Learning Module 8 of Quantitative Methods 11–14% of the exam

Portfolio Return and Risk

6 lessons. About 36 minutes in cram mode, 188 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary Practice this module

Lessons

1
Portfolio returns over time: buy-and-hold, drift and rebalancing
calculate 5 min cram, 26 min full
2
Portfolio expected return, variance and the role of correlation
calculate 8 min cram, 42 min full
3
How many assets does diversification need?
evaluate 4 min cram, 20 min full
4
The minimum-variance portfolio and the efficient frontier
calculate 6 min cram, 32 min full
5
Risk preferences, mean–variance utility and indifference curves
explain 5 min cram, 24 min full
6
From the capital allocation line to the capital market line and the CAPM
explain 8 min cram, 44 min full

What the exam expects you to do

Calculate, interpret, and evaluate the expected return, variance, standard deviation, covariance, and correlation of portfolio returns.Portfolio returns over time: buy-and-hold, drift and rebalancing
Describe, calculate, and interpret the minimum-variance portfolio and portfolios that lie on the efficient frontier.The minimum-variance portfolio and the efficient frontier
Explain the selection of an optimal portfolio, given an investor’s risk aversion and the capital allocation line, and how this extends to the market portfolio and the capital market line.Risk preferences, mean–variance utility and indifference curves

Module summary

The whole module on one sheet, organised by lesson: what to memorize, how the ideas fit together, the formulas and the key terms. No paragraphs, just what you need to revise.

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.

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

Known

MemorizePortfolio returns over time

Measure	Formula
Period return	$r_{P,t} = \sum_i w_{i,t-1} r_{i,t}$ (start-of-period weights)
Weight after a period	$w_{i,t-1}(1 + r_{i,t}) \div (1 + r_{P,t})$
Buy-and-hold, T periods	$\sum_i w_{i,0} \times$ asset cumulative return $= \prod_t (1 + r_{P,t}) - 1$
Geometric average	$(1 + r_{P,T})^{1/T} - 1$
Arithmetic average	$\frac{1}{T}\sum_t r_{P,t}$ , higher, does not compound

Memory hook Weights look back one period. Winners grow, losers shrink. Wealth multiplies, never adds.

Know it cold Rebalancing is not automatically better: it wins when returns reverse, loses when the trend continues.

Key terms

portfolio return Memorize
beginning-of-period weights Use it
buy-and-hold return Memorize
cumulative portfolio return Memorize
arithmetic average portfolio return Memorize
geometric average portfolio return Memorize
portfolio drift Know the idea
style drift Know the idea
rebalancing Know the idea
rebalancing frequency Know the idea
concentration risk Know the idea

Portfolio returns over several periods

Portfolio return, T periodsweights at the start of each period

Period return Memorize Σ w(i,t−1) × r(i,t)
Buy-and-hold cumulative Memorize Σ starting weight × asset cumulative return = product of (1 + r_P,t) − 1
Geometric average Memorize (1 + cumulative)^(1/T) − 1; matches ending wealth
Arithmetic average Memorize simple mean of period returns; higher, does not compound

Two kinds of drift

Drift

Portfolio drift Know the idea weights move with relative performance, no trades
Style drift Know the idea strategy moves away from plan; can happen with no weight change
- Remedy systematic, pre-defined rebalancing frequency

Formulas

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

Known

MemorizePortfolio return and risk at a glance

Quantity	Formula
Expected return	$w_1 E[r_1] + w_2 E[r_2]$
Variance	$w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho_{1,2}\sigma_1\sigma_2$
Covariance	$\sigma_{1,2} = \rho_{1,2}\sigma_1\sigma_2$
Correlation	$\rho_{1,2} = \sigma_{1,2} \div (\sigma_1\sigma_2)$
Scenarios	$E[r_P] = \sum p_s r_{P,s}$ ; $\sigma_P^2 = \sum p_s (r_{P,s} - E[r_P])^2$

Memory hook Square each weight with its own variance, then multiply the weights for the cross term and double it, because σ₁,₂ and σ₂,₁ are the same number.

Know it cold Return averages; risk does not. $\sigma_{i,i} = \sigma_i^2$ . Weights sum to 1. N assets: N variances and $(N^2 - N)/2$ distinct covariances. A matrix "scaled by $10^{-4}$ " is in %².

MemorizeCorrelation and diversification

Correlation	Portfolio σ	Benefit
+1	weighted average of the σs	none
between −1 and +1	below that average	some
−1	$\lvert w_1\sigma_1 - w_2\sigma_2\rvert$	risk can reach zero

Memory hook Correlation is the only dial. At +1 the cross term is as large as it can be and the formula collapses to a straight average.

Know it cold Lower correlation lowers σ_P and leaves E[r_P] unchanged. Compare pairs by correlation, never by covariance alone.

Key terms

expected portfolio return Memorize
covariance Memorize
correlation Memorize
portfolio variance Memorize
variance–covariance matrix Use it
correlation matrix Use it
scenario-based portfolio return and variance Use it
diversification benefit Use it
mean–variance analysis Know the idea

What goes into portfolio variance

Portfolio varianceweights, own risks and how the assets move together

Own-variance terms Memorize wᵢ²σᵢ², always positive
Covariance terms Memorize 2wᵢwⱼσᵢ,ⱼ for every pair
- Positive covariance adds risk; returns move together
- Zero covariance adds nothing; returns unrelated
- Negative covariance subtracts risk; this is the diversification benefit

How correlation sets two-asset risk

Correlation between the two assetsweights and standard deviations held fixed; expected return never changes

ρ = +1 Memorize σ_P is the weighted average of the two σs; no diversification benefit
ρ between −1 and +1 Memorize σ_P below the weighted average; the lower ρ, the lower σ_P
ρ = −1 Use it σ_P = |w₁σ₁ − w₂σ₂|; can reach zero

Formulas

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?

Known

MemorizeDiversifying with N assets

Item	Formula or fact
Equal weights, same σ	$\sigma_P^2 = \frac{\sigma^2}{N} + (1 - \frac{1}{N})\bar\rho\sigma^2$
Average-covariance form	$\frac{\bar\sigma^2}{N} + \frac{N-1}{N}\,\overline{\sigma_{i,j}}$
Large N	$\sigma_P^2 \approx \bar\rho\sigma^2$ , so $\sigma_P \approx \sigma\sqrt{\bar\rho}$
Removed	unsystematic (asset-specific) risk
Left	systematic (market) risk

Memory hook Own variances die like 1/N; covariances stay. The floor is ρ̄σ².

Know it cold Most of the benefit comes from the first 20 to 30 assets (the curriculum also says 30 or more); the lower the correlation, the bigger the benefit.

Key terms

equally weighted portfolio Use it
average correlation Memorize
average covariance Use it
number of assets Use it
diversification benefit Use it
unsystematic risk Memorize
systematic risk Memorize
diversification rule of thumb Know the idea

What diversification removes

Total risk of an asset

Unsystematic risk Memorize asset-specific; also non-systematic or idiosyncratic; shrinks like 1/N
Systematic risk Memorize market-wide; the ρ̄σ² floor; cannot be diversified away

What sets the benefit

Equally weighted portfolio risk

Number of assets N Use it own-variance part falls like 1/N
Average correlation ρ̄ Memorize sets the floor ρ̄σ²; lower is better
Asset volatility σ Know the idea scales everything

Formulas

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

Known

MemorizeFrontier formulas

Quantity	Formula
Minimum-variance weight	$w_A = \frac{\sigma_B^2 - \sigma_{A,B}}{\sigma_A^2 + \sigma_B^2 - 2\sigma_{A,B}}$
Target-return weight	$w_A = \frac{r_p - r_B}{r_A - r_B}$
Sharpe ratio	$\frac{E[r_P] - r_f}{\sigma_P}$
Feasible target (no short sales)	between the lowest and highest asset return

Memory hook Minimum-variance weight: the OTHER asset's variance on top. The riskier asset gets the smaller weight.

Know it cold Efficient = on the upper edge, from the minimum-variance portfolio up. Optimal risky (tangency) = highest Sharpe ratio, not lowest variance.

Key terms

feasible portfolios Memorize
minimum-variance portfolio Memorize
global minimum-variance portfolio Memorize
minimum-variance frontier Memorize
Markowitz efficient frontier Memorize
inefficient portfolio Memorize
no-short-selling constraint Use it
short selling Know the idea
target-return weights Memorize
optimal risky portfolio Memorize
tangency portfolio Memorize
Sharpe ratio Memorize
return-constrained optimization Know the idea
risk-constrained optimization Know the idea
risk-adjusted optimization Know the idea
optimization inputs Know the idea

Mapping the risk–return plane

Feasible portfoliosevery mix you can build

Minimum-variance frontier Memorize lowest risk for each return
- Efficient frontier Memorize upper part, from the minimum-variance portfolio up
- Inefficient part Memorize below the minimum-variance portfolio; dominated
Minimum-variance portfolio Memorize leftmost point; return plays no part
Optimal risky (tangency) portfolio Memorize highest Sharpe ratio; needs a risk-free asset

Three ways to optimize

Optimization

Return-constrained Know the idea minimize risk for a target return
Risk-constrained Know the idea maximize return for a target risk
Risk-adjusted Know the idea maximize the Sharpe ratio

Formulas

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

Known

MemorizeUtility in one line

A	Attitude	Indifference curves
A > 0	risk-averse	upward, convex; steeper = more averse
A = 0	risk-neutral	flat; U = E[r]
A < 0	risk-seeking	concave

Memory hook U = E[r] − ½Aσ²: half, A, VARIANCE, all in decimals. The bill scores its own yield.

Know it cold Pick the highest U, even if all are negative. A is seldom below 1; studies put it between 2 and 3.

Key terms

assumptions of portfolio theory Know the idea
risk-seeking Memorize
risk-neutral Memorize
risk-averse Memorize
utility function Memorize
coefficient of risk aversion Memorize
mean–variance utility Know the idea
risk tolerance Know the idea
absolute risk aversion Know the idea
relative risk aversion Know the idea
varying risk aversion Know the idea
indifference curve Memorize
marginal rate of substitution Know the idea

Three attitudes to risk

Investor

Risk-averse Memorize A > 0; takes the sure thing; convex indifference curves
Risk-neutral Memorize A = 0; U = E[r]; flat curves
Risk-seeking Memorize A < 0; risk adds utility; concave curves

Ways to describe risk aversion

Risk aversion

Absolute Know the idea utility change per unit of risk, wealth fixed
Relative Know the idea share of wealth put at risk
Varying Know the idea changes as wealth changes

Formulas

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

Known

MemorizeFrom the capital allocation line to the CAPM

Line or measure	Formula
Capital allocation line	$E[r_p] = r_f + \frac{E[r_P] - r_f}{\sigma_P}\sigma_p$
Weight for a target return	$w_P = \frac{r_p - r_f}{E[r_P] - r_f}$ ; $\sigma_p = w_P\sigma_P$
Capital market line	$E[r_p] = r_f + \frac{E[r_m] - r_f}{\sigma_m}\sigma_p$
Beta	$\frac{\sigma_{i,m}}{\sigma_m^2} = \frac{\rho_{i,m}\sigma_i}{\sigma_m}$
Capital asset pricing model (security market line)	$E[r_i] = r_f + \beta_i(E[r_m] - r_f)$

Memory hook Capital allocation and capital market lines: x = σ. Security market line: x = β. σ_p ÷ σ_m is beta only on the capital market line.

Know it cold Slope of a capital allocation line = Sharpe ratio, constant along the line. Weight above 100% = borrowing. Only systematic risk is priced.

Key terms

risk-free asset Use it
capital allocation line Memorize
Sharpe ratio Memorize
lending and borrowing portfolios Use it
optimal investor portfolio Use it
market portfolio Know the idea
capital market line Memorize
mutual fund theorem Know the idea
beta Memorize
systematic risk Memorize
non-systematic risk Memorize
CAPM assumptions Know the idea
security market line Memorize
capital asset pricing model Memorize
market risk premium Memorize
beta estimation by regression Use it
alpha Use it
limitations of portfolio theory Know the idea

Three lines, three jobs

Risk–return lines

Capital allocation line Memorize risk-free + one risky portfolio; slope = its Sharpe ratio
Capital market line Memorize risk-free + market portfolio; efficient portfolios only
Security market line Memorize expected return vs beta; every asset

Which risk is paid

Total risk σ_i

Systematic (market) risk Memorize measured by beta; earns the market risk premium
Non-systematic (security-specific) risk Memorize diversifiable; earns nothing

Formulas

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.