CFA Level I Portfolio Construction formulas

Portfolio risk and return, the capital market line, the CAPM and beta, the risk-adjusted performance ratios and the arithmetic of risk management.

67 formulas from the Portfolio Construction lessons (8–12% of the exam), 23 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 Portfolio Risk and Return: Part I

How the major asset classes compare on return and risk

Expected return build-up Memorize

1 + E(R) = (1 + r_{rf})\,\big[1 + E(\pi)\big]\,\big[1 + E(RP)\big]

Quick example

Real risk-free rate 0.5%, expected inflation 3.0%, expected risk premium 6.5%. 1.005 × 1.03 × 1.065 − 1 = 10.24%. Adding the three gives 10.00%, which is the wrong answer.

On your calculator

1.005×1.03×1.065−1= 0.1024

1.005ENTER1.03×1.065×1− 0.1024

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

Real return Use it

r_{real} = \frac{1 + R_{nominal}}{1 + \pi} - 1

Quick example

A stock index returns 9.0% in a year when inflation is 3.0%. 1.09 ÷ 1.03 − 1 = 5.83%, not 6.0%.

On your calculator

1.09÷1.03−1= 0.0583

1.09ENTER1.03÷1− 0.0583

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

Relative bid–ask spread Use it

\text{Spread} = \frac{\text{Ask} - \text{Bid}}{\text{Midpoint}}

Quick example

Bid 19.90, ask 20.10, midpoint 20.00. (20.10 − 19.90) ÷ 20.00 = 1.00% of the price, a cost paid on every round trip.

On your calculator

20.10−19.90=÷20= 0.01

20.10ENTER19.90−20÷ 0.01

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

Risk aversion, utility and indifference curves

Utility function Memorize

U = E(r) - \tfrac{1}{2}\,A\,\sigma^{2}

Quick example

E(r) = 8%, σ = 12%, A = 5. Use decimals: 0.08 − 0.5 × 5 × 0.12² = 0.08 − 0.036 = 0.044. A higher score is a better portfolio for this investor.

On your calculator

0.08−(0.5×5×0.12x²)= 0.044

x² square

0.12ENTER×0.5×5×CHS0.08+ 0.044

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

Risk-free equivalent return Use it

R_{eq} = U = E(r) - \tfrac{1}{2}\,A\,\sigma^{2}

Quick example

A portfolio with E(r) = 9% and σ = 16% scores 0.09 − 0.5 × 2.5 × 0.16² = 0.058 for A = 2.5. A sure 5.80% gives this investor the same utility.

On your calculator

0.09−(0.5×2.5×0.16x²)= 0.058

x² square

0.16ENTER×0.5×2.5×CHS0.09+ 0.058

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

Portfolio variance, covariance and correlation

Sample mean return Use it

\bar{R} = \frac{\sum_{t=1}^{T} R_t}{T}

Quick example

Five annual returns: 2%, 8%, 5%, 9%, 6%. The sum is 30 and T = 5, so the mean is 6%.

On your calculator

2+8+5+9+6=÷5= 6

2ENTER8+5+9+6+5÷ 6

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

Sample variance Use it

s^{2} = \frac{\sum_{t=1}^{T} (R_t - \bar{R})^{2}}{T - 1}

Quick example

Same five returns, mean 6. Deviations: −4, +2, −1, +3, 0. Squares: 16, 4, 1, 9, 0 sum to 30. 30 ÷ 4 = 7.5 (%²), so s = 2.74%.

On your calculator

4x²+2x²+1x²+3x²=÷4= 7.5

x² square

4ENTER×2ENTER×+1ENTER×+3ENTER×+4÷ 7.5

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

Sample covariance Memorize

\mathrm{Cov}_{1,2} = \frac{\sum_{t=1}^{T} (R_{1,t} - \bar{R}_1)(R_{2,t} - \bar{R}_2)}{T - 1}

Quick example

Four years. Fund A’s deviations from its mean: +3, −1, −4, +2. Fund B’s: +2, 0, −3, +1. Products: 6, 0, 12, 2 sum to 20. 20 ÷ 3 = 6.67 (%²). Positive: they tend to sit on the same side of their means.

On your calculator

3×2+(4+|-×3+|-)+(2×1)=÷3= 6.6667

+|- change sign

3ENTER2×4CHSENTER3CHS×+2ENTER1×+3÷ 6.6667

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

Correlation coefficient Memorize

\rho_{1,2} = \frac{\mathrm{Cov}_{1,2}}{\sigma_1\,\sigma_2}

Quick example

Covariance −0.0045, standard deviations 15% and 10%. −0.0045 ÷ (0.15 × 0.10) = −0.30. Always between −1 and +1.

On your calculator

0.0045+|-÷(0.15×0.1)= -0.3

+|- change sign

0.0045CHSENTER0.15ENTER0.1×÷ -0.3

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

Covariance from correlation Memorize

\mathrm{Cov}_{1,2} = \rho_{1,2}\,\sigma_1\,\sigma_2

Quick example

Correlation −0.35, standard deviations 14% and 9%. −0.35 × 0.14 × 0.09 = −0.00441.

On your calculator

0.35+|-×0.14×0.09= -0.00441

+|- change sign

0.35CHSENTER0.14×0.09× -0.00441

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

Portfolio expected return Memorize

E(R_p) = w_1\,E(R_1) + w_2\,E(R_2)

Quick example

25% in an asset expected to return 12% and 75% in one expected to return 6%. 0.25 × 12 + 0.75 × 6 = 7.5%, whatever the correlation.

On your calculator

0.25×12+(0.75×6)= 7.5

0.25ENTER12×0.75ENTER6×+ 7.5

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

40% in an asset with σ = 24% and 60% in one with σ = 8%, correlation 0.6. 0.009216 + 0.002304 + 0.00553 = 0.01705, so σp = √0.01705 = 13.06%. The weighted average of the σs is 14.4%.

On your calculator

0.4x²×0.24x²+(0.6x²×0.08x²)+(2×0.4×0.6×0.6×0.24×0.08)=√x 0.1306

x² square · √x square root

0.4ENTER×0.24ENTER××0.6ENTER×0.08ENTER××+2ENTER0.4×0.6×0.6×0.24×0.08×+g√x 0.1306

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

Zero-risk weight when correlation is −1 Use it

w_1 = \frac{\sigma_2}{\sigma_1 + \sigma_2}

Quick example

Perfectly negatively correlated assets with σ = 15% and 5%. Weight in the first: 5 ÷ (15 + 5) = 25%. Check: 0.25 × 15 = 3.75 and 0.75 × 5 = 3.75, which cancel.

On your calculator

5÷(15+5)= 0.25

5ENTER15ENTER5+÷ 0.25

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

Equally weighted portfolio of N assets Use it

\sigma_p^{2} = \frac{\bar{\sigma}^{2}}{N} + \frac{N-1}{N}\,\overline{\mathrm{Cov}}

Quick example

N = 25 assets, each σ = 20%, average correlation 0.4 (average covariance 0.016). 0.04 ÷ 25 + (24 ÷ 25) × 0.016 = 0.0016 + 0.01536 = 0.01696, so σp = 13.02%. The floor as N grows is √0.016 = 12.65%.

On your calculator

0.2x²÷25+(24÷25×0.016)=√x 0.1302

x² square · √x square root

0.2ENTER×25÷24ENTER25÷0.016×+g√x 0.1302

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

Portfolio variance, any number of assets Know the idea

\sigma_p^{2} = \sum_{i=1}^{N}\sum_{j=1}^{N} w_i\,w_j\,\mathrm{Cov}_{i,j}

Quick example

With three assets there are 3 variance terms and 3 distinct covariances, each covariance counted twice (9 terms in all). The covariance of an asset with itself is its variance.

The efficient frontier and the global minimum-variance portfolio

Condition for adding a new asset Use it

\frac{E(R_{new}) - R_f}{\sigma_{new}} > \frac{E(R_p) - R_f}{\sigma_p} \times \rho_{new,p}

Quick example

Portfolio: E = 10%, σ = 25%, Rf = 2%, so its Sharpe ratio is 0.32. New asset: E = 4.4%, σ = 10%, correlation 0.7. Its ratio (4.4 − 2) ÷ 10 = 0.24 beats 0.32 × 0.7 = 0.224, so add it.

On your calculator

4.4−2=÷10= 0.24
10−2=÷25=×0.7= 0.224

4.4ENTER2−10÷ 0.24
10ENTER2−25÷0.7× 0.224

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

Two-asset global minimum-variance weight Know the idea

w_1 = \frac{\sigma_2^{2} - \mathrm{Cov}_{1,2}}{\sigma_1^{2} + \sigma_2^{2} - 2\,\mathrm{Cov}_{1,2}}

Quick example

σ₁ = 18%, σ₂ = 8%, correlation 0.1, so Cov = 0.00144. w₁ = (0.0064 − 0.00144) ÷ (0.0324 + 0.0064 − 0.00288) = 0.00496 ÷ 0.03592 = 13.81% in asset 1.

On your calculator

(0.08x²−0.00144)÷(0.18x²+0.08x²−(2×0.00144))= 0.1381

x² square

0.08ENTER×0.00144−0.18ENTER×0.08ENTER×+0.00288−÷ 0.1381

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

The capital allocation line and the optimal portfolio

Return and risk with a risk-free asset Memorize

E(R_p) = w_f R_f + (1 - w_f)\,E(R_i), \qquad \sigma_p = (1 - w_f)\,\sigma_i

Quick example

30% in a bill yielding 1.5% and 70% in a risky portfolio expected to earn 7.5% with σ = 12%. E = 0.3 × 1.5 + 0.7 × 7.5 = 5.7%; σp = 0.7 × 12 = 8.4%. The bill adds no risk of its own.

On your calculator

0.3×1.5+(0.7×7.5)= 5.7
0.7×12= 8.4

0.3ENTER1.5×0.7ENTER7.5×+ 5.7
0.7ENTER12× 8.4

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

Capital allocation line Memorize

E(R_p) = R_f + \frac{E(R_i) - R_f}{\sigma_i}\,\sigma_p

Quick example

Rf = 1.5%, risky portfolio E = 7.5%, σ = 12%. An investor who wants σp = 9% expects 1.5 + (6 ÷ 12) × 9 = 6.0%.

On your calculator

7.5−1.5=÷12=×9+1.5= 6

7.5ENTER1.5−12÷9×1.5+ 6

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

Slope of the CAL (market price of risk) Use it

\text{Slope} = \frac{E(R_i) - R_f}{\sigma_i}

Quick example

Rf = 1.5%, risky portfolio E = 7.5%, σ = 12%. (7.5 − 1.5) ÷ 12 = 0.50: 0.50 points of extra return for each point of risk.

On your calculator

7.5−1.5=÷12= 0.5

7.5ENTER1.5−12÷ 0.5

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

Optimal weight in the risky portfolio Know the idea

y^{*} = \frac{E(R_i) - R_f}{A\,\sigma_i^{2}}

Quick example

Rf = 1.5%, risky portfolio E = 7.5%, σ = 12%, investor A = 5. y = 0.06 ÷ (5 × 0.0144) = 83.3% in the risky portfolio and 16.7% in the risk-free asset

On your calculator

0.06÷(5×0.12x²)= 0.8333

x² square

0.06ENTER5ENTER0.12ENTER××÷ 0.8333

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

Module 2 Portfolio Risk and Return: Part II (CAPM and Performance Measures)

The capital market line

Return and risk of a mix with the risk-free asset Use it

E(R_p) = w\,E(R_i) + (1 - w)\,R_f, \qquad \sigma_p = w\,\sigma_i

Quick example

75% in a risky portfolio expected to earn 8.5% with σ = 16%, 25% in a bill paying 2.5%. E = 0.75 × 8.5 + 0.25 × 2.5 = 7.0%; σp = 0.75 × 16 = 12%.

On your calculator

0.75×8.5+(0.25×2.5)= 7

0.75ENTER8.5×0.25ENTER2.5×+ 7

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

Capital market line Memorize

E(R_p) = R_f + \frac{E(R_m) - R_f}{\sigma_m}\,\sigma_p

Quick example

Rf = 4%, E(Rm) = 10%, σm = 20%. A portfolio with σp = 15% expects 4 + (6 ÷ 20) × 15 = 8.5%.

On your calculator

10−4=÷20=×15+4= 8.5

10ENTER4−20÷15×4+ 8.5

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

Slope of the CML (market price of risk) Memorize

\text{Slope} = \frac{E(R_m) - R_f}{\sigma_m}

Quick example

Rf = 4%, E(Rm) = 10%, σm = 20%. (10 − 4) ÷ 20 = 0.30: 0.30 points of extra return per point of risk, the market’s Sharpe ratio.

On your calculator

10−4=÷20= 0.3

10ENTER4−20÷ 0.3

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

Weight in the market portfolio Use it

w = \frac{\sigma_p}{\sigma_m}

Quick example

σm = 20% and a target σp of 25% need w = 25 ÷ 20 = 1.25: 125% in the market, 25% borrowed.

On your calculator

25÷20= 1.25

25ENTER20÷ 1.25

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

Leveraged portfolio with a borrowing rate above Rf Use it

E(R_p) = w\,E(R_m) - (w - 1)\,R_b, \qquad w > 1

Quick example

Rf = 4%, borrowing rate 6%, E(Rm) = 10%, w = 1.25. E = 1.25 × 10 − 0.25 × 6 = 11.0%, against 11.5% if borrowing were at Rf.

On your calculator

1.25×10−(0.25×6)= 11

1.25ENTER10×0.25ENTER6×− 11

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

Systematic vs nonsystematic risk and return-generating models

Multi-factor model Use it

E(R_i) - R_f = \sum_{j=1}^{k} \beta_{ij}\,E(F_j)

Quick example

Rf = 3%. A stock has market, size and value sensitivities of 1.2, 0.3 and −0.1, with factor premiums of 5%, 2% and 3%. E = 3 + 1.2 × 5 + 0.3 × 2 − 0.1 × 3 = 9.3%.

On your calculator

1.2×5+(0.3×2)−(0.1×3)+3= 9.3

1.2ENTER5×0.3ENTER2×+0.1ENTER3×−3+ 9.3

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

Market model Memorize

R_i = \alpha_i + \beta_i\,R_m + e_i

Quick example

α = 0.1%, β = 1.1. In a month the market falls 3%. The model gives 0.1 + 1.1 × (−3) = −3.2% before the firm-specific shock ei.

On your calculator

1.1×3+|-+0.1= −3.2

+|- change sign

1.1ENTER3CHS×0.1+ −3.2

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

Decomposition of total variance Memorize

\sigma_i^2 = \beta_i^2\,\sigma_m^2 + \sigma_e^2

Quick example

β = 1.1, σm = 10%, σi = 30%. Systematic variance = 1.21 × 0.01 = 0.0121; total = 0.09; nonsystematic = 0.0779, so σe = 27.91%.

On your calculator

0.3x²−(1.1x²×0.1x²)=√x 0.2791

x² square · √x square root

0.3ENTER×1.1ENTER×0.1ENTER××−g√x 0.2791

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

How to calculate and interpret beta

Beta from covariance Memorize

\beta_i = \frac{\text{Cov}(R_i, R_m)}{\sigma_m^2}

Quick example

Cov(Ri, Rm) = 0.0300 and σm = 25%. β = 0.0300 ÷ 0.0625 = 0.48.

On your calculator

0.03÷0.25x²= 0.48

x² square

0.03ENTER0.25ENTER×÷ 0.48

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

Covariance from correlation Use it

\text{Cov}(R_i, R_m) = \rho_{i,m}\,\sigma_i\,\sigma_m

Quick example

ρ = 0.5, σi = 30%, σm = 20%. Cov = 0.5 × 0.30 × 0.20 = 0.0300.

On your calculator

0.5×0.3×0.2= 0.03

0.5ENTER0.3×0.2× 0.03

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

Beta from correlation Memorize

\beta_i = \rho_{i,m}\,\frac{\sigma_i}{\sigma_m}

Quick example

ρ = 0.55, σi = 28%, σm = 22%. β = 0.55 × 28 ÷ 22 = 0.70.

On your calculator

0.55×28÷22= 0.7

0.55ENTER28×22÷ 0.7

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

Security characteristic line Use it

R_i - R_f = \alpha_i + \beta_i\,(R_m - R_f) + e_i

Quick example

Rf = 2%, β = 1.2, α = 0.5%. If the market returns 8%, the line gives 0.5 + 1.2 × (8 − 2) = 7.7% of excess return, so a return of 9.7% before the firm-specific part.

On your calculator

8−2=×1.2+0.5+2= 9.7

8ENTER2−1.2×0.5+2+ 9.7

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

Portfolio beta Memorize

\beta_p = \sum_{i=1}^{n} w_i\,\beta_i

Quick example

30% in a stock with β 1.6, 50% in the market (β 1) and 20% in T-bills (β 0). βp = 0.48 + 0.50 + 0 = 0.98.

On your calculator

0.3×1.6+(0.5×1)= 0.98

0.3ENTER1.6×0.5ENTER1×+ 0.98

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

Adjusted beta Know the idea

\beta_{adj} = \tfrac{2}{3}\,\beta_{raw} + \tfrac{1}{3}

Quick example

A raw beta of 1.8 adjusts to (2 × 1.8 + 1) ÷ 3 = 1.53, pulled toward 1.

On your calculator

1.8×2+1=÷3= 1.5333

1.8ENTER2×1+3÷ 1.5333

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

The CAPM and the security market line

Capital asset pricing model Memorize

E(R_i) = R_f + \beta_i\,[E(R_m) - R_f]

Quick example

Rf = 1.5%, E(Rm) = 8.0%, β = 1.2. E = 1.5 + 1.2 × (8.0 − 1.5) = 9.3%. Subtract Rf from the market return first.

On your calculator

8−1.5=×1.2+1.5= 9.3

8ENTER1.5−1.2×1.5+ 9.3

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

Expected alpha (forecast minus required return) Use it

\alpha_i = E_{forecast}(R_i) - \left[R_f + \beta_i\,(E(R_m) - R_f)\right]

Quick example

Rf = 4%, E(Rm) = 9%, β = 1.1, forecast 9.0%. Required = 4 + 1.1 × 5 = 9.5%, so α = −0.5%: below the SML, overvalued.

On your calculator

9−4=×1.1+4=+|-+9= −0.5

+|- change sign

9ENTER9ENTER4−1.1×4+− −0.5

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

Arbitrage pricing theory Know the idea

E(R_p) = R_f + \lambda_1\,\beta_{p,1} + \dots + \lambda_K\,\beta_{p,K}

Quick example

Rf = 3%. Factor 1: premium 4%, β 0.9. Factor 2: premium 2.5%, β 1.2. E = 3 + 3.6 + 3.0 = 9.6%.

On your calculator

0.9×4+(1.2×2.5)+3= 9.6

0.9ENTER4×1.2ENTER2.5×+3+ 9.6

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

Sharpe ratio, Treynor ratio, M² and Jensen's alpha

Sharpe ratio Memorize

S = \frac{R_p - R_f}{\sigma_p}

Quick example

Rp = 12%, Rf = 2%, σp = 16%. S = 10 ÷ 16 = 0.625 of excess return per point of total risk.

On your calculator

12−2=÷16= 0.625

12ENTER2−16÷ 0.625

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

Treynor ratio Memorize

T = \frac{R_p - R_f}{\beta_p}

Quick example

Rp = 12%, Rf = 2%, βp = 1.25. T = 10 ÷ 1.25 = 8.00 of excess return per unit of beta.

On your calculator

12−2=÷1.25= 8

12ENTER2−1.25÷ 8

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

M² Memorize

M^2 = (R_p - R_f)\,\frac{\sigma_m}{\sigma_p} + R_f

Quick example

Rp = 12%, Rf = 2%, σp = 16%, σm = 18%. M² = 10 × 18 ÷ 16 + 2 = 13.25%: the return the fund would have had at the market’s risk.

On your calculator

12−2=÷16=×18+2= 13.25

12ENTER2−16÷18×2+ 13.25

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

M² alpha Use it

M^2\ \text{alpha} = M^2 - R_m

Quick example

M² = 13.25% and the market returned 9%. M² alpha = +4.25%: the fund beat the market on a total-risk basis.

On your calculator

13.25−9= 4.25

13.25ENTER9− 4.25

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

Jensen’s alpha Memorize

\alpha_p = R_p - \left[R_f + \beta_p\,(R_m - R_f)\right]

Quick example

Rp = 12%, Rf = 2%, Rm = 9%, βp = 1.25. CAPM return = 2 + 1.25 × 7 = 10.75%, so α = +1.25%.

On your calculator

9−2=×1.25+2=+|-+12= 1.25

+|- change sign

12ENTER9ENTER2−1.25×2+− 1.25

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

Module 3 Portfolio Management: An Overview

Why invest through a portfolio

Diversification ratio Memorize

DR = \dfrac{\sigma_p}{\bar{\sigma}}

Quick example

An equally weighted portfolio has σ = 20.4%. The average σ of its holdings is 24.0%. 20.4 ÷ 24.0 = 0.85: the portfolio carries 85% of the risk of a typical single holding.

On your calculator

20.4÷24= 0.85

20.4ENTER24÷ 0.85

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

Ratio for N equal-risk holdings Know the idea

DR = \sqrt{\dfrac{1}{N} + \dfrac{N-1}{N}\,\rho}

Quick example

Twenty shares of equal risk, average pairwise correlation 0.20: √(1 ÷ 20 + 19 ÷ 20 × 0.20) = √0.24 = 0.49.

On your calculator

1÷20+(19÷20×0.2)=√x 0.4899

√x square root

1ENTER20÷19ENTER20÷0.2×+g√x 0.4899

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

The three steps of the portfolio management process

Portfolio weight after market moves Use it

w_{S,1} = \dfrac{w_S(1+r_S)}{w_S(1+r_S) + w_B(1+r_B)}

Quick example

A 50/50 stock/bond portfolio. Stocks gain 30%, bonds lose 10%, no trades. Stocks: 50 × 1.30 = 65. Bonds: 50 × 0.90 = 45. Stock weight = 65 ÷ 110 = 59.1%.

On your calculator

50×1.3=÷(50×1.3+(50×0.9))= 0.5909

50ENTER1.3×STO150ENTER0.9×+RCL1x<>y÷ 0.5909

ENTER enter (on the HP, puts the number on the stack) · STO store · RCL recall a stored number · x<>y swap the last two numbers

Types of investors, and DB vs DC pension plans

Defined benefit pension (final-salary formula) Use it

P = a \times Y \times S

Quick example

Plan pays 2.0% × years of service × final salary. 35 years, final salary 50,000. 0.02 × 35 × 50,000 = 35,000 a year, whatever the markets did.

On your calculator

0.02×35×50000= 35000

0.02ENTER35×50000× 35000

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

Mutual funds, ETFs and other pooled investment products

Shares issued at NAV Use it

N = \dfrac{I}{NAV}

Quick example

An investor puts USD 120,000 into an open-end fund with a NAV of USD 24.00. 120,000 ÷ 24 = 5,000 new shares.

On your calculator

120000÷24= 5000

120000ENTER24÷ 5000

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

Discount to NAV Use it

D = \dfrac{NAV - P}{NAV}

Quick example

A closed-end fund has a NAV of 18.00 and trades at 16.38. (18.00 − 16.38) ÷ 18.00 = 9.0% discount. A negative result is a premium.

On your calculator

18−16.38=÷18= 0.09

18ENTER16.38−18÷ 0.09

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

Module 4 Basics of Portfolio Planning and Construction

Risk and return objectives: willingness vs ability to take risk

Required return to fund spending and keep real value Use it

R_{req} = \left(1 + \frac{S}{V}\right)(1 + \pi) - 1

Quick example

Portfolio 600,000, spending 27,000 a year (4.5%), inflation 1.8%. 1.045 × 1.018 − 1 = 6.38% (adding gives 6.30%, which falls short).

On your calculator

27000÷600000+1=×1.018−1= 0.06381

27000ENTER600000÷1+1.018×1− 0.06381

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

Required return to reach a goal in today’s money Use it

R_{req} = \left(\frac{G}{V}\right)^{1/N}(1 + \pi) - 1

Quick example

Invest 150,000 now to have 420,000 in today’s money in 25 years, inflation 2.0%. (2.8)^(1/25) × 1.02 − 1 = 1.0420 × 1.02 − 1 = 6.29%.

On your calculator

420000÷150000=y^x(1÷25)=×1.02−1= 0.0629

y^x power

420000ENTER150000÷1ENTER25÷y^x1.02×1− 0.0629

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

IPS constraints: liquidity, time horizon, taxes, legal and unique circumstances

After-tax return Use it

r_{AT} = r\,(1 - t)

Quick example

A taxable bond yields 4.8% before tax; the investor’s tax rate on interest is 25%. 4.8% × (1 − 0.25) = 3.60%. A tax-free bond must yield more than this to win.

On your calculator

4.8×(1−0.25)= 3.6

4.8ENTER1ENTER0.25−× 3.6

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

Asset classes, strategic asset allocation and portfolio construction

Tactical asset allocation contribution Use it

C_{TAA} = \sum_i \left(w_i - w_i^{policy}\right) R_i

Quick example

Policy 45% equities / 55% bonds. Actual 50 / 50. Equities return 6%, bonds −1%. (0.50 − 0.45)(6%) + (0.50 − 0.55)(−1%) = 0.30% + 0.05% = +0.35%.

On your calculator

0.5−0.45=×6= 0.30
0.5−0.55=×1+|-= 0.05
0.30+0.05= 0.35

+|- change sign

0.5ENTER0.45−6× 0.30
0.5ENTER0.55−1CHS× 0.05
+ 0.35

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

Module 5 The Behavioral Biases of Individuals

Cognitive errors: belief perseverance and information processing

Probability of an event after a flag (base rate with Bayes) Use it

P(E \mid F) = \frac{P(E)\,P(F \mid E)}{P(E)\,P(F \mid E) + [1 - P(E)]\,P(F \mid \text{not } E)}

Quick example

Base rate 4%, hit rate 75%, false-positive rate 12%. (0.04 × 0.75) ÷ (0.04 × 0.75 + 0.96 × 0.12) = 0.03 ÷ 0.1452 = 20.7%, not 75%.

On your calculator

0.04×0.75=STO1 numerator 0.03
(1−0.04)×0.12+RCL1= denominator 0.1452
1/x×RCL1= 0.2066

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

0.04ENTER0.75×STO1 numerator 0.03
1ENTER0.04−0.12×RCL1+ denominator 0.1452
1/xRCL1× 0.2066

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

Emotional biases: loss aversion, overconfidence and four more

Gain needed to recover a loss Know the idea

G = \frac{1}{1 - L} - 1 = \frac{L}{1 - L}

Quick example

A position falls 20%. It needs 1 ÷ 0.80 − 1 = 25% from the lower price to get back to even (not 20%).

On your calculator

1÷0.8=−1= 0.25

1ENTER0.8÷1− 0.25

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

Module 6 Introduction to Risk Management

What risk management is, and the risk management framework

Risk exposure Use it

\text{Exposure} = \text{Position} \times \text{Driver}

Quick example

A firm holds USD 12 million of oil inventory (the position). If the oil price falls 15% (the driver), the exposure is 12 × (−0.15) = −USD 1.8 million.

On your calculator

12×0.15+|-= -1.8 (millions)

+|- change sign

12ENTER0.15CHS× -1.8 (millions)

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

Risk governance, risk tolerance and risk budgeting

Combined risk, uncorrelated sources Use it

\sigma_p = \sqrt{\sigma_1^{2} + \sigma_2^{2} + \cdots + \sigma_n^{2}}

Quick example

Two uncorrelated sources of active risk, 5% and 12%. √(25 + 144) = √169 = 13%, well below the simple sum of 17%.

On your calculator

5x²+12x²=√x 13

x² square · √x square root

5ENTER×12ENTER×+g√x 13

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

Combined risk with a common correlation Use it

\sigma_p = \sqrt{\sum_i \sigma_i^{2} + 2\rho\sum_{i<j}\sigma_i\sigma_j}

Quick example

Two sources of 6% and 8% with correlation 0.25. √(36 + 64 + 2 × 0.25 × 6 × 8) = √124 = 11.14%. At ρ = +1 the answer is the simple sum, 14%.

On your calculator

6x²+8x²+(2×0.25×6×8)=√x 11.1355

x² square · √x square root

6ENTER×8ENTER×+2ENTER0.25×6×8×+g√x 11.1355

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

Financial and non-financial risks, and how they interact

Credit-adjusted value of a claim Know the idea

V = \sum_{s} p_s \times X_s \times (1 - d_s)

Quick example

A claim pays 200 in a downturn, which has a 10% probability. The counterparty fails in 40% of downturns. V = 0.10 × 200 × 0.60 = 12, against 20 if default were ignored.

On your calculator

0.1×200×0.6= 12

0.1ENTER200×0.6× 12

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

Measuring risk: probability, VaR, CVaR and sensitivities

Interest-rate sensitivity (duration) Use it

\%\Delta P \approx -D \times \Delta y

Quick example

Duration 4.5 and yields fall 0.80%. The price rises about 4.5 × 0.008 = 3.6%.

On your calculator

4.5×0.008+|-=+|- 0.036

+|- change sign

4.5ENTER0.008CHS×CHS 0.036

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

Option price change from delta Use it

\Delta C \approx \delta \times \Delta S

Quick example

A call with delta 0.35 and an underlying that falls 3.00. The option falls about 0.35 × 3.00 = 1.05. Gamma matters for larger moves because delta itself changes.

On your calculator

0.35×3+|-= -1.05

+|- change sign

0.35ENTER3CHS× -1.05

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

Expected number of days beyond VaR Use it

N = T \times p

Quick example

A 5% one-day VaR over 252 trading days is exceeded on about 252 × 0.05 = 12.6 days.

On your calculator

252×0.05= 12.6

252ENTER0.05× 12.6

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

Modifying risk: prevention, acceptance, transfer and shifting

Expected annual loss Use it

EL = p \times L

Quick example

A cargo vessel worth USD 9,000,000 has a 0.3% annual probability of a total loss. EL = 0.003 × 9,000,000 = USD 27,000 a year.

On your calculator

0.003×9000000= 27000

0.003ENTER9000000× 27000

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

Premium above expected loss Know the idea

\text{Premium} - EL

Quick example

A premium of USD 90,000 against an expected loss of USD 60,000 costs USD 30,000 a year above the expected loss. That is the price of removing the chance of a large loss.

On your calculator

90000−60000= 30000

90000ENTER60000− 30000

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