12.

Continuous uniform probability Use it

P(x_1 \le X \le x_2) = \dfrac{x_2 - x_1}{b - a}

Quick example

Uniform on [10, 50]: P(20 ≤ X ≤ 30) = 10 ÷ 40 = 0.25.

On your calculator

30−20=÷(50−10)= 0.25

30ENTER20−50ENTER10−÷ 0.25

ENTER enter (on the HP, puts the number on the stack)

Normal ranges Memorize

P(\mu - \sigma \le X \le \mu + \sigma) \approx 0.68, \quad P(\mu - 2\sigma \le X \le \mu + 2\sigma) \approx 0.95

Quick example

N(5%, 10%²): about 95% of outcomes lie between 5 − 20 = −15% and 5 + 20 = 25%.

On your calculator

5−(2×10)= -15
5+(2×10)= 25

5ENTER2ENTER10×− -15
5ENTER2ENTER10×+ 25

ENTER enter (on the HP, puts the number on the stack)

Logistic mean and variance Memorize

E[X] = \mu, \quad \sigma^2 = \dfrac{s^2 \pi^2}{3}

Quick example

Scale s = 1.5: 2.25 × 9.8696 ÷ 3 = 7.40, so the standard deviation is 2.72.

On your calculator

1.5x²×9.8696÷3= 7.4022
√x 2.7207

x² square · √x square root

1.5ENTER×9.8696×3÷ 7.4022
g√x 2.7207

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Why asset prices are log-normal

Price from a continuously compounded return Memorize

S_T = S_0\,e^{r_{0,T}}, \qquad r_{0,T} = \ln\left(\frac{S_T}{S_0}\right)

Quick example

A 25.00 share has a continuously compounded return of 8% over the year. $S_T = 25 \times e^{0.08}$ = 27.08.

On your calculator

0.082NDe^x×25= 27.08

2ND second function (the shifted label above a key) · e^x e to the power x

0.08ge^x25× 27.08

g blue shift key · e^x e to the power x

Continuously compounded returns add Memorize

r_{0,T} = r_{0,1} + r_{1,2} + \cdots + r_{T-1,T}

Quick example

A share goes 40.00, 42.40, 41.00, 44.00. Daily returns 5.83% − 3.36% + 7.06% = 9.53%, which equals $\ln(44 \div 40)$ .

On your calculator

44÷40=LN 0.0953, the same total in one step

LN natural logarithm

44ENTER40÷gLN 0.0953

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · LN natural logarithm

Projected log price Memorize

\ln S_T = \ln S_0 + \left(\mu - \tfrac{\sigma^2}{2}\right)T, \quad S_T = S_0\,e^{(\mu - \sigma^2/2)T}

Quick example

S₀ = 25, μ = 6%, σ = 16%, T = 3: 25 × e^((0.06 − 0.0128) × 3) = 25 × e^0.1416 = 28.80.

On your calculator

0.16x²÷2=+|-+0.06=×3=2NDe^x×25= 28.80

x² square · +|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.06ENTER0.16ENTER×2÷−3×ge^x25× 28.80

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Standard deviation of the log return over T Use it

\sigma_T = \sigma\sqrt{T}

Quick example

σ = 22% a year, T = 3: 22 × √3 = 38.11%.

On your calculator

3√x×22= 38.1051

√x square root

3g√x22× 38.1051

g blue shift key · √x square root

Mean of a log-normal variable Use it

E[Y] = \mu_Y = e^{\mu_X + \sigma_X^{2}/2}

Quick example

Annual log return mean 6%, standard deviation 30%, price 100. Mean price $100 \times e^{0.105}$ = 111.07, above the median 106.18.

On your calculator

0.3x²÷2+0.06=2NDe^x×100= 111.07

x² square · 2ND second function (the shifted label above a key) · e^x e to the power x

0.3ENTER×2÷0.06+ge^x100× 111.07

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Variance of a log-normal variable Use it

\sigma_Y^2 = \left(e^{\sigma_X^2} - 1\right)e^{2\mu_X + \sigma_X^2}

Quick example

μ_X = 0.06, σ_X = 0.30: (e^0.09 − 1) × e^0.21 = 0.094174 × 1.233678 = 0.1162, standard deviation 0.3409.

On your calculator

0.3x²=STO1 0.09
2NDe^x−1=STO2 0.094174
2×0.06+RCL1=2NDe^x×RCL2= 0.116181

x² square · STO store · 2ND second function (the shifted label above a key) · e^x e to the power x · RCL recall a stored number

0.3ENTER×STO1 0.09
ge^x1−STO2 0.094174
0.06ENTER2×RCL1+ge^xRCL2× 0.116181

ENTER enter (on the HP, puts the number on the stack) · STO store · g blue shift key · e^x e to the power x · RCL recall a stored number

Probability trees, conditional expectations and conditional variance

Joint probability along a path Use it

P(A \text{ and } B) = P(B \mid A)\,P(A)

Quick example

A scenario has probability 0.65, and an outcome within it has conditional probability 0.75. The joint probability is 0.65 × 0.75 = 0.4875.

On your calculator

0.65×0.75= 0.4875

0.65ENTER0.75× 0.4875

ENTER enter (on the HP, puts the number on the stack)

Conditional probability Use it

P(A \mid B) = \dfrac{P(A \text{ and } B)}{P(B)}, \quad P(B) \neq 0

Quick example

Joint probability 0.4875, P(B) = 0.65. P(A | B) = 0.4875 ÷ 0.65 = 0.75.

On your calculator

0.4875÷0.65= 0.75

0.4875ENTER0.65÷ 0.75

ENTER enter (on the HP, puts the number on the stack)

Conditional expected value Memorize

E[X \mid S] = P(x_1 \mid S)\,x_1 + P(x_2 \mid S)\,x_2 + \cdots + P(x_n \mid S)\,x_n

Quick example

Given scenario S, an outcome of 15 has probability 0.9 and an outcome of 5 has 0.1. E[X | S] = 13.5 + 0.5 = 14.0.

On your calculator

0.9×15+(0.1×5)= 14

0.9ENTER15×0.1ENTER5×+ 14

ENTER enter (on the HP, puts the number on the stack)

Total probability rule for expected value Memorize

E[X] = E[X \mid S_1]\,P(S_1) + E[X \mid S_2]\,P(S_2) + \cdots + E[X \mid S_n]\,P(S_n)

Quick example

E[X | S₁] = 5.0 with P(S₁) = 0.65, and E[X | S₂] = −2.0 with P(S₂) = 0.35. E[X] = 3.25 − 0.70 = 2.55.

On your calculator

0.65×5+(0.35×2+|-)= 2.55

+|- change sign

0.65ENTER5×0.35ENTER2CHS×+ 2.55

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Conditional variance Use it

\mathrm{Var}(X \mid S) = E\big[(X - E[X \mid S])^2 \mid S\big] = E[X^2 \mid S] - \big(E[X \mid S]\big)^2

Quick example

Given S, outcomes 5 and −1 each have probability 0.5, so E[X | S] = 2. Variance = 0.5 × 9 + 0.5 × 9 = 9. Shortcut: E[X² | S] = 13, and 13 − 2² = 9.

On your calculator

(5−2)x²×0.5+((1+|-−2)x²×0.5)= 9

x² square · +|- change sign

5ENTER2−ENTER×0.5×1CHSENTER2−ENTER×0.5×+ 9

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Conditional covariance Know the idea

\mathrm{Cov}(X, Y \mid S) = E\big[(X - E[X \mid S])(Y - E[Y \mid S]) \mid S\big]

Quick example

Given S, X is 9 and Y is 11 with probability 0.75; X is 5 and Y is 3 with probability 0.25. E[X | S] = 8, E[Y | S] = 9. Cov = 0.75 × (1)(2) + 0.25 × (−3)(−6) = 6.0.

On your calculator

0.75×1×2+(0.25×3+|-×6+|-)= 6

+|- change sign

0.75ENTER1×2×0.25ENTER3CHS×6CHS×+ 6

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Covariance and correlation from a joint probability table

Expected return from scenarios Memorize

E[R] = \sum_{i} P(s_i)\, R_i

Quick example

Probabilities 0.25, 0.50, 0.25 with returns 12%, 6% and −4%. 0.25 × 12 + 0.50 × 6 + 0.25 × (−4) = 3 + 3 − 1 = 5%.

On your calculator

0.25×12+(0.5×6)+(0.25×4+|-)= 5

+|- change sign

0.25ENTER12×0.5ENTER6×+0.25ENTER4CHS×+ 5

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Covariance shortcut Memorize

\sigma_{AB} = \mathrm{Cov}(R_A, R_B) = E[R_A R_B] - E[R_A]\,E[R_B]

Quick example

Two states, probabilities 0.6 and 0.4: A returns 10% and −5% (mean 4%), B returns 6% and 2% (mean 4.4%). E[R_A R_B] = 0.6 × 60 + 0.4 × (−10) = 32. Cov = 32 − 4 × 4.4 = 14.4 (%²).

On your calculator

0.6×60+(0.4×10+|-)=−(4×4.4)= 14.4

+|- change sign

0.6ENTER60×0.4ENTER10CHS×+4ENTER4.4×− 14.4

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Covariance from a joint probability table Use it

\mathrm{Cov}(R_A, R_B) = \sum_{i} P(s_i)\,\big[R_{A,i} - E[R_A]\big]\big[R_{B,i} - E[R_B]\big]

Quick example

Two states with probabilities 0.6 and 0.4. A returns 10% and −5% (mean 4%); B returns 6% and 2% (mean 4.4%). 0.6 × (6)(1.6) + 0.4 × (−9)(−2.4) = 5.76 + 8.64 = 14.4 (%²).

On your calculator

0.6×(10−4)×(6−4.4)+(0.4×(5+|-−4)×(2−4.4))= 14.4

+|- change sign

10ENTER4−6ENTER4.4−×0.6×5CHSENTER4−2ENTER4.4−×0.4×+ 14.4

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Correlation from scenario returns Memorize

\rho_{XY} = \dfrac{\sigma_{XY}}{\sigma_X\,\sigma_Y} = \dfrac{\mathrm{Cov}(X, Y)}{\sigma_X\,\sigma_Y}

Quick example

Covariance −12, standard deviations 6% and 4%. −12 ÷ (6 × 4) = −0.50.

On your calculator

12+|-÷(6×4)= -0.5

+|- change sign

12CHSENTER6ENTER4×÷ -0.5

CHS change sign · ENTER enter (on the HP, puts the number on the stack)

Joint distribution function Know the idea

F_{12}(x_1, x_2) = P(X_1 \le x_1,\; X_2 \le x_2)

Quick example

In the grid below, P(R_A ≤ 7% and R_B ≤ 5%) adds the cells (7, 5), (7, −2), (−4, 5) and (−4, −2): 0.05 + 0 + 0.20 + 0.25 = 0.50.

On your calculator

0.05+0.20+0.25= 0.5

0.05ENTER0.20+0.25+ 0.5

ENTER enter (on the HP, puts the number on the stack)

Independence Use it

P(X, Y) = P(X)\,P(Y)

Quick example

P(X = 1) = 0.35 and P(Y = 1) = 0.60, with X and Y independent. P(X = 1 and Y = 1) = 0.35 × 0.60 = 0.21.

On your calculator

0.35×0.6= 0.21

0.35ENTER0.6× 0.21

ENTER enter (on the HP, puts the number on the stack)

Expected product of uncorrelated returns Use it

E[XY] = E[X]\,E[Y] \quad \text{if } X, Y \text{ are uncorrelated}

Quick example

Revenue is price times quantity. If expected price is 20 and expected quantity is 300 and they are uncorrelated, expected revenue is 20 × 300 = 6,000.

On your calculator

20×300= 6000

20ENTER300× 6000

ENTER enter (on the HP, puts the number on the stack)

Bayes' theorem: updating a probability with new information

Bayes' theorem Memorize

P(A \mid B) = \dfrac{P(B \mid A)}{P(B)}\,P(A)

Quick example

P(A) = 0.35, P(B | A) = 0.85, P(B) = 0.50. P(A | B) = 0.85 × 0.35 ÷ 0.50 = 0.595.

On your calculator

0.85×0.35=÷0.5= 0.595

0.85ENTER0.35×0.5÷ 0.595

ENTER enter (on the HP, puts the number on the stack)

Marginal probability of the information Memorize

P(B) = P(B \mid A)\,P(A) + P(B \mid A^c)\,P(A^c)

Quick example

P(A) = 0.55, P(B | A) = 0.75, P(B | not A) = 0.35. P(B) = 0.4125 + 0.1575 = 0.57.

On your calculator

0.55×0.75+(0.45×0.35)= 0.57

0.55ENTER0.75×0.45ENTER0.35×+ 0.57

ENTER enter (on the HP, puts the number on the stack)

Missing likelihood from the marginal probability Use it

P(B \mid A^c) = \dfrac{P(B) - P(B \mid A)\,P(A)}{P(A^c)}

Quick example

P(B) = 0.34, P(B | A) = 0.55, P(A) = 0.25. P(B | not A) = (0.34 − 0.1375) ÷ 0.75 = 0.27.

On your calculator

0.34−(0.55×0.25)=÷0.75= 0.27

0.34ENTER0.55ENTER0.25×−0.75÷ 0.27

ENTER enter (on the HP, puts the number on the stack)

Bayes' theorem with several scenarios Use it

P(A_i \mid B) = \dfrac{P(B \mid A_i)\,P(A_i)}{\sum_{j} P(B \mid A_j)\,P(A_j)}

Quick example

Priors 0.55, 0.35, 0.10 and likelihoods 0.25, 0.45, 0.75. P(B) = 0.1375 + 0.1575 + 0.075 = 0.37. Posterior of the first: 0.1375 ÷ 0.37 = 0.372.

On your calculator

0.55×0.25=STO1
+(0.35×0.45)+(0.1×0.75)= 0.37
1/x×RCL1= 0.3716

STO store · 1/x reciprocal · RCL recall a stored number

0.55ENTER0.25×STO1
0.35ENTER0.45×+0.1ENTER0.75×+ 0.37
1/xRCL1× 0.3716

ENTER enter (on the HP, puts the number on the stack) · STO store · 1/x reciprocal · RCL recall a stored number

Module 7 Estimation and Hypothesis Testing

The central limit theorem and the standard error of the mean

Distribution of the sample mean Memorize

\bar{X} \sim N\!\left(\mu,\ \dfrac{\sigma^2}{n}\right)

Quick example

Returns have mean 8% and standard deviation 20%. With n = 100, the sample mean is approximately normal with mean 8% and variance 20² ÷ 100 = 4 (%²), so a standard deviation of 2%.

On your calculator

20x²÷100= 4

x² square

20ENTER×100÷ 4

ENTER enter (on the HP, puts the number on the stack)

Standard error (population σ known) Memorize

\sigma_{\bar{X}} = \dfrac{\sigma}{\sqrt{n}}

Quick example

σ = 20%, n = 100. 20 ÷ √100 = 20 ÷ 10 = 2.0%.

On your calculator

20÷100√x= 2

√x square root

20ENTER100g√x÷ 2

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Standard error (estimated from the sample) Memorize

s_{\bar{X}} = \dfrac{s}{\sqrt{n}}

Quick example

A sample of 25 monthly returns has s = 15%. 15 ÷ √25 = 15 ÷ 5 = 3.0%.

On your calculator

15÷25√x= 3

√x square root

15ENTER25g√x÷ 3

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Sample size for a target standard error Use it

n = \left(\dfrac{s}{s_{\bar{X}}}\right)^2

Quick example

Returns have s = 12%. For a standard error of 1.5%, n = (12 ÷ 1.5)² = 8² = 64.

On your calculator

12÷1.5=x² 64

x² square

12ENTER1.5÷ENTER× 64

ENTER enter (on the HP, puts the number on the stack)

Confidence intervals for a mean return

Confidence interval for a mean Memorize

\bar{x} \pm Z_{\alpha/2} \times \dfrac{s}{\sqrt{n}}

Quick example

100 monthly returns, mean 1.0%, s = 5.0%. Standard error 5.0 ÷ √100 = 0.5%; margin 1.96 × 0.5 = 0.98%. 95% interval: 0.02% to 1.98%.

On your calculator

5÷100√x×1.96= 0.98, the margin of error

√x square root

5ENTER100g√x÷1.96× 0.98, the margin of error

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Margin of error Memorize

E = Z_{\alpha/2} \times \dfrac{s}{\sqrt{n}}

Quick example

64 returns with s = 4.0%. Standard error 4.0 ÷ 8 = 0.5%. At 99%, the margin is 2.576 × 0.5 = 1.288%.

On your calculator

4÷64√x×2.576= 1.288

√x square root

4ENTER64g√x÷2.576× 1.288

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Value at risk from the normal distribution

Parametric value at risk Memorize

\text{VaR} = Z \times \sigma \times P

Quick example

USD 4 million portfolio, daily σ 1.5%, 95%: 1.645 × 0.015 × 4,000,000 = USD 98,700. A 5% chance of losing more than that in one day.

On your calculator

1.645×0.015×4000000= 98,700

1.645ENTER0.015×4000000× 98,700

ENTER enter (on the HP, puts the number on the stack)

Sampling methods and sampling error

Sampling error Memorize

\text{Sampling error} = \bar{X} - \mu

Quick example

A sample of 40 funds has a mean return of 9.1%. The population mean is 8.4%. Sampling error = 9.1 − 8.4 = 0.7 percentage points.

On your calculator

9.1−8.4= 0.7

9.1ENTER8.4− 0.7

ENTER enter (on the HP, puts the number on the stack)

Items drawn from one stratum Use it

n_i = n \times \dfrac{N_i}{N}

Quick example

A population of 5,000 stocks has 1,250 in one cell. A proportional stratified sample of 200 takes 200 × 1,250 ÷ 5,000 = 50 from that cell.

On your calculator

200×1250÷5000= 50

200ENTER1250×5000÷ 50

ENTER enter (on the HP, puts the number on the stack)

Systematic sampling interval Use it

k = \dfrac{N}{n}

Quick example

To pull 80 trades from a log of 2,000, take every 2,000 ÷ 80 = 25th trade.

On your calculator

2000÷80= 25

2000ENTER80÷ 25

ENTER enter (on the HP, puts the number on the stack)

How a hypothesis test works: four steps from claim to decision

Test statistic Memorize

\text{Test statistic} = \dfrac{\text{Sample statistic} - \text{Hypothesized value}}{\text{Standard error of the sample statistic}}

Quick example

A sample mean of 7.2% against a hypothesized 6%, with a standard error of 0.8%: (7.2 − 6) ÷ 0.8 = 1.50 standard errors above the null.

On your calculator

(7.2−6)÷0.8= 1.5

7.2ENTER6−0.8÷ 1.5

ENTER enter (on the HP, puts the number on the stack)

Test statistic for a single mean Memorize

t = \dfrac{\bar{X} - \mu_0}{s \,/\, \sqrt{n}},\quad df = n - 1

Quick example

16 returns average 1.5% with s = 3%. For H₀: μ = 0.5%, t = (1.5 − 0.5) ÷ (3 ÷ √16) = 1 ÷ 0.75 = 1.33, with 15 degrees of freedom.

On your calculator

(1.5−0.5)÷(3÷16√x)= 1.3333

√x square root

1.5ENTER0.5−3ENTER16g√x÷÷ 1.3333

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Confidence level Use it

\text{Confidence level} = 1 - \alpha

Quick example

A 1% significance level is a 99% confidence level.

Type I and Type II errors and the power of a test

Power of a test Memorize

\text{Power} = 1 - \beta

Quick example

A test misses a false null 20% of the time (β = 0.20). Power = 1 − 0.20 = 0.80.

t-tests of a mean, a difference in means and paired differences

Pooled variance Memorize

s_p^2 = \dfrac{(n_1 - 1)s_1^2 + (n_2 - 1)s_2^2}{n_1 + n_2 - 2}

Quick example

n₁ = 20, s₁ = 3%; n₂ = 15, s₂ = 2.5%. s²ₚ = (19 × 9 + 14 × 6.25) ÷ 33 = 258.5 ÷ 33 = 7.83 (%²).

On your calculator

(19×3x²)+(14×2.5x²)=÷33= 7.8333

x² square

19ENTER3ENTER××14ENTER2.5ENTER××+33÷ 7.8333

ENTER enter (on the HP, puts the number on the stack)

Test of the difference in means (independent samples, equal variances) Memorize

t = \dfrac{(\bar{X}_1 - \bar{X}_2) - \Delta_0}{\sqrt{\dfrac{s_p^2}{n_1} + \dfrac{s_p^2}{n_2}}},\quad df = n_1 + n_2 - 2

Quick example

Means 8.0% and 6.5%, Δ₀ = 0, s²ₚ = 7.83, n₁ = 20, n₂ = 15. Standard error = √(7.83 × (1/20 + 1/15)) = 0.956, so t = (1.5 − 0) ÷ 0.956 = 1.57 with 33 degrees of freedom.

On your calculator

201/x+151/x=×7.8333=√xSTO1 0.956
(8−6.5)÷RCL1= 1.569

1/x reciprocal · √x square root · STO store · RCL recall a stored number

201/x151/x+7.8333×g√xSTO1 0.956
8ENTER6.5−RCL1÷ 1.569

1/x reciprocal · g blue shift key · √x square root · STO store · ENTER enter (on the HP, puts the number on the stack) · RCL recall a stored number

Mean of the differences Memorize

\bar{d} = \dfrac{1}{n}\sum_{i=1}^{n} d_i,\quad d_i = X_{1i} - X_{2i}

Quick example

Four monthly differences of 0.9, −0.2, 0.4 and 0.5 (percentage points): (0.9 − 0.2 + 0.4 + 0.5) ÷ 4 = 0.40.

On your calculator

0.9−0.2+0.4+0.5=÷4= 0.4

0.9ENTER0.2−0.4+0.5+4÷ 0.4

ENTER enter (on the HP, puts the number on the stack)

Test of paired differences (paired t-test) Memorize

t = \dfrac{\bar{d} - \mu_0}{s_d \,/\, \sqrt{n}},\quad df = n - 1

Quick example

25 paired months, d̄ = 0.8%, s_d = 1.5%, H₀: μ_d = 0. t = 0.8 ÷ (1.5 ÷ √25) = 0.8 ÷ 0.3 = 2.67 with 24 degrees of freedom.

On your calculator

0.8÷(1.5÷25√x)= 2.6667

√x square root

0.8ENTER1.5ENTER25g√x÷÷ 2.6667

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Chi-square and F-tests of variance

Chi-square test of a single variance Memorize

\chi^2 = \dfrac{(n-1)\,s^2}{\sigma_0^2},\quad df = n - 1

Quick example

16 monthly returns with s = 3.0%, tested against σ₀ = 2.0%. χ² = 15 × 9 ÷ 4 = 33.75, with 15 degrees of freedom.

On your calculator

15×3x²÷2x²= 33.75

x² square

15ENTER3ENTER××2ENTER×÷ 33.75

ENTER enter (on the HP, puts the number on the stack)

F-test of the equality of two variances Memorize

F = \dfrac{s_1^2}{s_2^2}\quad(s_1^2 \ge s_2^2),\quad df = (n_1 - 1,\ n_2 - 1)

Quick example

Standard deviations of 8% and 5%. F = 64 ÷ 25 = 2.56, larger variance on top.

On your calculator

8x²÷5x²= 2.56

x² square

8ENTER×5ENTER×÷ 2.56

ENTER enter (on the HP, puts the number on the stack)

Is a correlation significant? The t-test of correlation

t-statistic for a correlation Memorize

t = \dfrac{r\sqrt{n-2}}{\sqrt{1-r^2}},\quad df = n - 2

Quick example

A correlation of 0.52 from 34 observations. t = 0.52 × √32 ÷ √(1 − 0.2704) = 2.94 ÷ 0.854 = 3.44, with 32 degrees of freedom.

On your calculator

0.52×32√x÷(1−0.52x²)√x= 3.4438

√x square root · x² square

0.52ENTER32g√x×1ENTER0.52ENTER×−g√x÷ 3.4438

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

P-values, p-hacking and critical values from software

Right-tail p-value Memorize

p = P(T \ge t_{obs} \mid H_0)

Quick example

A one-tailed test of a mean gives t = 2.25 with 35 degrees of freedom. The area to its right is p = 0.015: reject at 5%, not at 1%.

Left-tail p-value Memorize

p = P(T \le t_{obs} \mid H_0)

Quick example

A left-tail test gives t = −1.80 with 24 degrees of freedom. The area to its left is p = 0.042: reject at 5%, not at 1%.

Two-tailed p-value Memorize

p = 2 \times P(T \ge |t_{obs}| \mid H_0)

Quick example

Software gives a right-tail p-value of 0.018 for t = 2.15. For a two-sided alternative, p = 2 × 0.018 = 0.036.

On your calculator

0.018×2= 0.036

0.018ENTER2× 0.036

ENTER enter (on the HP, puts the number on the stack)

Spearman rank correlation: the non-parametric test

Spearman rank correlation Memorize

r_s = 1 - \dfrac{6\sum d_i^2}{n\,(n^2 - 1)}

Quick example

8 pairs of rankings with Σd² = 14. rₛ = 1 − 6 × 14 ÷ (8 × 63) = 1 − 84 ÷ 504 = 0.833.

On your calculator

1−(6×14÷(8×(8x²−1)))= 0.8333

x² square

6ENTER14×8ENTERENTER×1−×÷CHS1+ 0.8333

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

t-statistic for large samples (n > 30) Memorize

t = \dfrac{r_s\sqrt{n-2}}{\sqrt{1-r_s^2}},\quad df = n - 2

Quick example

rₛ = 0.38 from 50 pairs. t = 0.38 × √48 ÷ √(1 − 0.1444) = 2.85, against a critical value of 2.011.

On your calculator

0.38×48√x÷(1−0.38x²)√x= 2.8462

√x square root · x² square

0.38ENTER48g√x×1ENTER0.38ENTER×−g√x÷ 2.8462

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Chi-square test of independence with a contingency table

Expected frequency Memorize

E_{ij} = \dfrac{(\text{row } i \text{ total}) \times (\text{column } j \text{ total})}{\text{grand total}} = (\text{row } i \text{ share}) \times (\text{column } j \text{ share}) \times N

Quick example

Row total 90, column total 60, grand total 300. E = 90 × 60 ÷ 300 = 18. In shares: 0.30 × 0.20 × 300 = 18.

On your calculator

90×60÷300= 18
0.3×0.2×300= 18

90ENTER60×300÷ 18
0.3ENTER0.2×300× 18

ENTER enter (on the HP, puts the number on the stack)

Chi-square statistic Memorize

\chi^2 = \sum_{i,j} \dfrac{(O_{ij} - E_{ij})^2}{E_{ij}}

Quick example

A 2 × 2 table with observed 40, 60, 60, 40 and every expected count 50. Each cell adds 100 ÷ 50 = 2, so χ² = 8.0.

On your calculator

(40−50)x²÷50×4= 8

x² square

40ENTER50−ENTER×50÷4× 8

ENTER enter (on the HP, puts the number on the stack)

Degrees of freedom Memorize

df = (r - 1)(c - 1)

Quick example

A table with 3 rows and 4 columns. df = 2 × 3 = 6.

On your calculator

(3−1)×(4−1)= 6

3ENTER1−4ENTER1−× 6

ENTER enter (on the HP, puts the number on the stack)

Standardized residual Memorize

\text{standardized residual} = \dfrac{O_{ij} - E_{ij}}{\sqrt{E_{ij}}}

Quick example

A cell with O = 26 and E = 16. (26 − 16) ÷ √16 = +2.5: beyond ±2, so well above what independence predicts.

On your calculator

26−16=÷16√x= 2.5

√x square root

26ENTER16−16g√x÷ 2.5

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Module 8 Portfolio Return and Risk

Portfolio returns over time: buy-and-hold, drift and rebalancing

Portfolio return in a period Memorize

r_{P,t} = \sum_{i=1}^{N} w_{i,t-1}\, r_{i,t}

Quick example

Start-of-period weights 55% in A and 45% in B. A returns 9%, B returns −3%. 0.55 × 9 + 0.45 × (−3) = 4.95 − 1.35 = 3.6%.

On your calculator

0.55×9+(0.45×3+|-)= 3.6

+|- change sign

0.55ENTER9×0.45ENTER3CHS×+ 3.6

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Weight after one period Use it

w_{i,t} = \dfrac{w_{i,t-1}\,(1 + r_{i,t})}{1 + r_{P,t}}

Quick example

Start 50/50. A returns 20%, B returns 4%. w_A = 0.5 × 1.20 ÷ (0.5 × 1.20 + 0.5 × 1.04) = 0.60 ÷ 1.12 = 53.57%.

On your calculator

0.5×1.2=STO1 0.6
0.5×1.04+RCL1= 1.12, the portfolio's growth
1/x×RCL1= 0.5357

STO store · RCL recall a stored number · 1/x reciprocal

0.5ENTER1.2×STO1 0.6
0.5ENTER1.04×+ 1.12, the portfolio's growth
1/xRCL1× 0.5357

ENTER enter (on the HP, puts the number on the stack) · STO store · 1/x reciprocal · RCL recall a stored number

Buy-and-hold cumulative return Memorize

r_{P,T} = \sum_{i=1}^{N} w_{i,0}\left[\prod_{t=1}^{T}(1 + r_{i,t}) - 1\right]

Quick example

Start 30% in X and 70% in Y. Over three years X gains 40% in total and Y gains 10%. 0.3 × 40 + 0.7 × 10 = 19.0%.

On your calculator

0.3×40+(0.7×10)= 19

0.3ENTER40×0.7ENTER10×+ 19

ENTER enter (on the HP, puts the number on the stack)

Geometric average portfolio return Memorize

\bar{r}_{gP} = \left[\prod_{t=1}^{T}(1 + r_{P,t})\right]^{1/T} - 1 = (1 + r_{P,T})^{1/T} - 1

Quick example

Portfolio returns 12%, −5% and 7%. (1.12 × 0.95 × 1.07)^(1/3) − 1 = 1.13848^(1/3) − 1 = 4.42% a year. The arithmetic average, (12 − 5 + 7) ÷ 3, is 4.67%.

On your calculator

1.12×0.95×1.07= 1.1385
y^x31/x= 1.0442
−1= 0.0442

y^x power · 1/x reciprocal

1.12ENTER0.95×1.07× 1.1385
31/xy^x 1.0442
1− 0.0442

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · y^x power

Arithmetic average portfolio return Memorize

\bar{r}_P = \frac{1}{T}\sum_{t=1}^{T} r_{P,t}

Quick example

Portfolio returns 12%, −5% and 7%. (12 − 5 + 7) ÷ 3 = 4.67%, above the geometric average of 4.42%.

On your calculator

12−5+7=÷3= 4.67

12ENTER5−7+3÷ 4.67

ENTER enter (on the HP, puts the number on the stack)

Portfolio expected return, variance and the role of correlation

Portfolio expected return Memorize

E[r_P] = w_1 E[r_1] + w_2 E[r_2] + \cdots + w_N E[r_N] = \sum_{i=1}^{N} w_i\, E[r_i]

Quick example

Weights 0.45, 0.45 and 0.10 in assets expected to return 9%, 5% and 2%. 0.45 × 9 + 0.45 × 5 + 0.10 × 2 = 6.5%.

On your calculator

0.45×9+(0.45×5)+(0.1×2)= 6.5

0.45ENTER9×0.45ENTER5×+0.1ENTER2×+ 6.5

ENTER enter (on the HP, puts the number on the stack)

Covariance of two returns Use it

\sigma_{i,j} = E\big[(r_i - E[r_i])(r_j - E[r_j])\big]

Quick example

Two equally likely periods. In the first, asset 1 is 3 above its mean and asset 2 is 2 above. In the second, they are 1 and 4 below. $\sigma_{1,2}$ = 0.5 × (3 × 2) + 0.5 × (−1 × −4) = 5 (%²). Positive: the returns sit on the same side of their means.

On your calculator

0.5×3×2+(0.5×1+|-×4+|-)= 5

+|- change sign

0.5ENTER3×2×0.5ENTER1CHS×4CHS×+ 5

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Covariance from correlation Memorize

\sigma_{1,2} = \rho_{1,2}\, \sigma_1\, \sigma_2

Quick example

Correlation 0.8, standard deviations 25% and 14%. 0.8 × 0.25 × 0.14 = 0.028.

On your calculator

0.8×0.25×0.14= 0.028

0.8ENTER0.25×0.14× 0.028

ENTER enter (on the HP, puts the number on the stack)

Correlation Memorize

\rho_{1,2} = \frac{\sigma_{1,2}}{\sigma_1\, \sigma_2}

Quick example

Covariance 0.0336, standard deviations 28% and 16%. 0.0336 ÷ (0.28 × 0.16) = 0.75.

On your calculator

0.0336÷(0.28×0.16)= 0.75

0.0336ENTER0.28ENTER0.16×÷ 0.75

ENTER enter (on the HP, puts the number on the stack)

Two-asset portfolio variance Memorize

\sigma_P^2 = w_1^2 \sigma_1^2 + w_2^2 \sigma_2^2 + 2\, w_1 w_2\, \rho_{1,2}\, \sigma_1 \sigma_2

Quick example

50/50 mix, standard deviations 14% and 8%, correlation 0.5. 0.25 × 0.0196 + 0.25 × 0.0064 + 2 × 0.25 × 0.5 × 0.14 × 0.08 = 0.0049 + 0.0016 + 0.0028 = 0.0093.

On your calculator

0.5x²×0.14x²+(0.5x²×0.08x²)+(2×0.5×0.5×0.5×0.14×0.08)= 0.0093

x² square

0.5ENTER×0.14ENTER××0.5ENTER×0.08ENTER××+2ENTER0.5×0.5×0.5×0.14×0.08×+ 0.0093

ENTER enter (on the HP, puts the number on the stack)

Portfolio standard deviation Memorize

\sigma_P = \sqrt{\sigma_P^2}

Quick example

Variance 0.0093. √0.0093 = 9.64%. The weighted average of 14% and 8% would be 11%, so the mix is less risky than its parts.

On your calculator

0.0093√x 0.0964

√x square root

0.0093g√x 0.0964

g blue shift key · √x square root

Portfolio variance, any number of assets Use it

\sigma_P^2 = \sum_{i=1}^{N} \sum_{j=1}^{N} w_i\, w_j\, \sigma_{i,j}

Quick example

Weights 0.45, 0.45, 0.10. Variances (%²) 400, 100, 225; covariances 60 (1,2), 90 (1,3), 30 (2,3). Own terms 81 + 20.25 + 2.25 = 103.5; cross terms 2 × (12.15 + 4.05 + 1.35) = 35.1. Variance 138.6 %², so $\sigma_P$ = 11.77%. ( $\sigma_{i,i}$ is asset i's variance.)

On your calculator

0.45x²×400+(0.45x²×100)+(0.1x²×225)+(2×0.45×0.45×60)+(2×0.45×0.1×90)+(2×0.45×0.1×30)= 138.6
√x 11.77

x² square · √x square root

0.45ENTER×400×0.45ENTER×100×+0.1ENTER×225×+2ENTER0.45×0.45×60×+2ENTER0.45×0.1×90×+2ENTER0.45×0.1×30×+ 138.6
g√x 11.77

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Distinct covariances for N assets Memorize

\frac{N^2 - N}{2} = \frac{N(N-1)}{2}

Quick example

Ten assets need 10 variances and 10 × 9 ÷ 2 = 45 distinct covariances.

On your calculator

10×9÷2= 45

10ENTER9×2÷ 45

ENTER enter (on the HP, puts the number on the stack)

Portfolio return in a scenario Use it

r_{P,s} = \sum_{i=1}^{N} w_i\, r_{i,s}

Quick example

Weights 75/25. In a boom the two assets return 12% and 2%. 0.75 × 12 + 0.25 × 2 = 9.5%.

On your calculator

0.75×12+(0.25×2)= 9.5

0.75ENTER12×0.25ENTER2×+ 9.5

ENTER enter (on the HP, puts the number on the stack)

Expected return from scenarios Use it

E[r_P] = \sum_{s} p_s\, r_{P,s}

Quick example

Probabilities 0.45 and 0.55 with portfolio returns 10% and −4%. 0.45 × 10 + 0.55 × (−4) = 4.5 − 2.2 = 2.3%.

On your calculator

0.45×10+(0.55×4+|-)= 2.3

+|- change sign

0.45ENTER10×0.55ENTER4CHS×+ 2.3

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Variance from scenarios Use it

\sigma_P^2 = \sum_{s} p_s \left(r_{P,s} - E[r_P]\right)^2

Quick example

Same two scenarios, mean 2.3%. Deviations 7.7 and −6.3: 0.45 × 59.29 + 0.55 × 39.69 = 26.6805 + 21.8295 = 48.51 %², so $\sigma_P$ = 6.96%.

On your calculator

7.7x²×0.45+(6.3x²×0.55)= 48.51
√x 6.96

x² square · √x square root

7.7ENTER×0.45×6.3ENTER×0.55×+ 48.51
g√x 6.96

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

How many assets does diversification need?

Many-asset variance, equal weights Memorize

\sigma_P^2 = \frac{\sigma^2}{N} + \left(1 - \frac{1}{N}\right)\bar{\rho}\,\sigma^2

Quick example

N = 8, σ = 20%, average correlation 0.40. 0.04 ÷ 8 + (7/8) × 0.4 × 0.04 = 0.005 + 0.014 = 0.019, so $\sigma_P$ = 13.78%.

On your calculator

0.2x²÷8+(7÷8×0.4×0.2x²)= 0.019
√x 0.1378

x² square · √x square root

0.2ENTER×8÷7ENTER8÷0.4×0.2ENTER××+ 0.019
g√x 0.1378

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Many-asset variance, average covariance Use it

\sigma_P^2 = \frac{\bar{\sigma}^2}{N} + \frac{N - 1}{N}\,\overline{\sigma_{i,j}}

Quick example

Average variance 0.0625, average covariance 0.015, N = 20. 0.0625 ÷ 20 + (19/20) × 0.015 = 0.003125 + 0.01425 = 0.017375, so $\sigma_P$ = 13.18%.

On your calculator

0.0625÷20+(19÷20×0.015)= 0.017375
√x 0.1318

√x square root

0.0625ENTER20÷19ENTER20÷0.015×+ 0.017375
g√x 0.1318

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Large-N limit Memorize

\sigma_P^2 \approx \bar{\rho}\,\sigma^2 \quad\text{so}\quad \sigma_P \approx \sigma\sqrt{\bar{\rho}}

Quick example

σ = 20%, average correlation 0.40. $\sigma_P \approx \sqrt{0.4 \times 0.04}$ = √0.016 = 12.65%.

On your calculator

0.4×0.04= 0.016
√x 0.1265

√x square root

0.4ENTER0.04× 0.016
g√x 0.1265

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

The minimum-variance portfolio and the efficient frontier

Minimum-variance problem Use it

\min \sum_{i=1}^{N}\sum_{j=1}^{N} w_i w_j \sigma_{i,j} \quad\text{subject to}\quad \sum_i w_i = 1,\ w_i \ge 0,\ E[r_P] = k

Quick example

Assets returning 6% and 13%, no short selling. A target return k of 10% is feasible. Targets of 5% and 14% are not, because k must lie between the lowest and the highest asset return.

Minimum-variance weights, two assets Memorize

w_A = \dfrac{\sigma_B^2 - \sigma_{A,B}}{\sigma_A^2 + \sigma_B^2 - 2\sigma_{A,B}}, \quad w_B = 1 - w_A

Quick example

σ_A = 9%, σ_B = 15%, σ_A,B = 0.0027 (correlation 0.2). w_A = (0.0225 − 0.0027) ÷ (0.0081 + 0.0225 − 0.0054) = 0.0198 ÷ 0.0252 = 78.57%.

On your calculator

0.15x²−0.0027=STO1 0.0198
0.09x²+0.15x²−(2×0.0027)= 0.0252
1/x×RCL1= 0.7857

x² square · STO store · 1/x reciprocal · RCL recall a stored number

0.15ENTER×0.0027−STO1 0.0198
0.09ENTER×0.15ENTER×+2ENTER0.0027×− 0.0252
1/xRCL1× 0.7857

ENTER enter (on the HP, puts the number on the stack) · STO store · 1/x reciprocal · RCL recall a stored number

Minimum-variance weight from correlation Use it

w_A = \dfrac{\sigma_B^2 - \rho_{A,B}\sigma_A\sigma_B}{\sigma_A^2 + \sigma_B^2 - 2\rho_{A,B}\sigma_A\sigma_B}

Quick example

The same weight when the correlation is given: σ_A = 9%, σ_B = 15%, ρ = 0.2 gives σ_A,B = 0.2 × 0.09 × 0.15 = 0.0027, and the weight is 78.57% again.

Weights for a target return Memorize

w_A = \dfrac{r_p - r_B}{r_A - r_B}, \quad w_B = \dfrac{r_A - r_p}{r_A - r_B}

Quick example

r_A = 4%, r_B = 10%, target 8%. w_A = (8 − 10) ÷ (4 − 10) = 33.3%, so w_B = 66.7%.

On your calculator

(8−10)÷(4−10)= 0.3333

8ENTER10−4ENTER10−÷ 0.3333

ENTER enter (on the HP, puts the number on the stack)

Two-asset minimum-variance frontier Know the idea

\sigma_p^2 = \dfrac{(r_p - r_B)^2\sigma_A^2 + (r_A - r_p)^2\sigma_B^2 + 2(r_p - r_B)(r_A - r_p)\sigma_{A,B}}{(r_A - r_B)^2}

Quick example

Funds A (5%, σ 10%) and B (12%, σ 20%), σ_A,B = 0.005, target 9% (units in %²). (9 × 0.01 + 16 × 0.04 + 2 × (−3) × (−4) × 0.005) ÷ 49 = (0.09 + 0.64 + 0.12) ÷ 49 = 0.85 ÷ 49 = 0.017347, the variance of the 9% mix.

On your calculator

(9−12)x²×0.01+((5−9)x²×0.04)+(2×(9−12)×(5−9)×0.005)=÷(5−12)x²= 0.017347

x² square

9ENTER12−ENTER×0.01×5ENTER9−ENTER×0.04×+2ENTER9ENTER12−×5ENTER9−×0.005×+5ENTER12−ENTER×÷ 0.017347

ENTER enter (on the HP, puts the number on the stack)

Sharpe ratio Memorize

\text{Sharpe ratio} = \dfrac{E[r_P] - r_f}{\sigma_P}

Quick example

Expected return 9%, risk-free rate 2%, standard deviation 14%. (9 − 2) ÷ 14 = 0.50.

On your calculator

(9−2)÷14= 0.5

9ENTER2−14÷ 0.5

ENTER enter (on the HP, puts the number on the stack)

Risk preferences, mean–variance utility and indifference curves

Utility Memorize

U = E[r] - \tfrac{1}{2} A \sigma^2

Quick example

E[r] = 8%, σ = 15%, A = 2. 0.08 − 0.5 × 2 × 0.0225 = 0.0575 (5.75%). Use decimals, with σ² the variance.

On your calculator

0.5×2×0.15x²= 0.0225
+|-+0.08= 0.0575

x² square · +|- change sign

0.08ENTER0.5ENTER2×0.15ENTER××− 0.0575

ENTER enter (on the HP, puts the number on the stack)

Indifference curve Use it

E[r] = U + \tfrac{1}{2} A \sigma^2

Quick example

A = 4 and U = 0.03. At σ = 20% the curve sits at 0.03 + 2 × 0.04 = 11.0%. A higher σ needs a higher E[r] to keep U unchanged.

On your calculator

2×0.2x²+0.03= 0.11

x² square

2ENTER0.2ENTER××0.03+ 0.11

ENTER enter (on the HP, puts the number on the stack)

From the capital allocation line to the capital market line and the CAPM

Return of a risk-free and risky mix Memorize

E[r_p] = w_f\, r_f + (1 - w_f)\, E[r_P]

Quick example

40% in a 3% risk-free asset, 60% in a risky portfolio with E[r] of 11%. 0.4 × 3 + 0.6 × 11 = 7.8%.

On your calculator

0.4×3+(0.6×11)= 7.8

0.4ENTER3×0.6ENTER11×+ 7.8

ENTER enter (on the HP, puts the number on the stack)

Risk of a risk-free and risky mix Memorize

\sigma_p = (1 - w_f)\, \sigma_P

Quick example

60% in a risky portfolio with σ of 16% and 40% in the risk-free asset. 0.6 × 16 = 9.6%.

On your calculator

0.6×16= 9.6

0.6ENTER16× 9.6

ENTER enter (on the HP, puts the number on the stack)

Capital allocation line Memorize

E[r_p] = r_f + \dfrac{E[r_P] - r_f}{\sigma_P}\, \sigma_p

Quick example

r_f 3%, risky portfolio E[r] 11% and σ 16%. At σ_p = 9.6%: 3 + (8 ÷ 16) × 9.6 = 7.8%. The slope, 8 ÷ 16 = 0.50, is the Sharpe ratio.

On your calculator

(11−3)÷16×9.6+3= 7.8

11ENTER3−16÷9.6×3+ 7.8

ENTER enter (on the HP, puts the number on the stack)

Risky weight for a target return Memorize

w_P = \dfrac{r_p - r_f}{E[r_P] - r_f}, \quad \sigma_p = w_P\, \sigma_P

Quick example

r_f 2%, risky portfolio 8%, target 10%. w = (10 − 2) ÷ (8 − 2) = 133.3%, so borrow 33.3% of your own money.

On your calculator

(10−2)÷(8−2)= 1.3333

10ENTER2−8ENTER2−÷ 1.3333

ENTER enter (on the HP, puts the number on the stack)

Capital market line Memorize

E[r_p] = r_f + \dfrac{E[r_m] - r_f}{\sigma_m}\, \sigma_p

Quick example

r_f 3%, E[r_m] 9%, σ_m 15%, σ_p 9%. 3 + (6 ÷ 15) × 9 = 6.6%.

On your calculator

(9−3)÷15×9+3= 6.6

9ENTER3−15÷9×3+ 6.6

ENTER enter (on the HP, puts the number on the stack)

Beta Memorize

\beta_i = \dfrac{\sigma_{i,m}}{\sigma_m^2} = \dfrac{\rho_{i,m}\,\sigma_i\,\sigma_m}{\sigma_m^2} = \dfrac{\rho_{i,m}\,\sigma_i}{\sigma_m}

Quick example

ρ = 0.5, σ_i = 24%, σ_m = 16%: β = 0.5 × 24 ÷ 16 = 0.75. From a covariance: σ_i,m = 0.0288 with σ_m = 16% gives 0.0288 ÷ 0.0256 = 1.125.

On your calculator

0.5×24÷16= 0.75
0.0288÷0.16x²= 1.125

x² square

0.5ENTER24×16÷ 0.75
0.0288ENTER0.16ENTER×÷ 1.125

ENTER enter (on the HP, puts the number on the stack)

Security market line Memorize

E[r_i] = r_f + \dfrac{E[r_m] - r_f}{\sigma_m^2}\, \sigma_{i,m}

Quick example

r_f 2%, E[r_m] 8%, σ_m 20%, σ_i,m = 0.052. (8 − 2) ÷ 0.04 × 0.052 + 2 = 7.8 + 2 = 9.8%. The same as a beta of 1.3.

On your calculator

(8−2)÷0.2x²×0.052+2= 9.8

x² square

8ENTER2−0.2ENTER×÷0.052×2+ 9.8

ENTER enter (on the HP, puts the number on the stack)

Capital asset pricing model Memorize

E[r_i] = r_f + \beta_i \left(E[r_m] - r_f\right)

Quick example

r_f 2.5%, E[r_m] 8.5%, β 0.75. 2.5 + 0.75 × (8.5 − 2.5) = 7.0%.

On your calculator

0.75×(8.5−2.5)+2.5= 7

8.5ENTER2.5−0.75×2.5+ 7

ENTER enter (on the HP, puts the number on the stack)

Beta regression Use it

r_i - r_f = \alpha_i + \beta_i \left(r_m - r_f\right) + \varepsilon_i

Quick example

Regress a stock's monthly excess returns on the market's. A slope of 1.35 and an intercept of −0.05% a month mean a beta of 1.35 and an alpha of −0.05% a month.

Module 9 Simulating Prices and Returns

Historical, bootstrap or Monte Carlo: the four-step engine

Portfolio return in one scenario Use it

r_{p,s} = \sum_{i} w_i\, r_{i,s}

Quick example

Weights 70% shares and 30% bonds; a scenario has shares +1.5% and bonds −0.5%. $0.7 \times 1.5 + 0.3 \times (-0.5)$ = 0.90%.

On your calculator

0.7×1.5=STO1
0.3×0.5+|-+RCL1= 0.9

STO store · +|- change sign · RCL recall a stored number

0.7ENTER1.5×0.3ENTER0.5CHS×+ 0.9

ENTER enter (on the HP, puts the number on the stack) · CHS change sign

Historical simulation and historical value at risk

Scenario profit or loss from a log return Use it

\text{P/L} = D\left(e^{\tilde r} - 1\right)

Quick example

A GBP 2,500,000 position and a past day with a log return of −1.80%. $2{,}500{,}000 \times (e^{-0.018} - 1)$ = −GBP 44,597.

On your calculator

0.018+|-2NDe^x−1=×2500000= −44,597.42

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.018CHSge^x1−2500000× −44,597.42

CHS change sign · g blue shift key · e^x e to the power x

Rank of the VaR scenario Use it

k = (1 - c)\,n

Quick example

300 daily scenarios, 99% VaR: $k = 0.01 \times 300 = 3$ , so VaR is the 3rd largest loss. At 95%, $k = 15$ .

Interpolated value at risk Use it

\text{VaR} = L_{\lfloor k \rfloor} + (k - \lfloor k \rfloor)\left(L_{\lceil k \rceil} - L_{\lfloor k \rfloor}\right)

Quick example

185 days, 95%: $k = 9.25$ , between the 9th loss (520,000) and the 10th (496,000). VaR = 520,000 + 0.25 × (496,000 − 520,000) = 514,000.

On your calculator

496000−520000=×0.25+520000= 514,000

496000ENTER520000−0.25×520000+ 514,000

ENTER enter (on the HP, puts the number on the stack)

Two-position standard deviation in currency Know the idea

\sigma_p = \sqrt{D_1^2\sigma_1^2 + D_2^2\sigma_2^2 + 2D_1D_2\rho\,\sigma_1\sigma_2}

Quick example

USD 3 million (σ 1.0%) and USD 2 million (σ 1.5%), ρ = 0.5: $\sqrt{30{,}000^2 + 30{,}000^2 + 2 \times 0.5 \times 30{,}000^2}$ = USD 51,962. One-tailed 95% parametric VaR = 1.645 × 51,962 = USD 85,477.

On your calculator

30000x²×3=√x 51,961.52
×1.645= 85,476.71

x² square · √x square root

30000ENTER×3×g√x 51,961.52
1.645× 85,476.71

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Bootstrapping: simulating from the data you have

End price of a bootstrap path Use it

P_T = P_0\, e^{\sum_{i=1}^{k} \tilde r_i}

Quick example

Start at 45.00; three draws of log returns 0.9%, −1.5% and 2.2% sum to 1.6%. $45 \times e^{0.016}$ = 45.73.

On your calculator

0.9+1.5+|-+2.2=÷100=2NDe^x×45= 45.73

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

0.9ENTER1.5CHS+2.2+100÷ge^x45× 45.73

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · g blue shift key · e^x e to the power x

Monte Carlo simulation

Risk-neutral log-return step Use it

\tilde r_t = \left(r_f - \tfrac{\sigma^2}{2}\right)\Delta t + \sigma\sqrt{\Delta t}\,Z, \qquad P_t = P_{t-1}\,e^{\tilde r_t}

Quick example

Price 80.00, σ = 25%, $r_f$ = 3%, a half-year step, $Z$ = 1.2. Drift $(0.03 - 0.03125) \times 0.5 = -0.000625$ ; shock $0.25 \times \sqrt{0.5} \times 1.2 = 0.2121$ ; $80 \times e^{0.2115}$ = 98.84.

On your calculator

0.25x²÷2=+|-+0.03=×0.5=STO1 −0.000625
0.5√x×0.25×1.2+RCL1= 0.2115
2NDe^x×80= 98.84

x² square · +|- change sign · STO store · √x square root · RCL recall a stored number · 2ND second function (the shifted label above a key) · e^x e to the power x

0.03ENTER0.25ENTER×2÷−0.5×0.5g√x0.25×1.2×+ 0.2115
ge^x80× 98.84

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root · e^x e to the power x

Asian option payoff, one path Use it

V_T = \max\left(P_T - \bar P,\, 0\right)

Quick example

Final price 41.00, average over the life 38.50: payoff = 2.50. A path ending below its average pays 0.

Monte Carlo estimate of value Use it

\hat V_0 = e^{-r_f T}\,\frac{1}{N}\sum_{i=1}^{N}\text{payoff}_i

Quick example

Five payoffs at a six-month expiry: 1.86, 0, 0.87, 0, 2.50; average 5.23 ÷ 5 = 1.046; $r_f$ = 4%: $1.046 \times e^{-0.02}$ = 1.03. A real run uses thousands.

On your calculator

1.86+0.87+2.5=÷5=×0.02+|-2NDe^x= 1.0253

+|- change sign · 2ND second function (the shifted label above a key) · e^x e to the power x

1.86ENTER0.87+2.5+5÷0.02CHSge^x× 1.0253

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · g blue shift key · e^x e to the power x

Simulating two correlated assets: the Cholesky step

First simulated variable Memorize

X_1 = \sigma_1 Z_1

Quick example

σ₁ = 2%, Z₁ = 1.0: $X_1$ = 2.0%.

Second, correlated variable (Cholesky) Memorize

X_2 = \sigma_2\left(\rho Z_1 + \sqrt{1 - \rho^2}\,Z_2\right)

Quick example

σ₂ = 3%, ρ = 0.28, Z₁ = 1.0, Z₂ = 0.5. $\sqrt{1 - 0.0784} = 0.96$ ; $X_2 = 3\% \times (0.28 + 0.96 \times 0.5)$ = 2.28%.

On your calculator

1−0.28x²=√x×0.5+0.28=×3= 2.28

x² square · √x square root

1ENTER0.28ENTER×−g√x0.5×0.28+3× 2.28

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

Covariance of the simulated pair Know the idea

\operatorname{Cov}(X_1, X_2) = \rho\,\sigma_1\sigma_2

Quick example

σ₁ = 2%, σ₂ = 3%, ρ = 0.28: covariance = 0.28 × 2 × 3 = 1.68 (%²), that is 0.000168.

Module 10 Simple Linear Regression in Finance

The regression line: slope, intercept and least squares

Simple linear regression model Memorize

Y_i = b_0 + b_1 X_i + \varepsilon_i

Quick example

Revenue growth (Y, %) on marketing spend (X, % of sales): Ŷ = 1.39 + 1.29X. A retailer spending 4% of sales is fitted at 1.39 + 1.29 × 4 = 6.55%; the gap to its actual growth is its residual.

On your calculator

1.29×4+1.39= 6.55

1.29ENTER4×1.39+ 6.55

ENTER enter (on the HP, puts the number on the stack)

Slope coefficient Memorize

\hat{b}_1 = \dfrac{\sum (X_i-\bar{X})(Y_i-\bar{Y})}{\sum (X_i-\bar{X})^2} = \dfrac{\text{Cov}(X,Y)}{\text{Var}(X)}

Quick example

Cov(X, Y) = 0.0036 and Var(X) = 0.0045. Slope = 0.0036 ÷ 0.0045 = 0.80.

On your calculator

0.0036÷0.0045= 0.8

0.0036ENTER0.0045÷ 0.8

ENTER enter (on the HP, puts the number on the stack)

Intercept Memorize

\hat{b}_0 = \bar{Y} - \hat{b}_1\bar{X}

Quick example

Ȳ = 7, X̄ = 6 and slope 0.80. Intercept = 7 − 0.80 × 6 = 2.20.

On your calculator

7−(0.8×6)= 2.2

7ENTER0.8ENTER6×− 2.2

ENTER enter (on the HP, puts the number on the stack)

Slope from the correlation Know the idea

\hat{b}_1 = r\,\dfrac{s_Y}{s_X}

Quick example

Correlation 0.50, standard deviation of Y 8%, of X 5%. Slope = 0.50 × 8 ÷ 5 = 0.80.

On your calculator

0.5×8÷5= 0.8

0.5ENTER8×5÷ 0.8

ENTER enter (on the HP, puts the number on the stack)

Indicator variable regression Use it

Y_i = b_0 + b_1 I_i + \varepsilon_i, \quad I_i = 1 \text{ or } 0

Quick example

A lender's mean annual default rate was 1.8% before a rule change and 1.1% after. With AFTER = 1 after the change, the fitted line is Y = 1.8 − 0.7 × AFTER: the slope −0.7 is the change in the mean.

On your calculator

1.1−1.8= −0.7

1.1ENTER1.8− −0.7

ENTER enter (on the HP, puts the number on the stack)

The four regression assumptions and what residual plots reveal

Homoskedasticity Know the idea

E(\varepsilon_i^2) = \sigma_\varepsilon^2 \quad \text{for every } i

Quick example

Residual variance about 4 in the first half of the sample and about 4 in the second: homoskedastic. About 1, then about 9: heteroskedastic, often two regimes pooled into one regression.

Goodness of fit and testing the regression coefficients

Coefficient of determination Memorize

R^2 = \dfrac{SSR}{SST} = 1 - \dfrac{SSE}{SST}

Quick example

SST = 200 and SSE = 52, so SSR = 148. R² = 148 ÷ 200 = 0.74: the model explains 74% of the variation in Y.

On your calculator

200−52=÷200= 0.74

200ENTER52−200÷ 0.74

ENTER enter (on the HP, puts the number on the stack)

Correlation from R² Use it

r = \pm\sqrt{R^2} \quad (\text{same sign as } \hat{b}_1)

Quick example

R² = 0.81 and a negative slope: r = −√0.81 = −0.90.

On your calculator

0.81√x+|- −0.9

√x square root · +|- change sign

0.81g√xCHS −0.9

g blue shift key · √x square root · CHS change sign

Standard error of the slope coefficient Use it

s_{\hat{b}_1} = \dfrac{s_e}{\sqrt{\sum (X_i-\bar{X})^2}}

Quick example

s_e = 2.0 and Σ(X − X̄)² = 64. s_b̂₁ = 2.0 ÷ √64 = 2.0 ÷ 8 = 0.25.

On your calculator

2÷64√x= 0.25

√x square root

2ENTER64g√x÷ 0.25

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

t-statistic for a regression coefficient Memorize

t = \dfrac{\hat{b}_1 - B_1}{s_{\hat{b}_1}},\quad df = n-2

Quick example

Slope 1.35, standard error 0.20, testing H₀: B₁ = 1. t = (1.35 − 1) ÷ 0.20 = 1.75. With a critical value near 2.0 you fail to reject.

On your calculator

1.35−1=÷0.2= 1.75

1.35ENTER1−0.2÷ 1.75

ENTER enter (on the HP, puts the number on the stack)

Standard error of the intercept Use it

s_{\hat{b}_0} = s_e\sqrt{\dfrac{1}{n} + \dfrac{\bar{X}^2}{\sum (X_i-\bar{X})^2}}

Quick example

s_e = 2.0, n = 25, X̄ = 3, Σ(X − X̄)² = 100: 2.0 × √(0.04 + 0.09) = 2.0 × √0.13 = 0.72.

On your calculator

3x²÷100+251/x=√x×2= 0.7211

x² square · 1/x reciprocal · √x square root

3ENTER×100÷251/x+g√x2× 0.7211

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · g blue shift key · √x square root

The ANOVA table and the standard error of the estimate

Decomposition of variation Memorize

SST = SSR + SSE

Quick example

SSR = 36 and SSE = 50 give SST = 86. Degrees of freedom split the same way: n − 1 = 1 + (n − 2).

Mean square regression Memorize

MSR = \dfrac{SSR}{k} = SSR \ (k = 1)

Quick example

SSR = 36 with one independent variable: MSR = 36.

Mean square error Memorize

MSE = \dfrac{SSE}{n-k-1} = \dfrac{SSE}{n-2}

Quick example

n = 27 and SSE = 50. MSE = 50 ÷ 25 = 2.00.

On your calculator

50÷(27−2)= 2

50ENTER27ENTER2−÷ 2

ENTER enter (on the HP, puts the number on the stack)

Standard error of the estimate Memorize

s_e = \sqrt{MSE} = \sqrt{\dfrac{SSE}{n-2}}

Quick example

MSE = 2.00, so s_e = √2.00 = 1.41, in the units of Y.

On your calculator

50÷(27−2)=√x 1.4142

√x square root

50ENTER27ENTER2−÷g√x 1.4142

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · √x square root

F-statistic Memorize

F = \dfrac{MSR}{MSE} = \dfrac{SSR \,/\, k}{SSE \,/\, (n-k-1)}

Quick example

SSR = 36, SSE = 50, n = 27. MSR = 36, MSE = 2.00, so F = 36 ÷ 2.00 = 18.0.

On your calculator

36÷(50÷25)= 18

36ENTER50ENTER25÷÷ 18

ENTER enter (on the HP, puts the number on the stack)

Variance of Y from the table Use it

s_Y^2 = \dfrac{SST}{n-1}

Quick example

SST = 150 and n = 22: 150 ÷ 21 = 7.14.

On your calculator

150÷21= 7.1429

150ENTER21÷ 7.1429

ENTER enter (on the HP, puts the number on the stack)

Predicting Y and building a prediction interval

Predicted value Memorize

\hat{Y}_f = \hat{b}_0 + \hat{b}_1 X_f

Quick example

Intercept 2.0, slope 0.75, forecast X = 6. Ŷ = 2.0 + 0.75 × 6 = 6.5.

On your calculator

0.75×6+2= 6.5

0.75ENTER6×2+ 6.5

ENTER enter (on the HP, puts the number on the stack)

Standard error of the forecast Use it

s_f = s_e\sqrt{1 + \dfrac{1}{n} + \dfrac{(X_f-\bar{X})^2}{\sum (X_i-\bar{X})^2}}

Quick example

s_e = 3.0, n = 25 and X_f = X̄. s_f = 3.0 × √(1 + 1/25) = 3.06, just above s_e.

On your calculator

3×(1+251/x)√x= 3.0594

1/x reciprocal · √x square root

251/x1+g√x3× 3.0594

1/x reciprocal · g blue shift key · √x square root

Standard error of the forecast (variance form) Use it

s_f^2 = s_e^2\left[1 + \dfrac{1}{n} + \dfrac{(X_f-\bar{X})^2}{(n-1)\,s_X^2}\right]

Quick example

s_e = 1.5, n = 31, s_X = 2.0 and X_f is 3 above X̄: s_f = 1.5 × √(1 + 1/31 + 9 ÷ 120) = 1.5 × 1.0523 = 1.58.

On your calculator

9÷120+311/x+1=√x×1.5= 1.5784

1/x reciprocal · √x square root

9ENTER120÷311/x+1+g√x1.5× 1.5784

ENTER enter (on the HP, puts the number on the stack) · 1/x reciprocal · g blue shift key · √x square root

Prediction interval Memorize

\hat{Y}_f \pm t_c \times s_f,\quad df = n-2

Quick example

Ŷ = 8.4, t_c = 2.04, s_f = 1.4. Interval = 8.4 ± 2.04 × 1.4 = 8.4 ± 2.86 = 5.54 to 11.26.

On your calculator

2.04×1.4=STO1 2.856
8.4−RCL1= 5.544
8.4+RCL1= 11.256

STO store · RCL recall a stored number

2.04ENTER1.4×STO1 2.856
8.4RCL1− 5.544
8.4RCL1+ 11.256

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Functional forms: log-lin, lin-log and log-log models

Log-lin model Memorize

\ln Y_i = b_0 + b_1 X_i + \varepsilon_i

Quick example

ln Y = 1.0 + 0.10X. At X = 3, ln Y = 1.30, so Y = e^1.30 = 3.67. The slope says Y grows about 10% per unit of X.

On your calculator

0.1×3+1=2NDe^x 3.6693

2ND second function (the shifted label above a key) · e^x e to the power x

0.1ENTER3×1+ge^x 3.6693

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Lin-log model Memorize

Y_i = b_0 + b_1 \ln X_i + \varepsilon_i

Quick example

Y = 4 + 2 ln X. X rises from 50 to 60. Y changes by 2 × ln(60 ÷ 50) = 2 × 0.1823 = 0.36.

On your calculator

60÷50=LN×2= 0.3646

LN natural logarithm

60ENTER50÷gLN2× 0.3646

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · LN natural logarithm

Log-log model Memorize

\ln Y_i = b_0 + b_1 \ln X_i + \varepsilon_i

Quick example

Price elasticity of demand b₁ = −1.2. A 5% price rise goes with a change in quantity of about −1.2 × 5% = −6%.

On your calculator

1.2+|-×5= −6

+|- change sign

1.2CHSENTER5× −6

CHS change sign · ENTER enter (on the HP, puts the number on the stack)

Percentage change in Y in a log-lin model Use it

\%\Delta Y = e^{\hat{b}_1 \Delta X} - 1

Quick example

b̂₁ = 0.04 and X rises by 5: e^0.20 − 1 = 22.14% (0.04 × 5 = 20% is only the approximation).

On your calculator

0.04×5=2NDe^x−1= 0.2214

2ND second function (the shifted label above a key) · e^x e to the power x

0.04ENTER5×ge^x1− 0.2214

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Converting a log-lin prediction back to Y Memorize

\hat{Y} = e^{\hat{b}_0 + \hat{b}_1 X}

Quick example

ln Y = 2.0 + 0.05X. At X = 20: ln Y = 3.0, so Y = e^3.0 = 20.09.

On your calculator

0.05×20+2=2NDe^x 20.0855

2ND second function (the shifted label above a key) · e^x e to the power x

0.05ENTER20×2+ge^x 20.0855

ENTER enter (on the HP, puts the number on the stack) · g blue shift key · e^x e to the power x

Estimating beta with a CAPM regression

Capital asset pricing model Memorize

E[r_i] = r_f + \beta_i\left(E[r_m] - r_f\right)

Quick example

r_f = 2.0%, E[r_m] = 7.5%, β = 0.9: 2.0 + 0.9 × 5.5 = 6.95%.

On your calculator

7.5−2=×0.9+2= 6.95

7.5ENTER2−0.9×2+ 6.95

ENTER enter (on the HP, puts the number on the stack)

Excess returns Memorize

Z_{i,t} = r_{i,t} - r_{f,t}, \quad Z_{m,t} = r_{m,t} - r_{f,t}

Quick example

A stock returns 1.8% in a month, the index 1.1% and the bill 0.3%: Z_i = 1.8 − 0.3 = 1.5%, Z_m = 1.1 − 0.3 = 0.8%.

On your calculator

1.8−0.3= 1.5
1.1−0.3= 0.8

1.8ENTER0.3− 1.5
1.1ENTER0.3− 0.8

ENTER enter (on the HP, puts the number on the stack)

CAPM regression Memorize

Z_{i,t} = b_0 + b_1 Z_{m,t} + \varepsilon_{i,t}

Quick example

Regress 60 monthly stock excess returns on the index's: the slope b̂₁ is the beta estimate, here 1.24.

Confidence interval for beta Use it

\hat{b}_1 \pm t_c \times s_{\hat{b}_1}

Quick example

β̂ = 0.75, standard error 0.10, t_c = 2.01: 0.75 ± 0.201 = 0.55 to 0.95.

On your calculator

2.01×0.1=STO1 0.201
0.75−RCL1= 0.549
0.75+RCL1= 0.951

STO store · RCL recall a stored number

2.01ENTER0.1×STO1 0.201
0.75RCL1− 0.549
0.75RCL1+ 0.951

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number

Change in expected excess return Use it

\Delta E[r_i - r_f] = \hat{\beta}_i \times \Delta\left(E[r_m] - r_f\right)

Quick example

β̂ = 0.8 and the market risk premium falls by 1.0 point: 0.8 × (−1.0) = −0.8 point.

On your calculator

0.8×1+|-= −0.8

+|- change sign

0.8ENTER1CHS× −0.8

ENTER enter (on the HP, puts the number on the stack) · CHS change sign