You are reading the 2026 curriculum (exams up to November 2026). Sitting in February 2027 or later? Go to the 2027 Quantitative Methods guide.

Learning Module 5 of Quantitative Methods 6–9% of the exam

Portfolio Mathematics

3 lessons. About 20 minutes in cram mode, 110 minutes in full.

Start with lesson 1 Learn with Sensei Read the module summary

Lessons

1
Portfolio expected return, variance and the role of correlation
calculate 8 min cram, 40 min full
2
Covariance and correlation from a joint probability table
calculate 6 min cram, 30 min full
3
Shortfall risk and Roy's safety-first ratio
calculate 6 min cram, 40 min full

What the exam expects you to do

Calculate and interpret the expected value, variance, standard deviation, covariances, and correlations of portfolio returns.Portfolio expected return, variance and the role of correlation
Calculate and interpret the covariance and correlation of portfolio returns using a joint probability function for returns.Covariance and correlation from a joint probability table
Define shortfall risk, calculate the safety-first ratio, and identify an optimal portfolio using Roy’s safety-first criterion.Shortfall risk and Roy's safety-first ratio

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 expected return, variance and the role of correlation

Known

MemorizeTwo-asset portfolio in four lines

Quantity	Formula
Expected return	$w_1 E(R_1) + w_2 E(R_2)$
Variance	$w_1^2\sigma_1^2 + w_2^2\sigma_2^2 + 2w_1w_2\rho\sigma_1\sigma_2$
Covariance	$\rho\,\sigma_1\sigma_2$
Correlation	$\mathrm{Cov} \div (\sigma_1\sigma_2)$

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

Know it cold Return averages; risk does not. Cov(R, R) = σ². Weights sum to 1. With n assets: n variances and n(n − 1)/2 distinct covariances.

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 σₚ and leaves E(Rₚ) unchanged. Compare pairs by correlation, never by covariance alone.

Key terms

expected return on the portfolio Memorize
covariance Memorize
correlation Memorize
portfolio variance Memorize
covariance matrix 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 2wᵢwⱼCov 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 σₚ is the weighted average of the two σs; no diversification benefit
ρ between −1 and +1 Memorize σₚ below the weighted average; the lower ρ, the lower σₚ
ρ = −1 Use it σₚ = |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 0.5, 0.3 and 0.2 in assets expected to return 10%, 6% and 3%. 0.5 × 10 + 0.3 × 6 + 0.2 × 3 = 7.4%.

On your calculator

0.5×10+(0.3×6)+(0.2×3)= 7.4

0.5ENTER10×0.3ENTER6×+0.2ENTER3×+ 7.4

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

Covariance of two returns Use it

\mathrm{Cov}(R_i, R_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. Cov = 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

\mathrm{Cov}(R_1, R_2) = \rho_{12}\, \sigma_1\, \sigma_2

Quick example

Correlation 0.4, standard deviations 20% and 10%. 0.4 × 0.20 × 0.10 = 0.008.

On your calculator

0.4×0.2×0.1= 0.008

0.4ENTER0.2×0.1× 0.008

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

Correlation Memorize

\rho_{12} = \frac{\mathrm{Cov}(R_1, R_2)}{\sigma_1\, \sigma_2}

Quick example

Covariance 0.0108, standard deviations 24% and 15%. 0.0108 ÷ (0.24 × 0.15) = 0.30.

On your calculator

0.0108÷(0.24×0.15)= 0.3

0.0108ENTER0.24ENTER0.15×÷ 0.3

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_{12}\, \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\, \mathrm{Cov}(R_i, R_j)

Quick example

Weights 0.5, 0.3, 0.2. Variances (%²) 400, 100, 225; covariances 60 (1,2), 90 (1,3), 30 (2,3). Own terms 100 + 9 + 9 = 118; cross terms 2 × (9 + 9 + 1.8) = 39.6. Variance 157.6 %², so σₚ = 12.55%.

On your calculator

0.5x²×400+(0.3x²×100)+(0.2x²×225)+(2×0.5×0.3×60)+(2×0.5×0.2×90)+(2×0.3×0.2×30)= 157.6
√x 12.55

x² square · √x square root

0.5ENTER×400×0.3ENTER×100×+0.2ENTER×225×+2ENTER0.5×0.3×60×+2ENTER0.5×0.2×90×+2ENTER0.3×0.2×30×+ 157.6
g√x 12.55

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(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)

Covariance and correlation from a joint probability table

Known

MemorizeReading the signs of the products

R_A vs mean	R_B vs mean	Product	Pushes covariance
above	above	+	up
below	below	+	up
above	below	−	down
below	above	−	down

Memory hook Each scenario casts a vote: same side of the means is a plus, opposite sides a minus, and the probability is the size of the vote.

Know it cold $\mathrm{Cov} = \sum P \times \text{dev}_A \times \text{dev}_B$ , or $E(R_AR_B) - E(R_A)E(R_B)$ . Independent: $P(X,Y) = P(X)P(Y)$ and Cov = 0.

Key terms

joint probability function Use it
scenario analysis Know the idea
independent random variables Use it
uncorrelated returns Use it

From a scenario table to correlation

Joint table: probability, R_A and R_B in each stateprobabilities add to 1; no n − 1

1. Expected returns Memorize E(R_A) = Σ P × R_A and E(R_B) = Σ P × R_B
2. Covariance Memorize Σ P × (R_A − E_A) × (R_B − E_B)
- Shortcut E(R_A R_B) − E(R_A) E(R_B)
3. Standard deviations Use it √ of Σ P × (R − E)²
4. Correlation Memorize Cov ÷ (σ_A σ_B), between −1 and +1

Independent, uncorrelated and dependent

Two random returnscovariance only sees straight-line links

Independent Use it P(X, Y) = P(X) P(Y); knowing one tells you nothing about the other
- Implies covariance 0, correlation 0, E(XY) = E(X)E(Y)
Uncorrelated Use it covariance 0 but a nonlinear link may remain
Positive covariance Memorize returns tend to sit on the same side of their means
Negative covariance Memorize one tends to be above while the other is below

Formulas

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 from a joint probability table Memorize

\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

Covariance shortcut Use it

\mathrm{Cov}(R_A, R_B) = E(R_A R_B) - E(R_A)\,E(R_B)

Quick example

Same two states. 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

Correlation from scenario returns Memorize

\rho_{AB} = \frac{\mathrm{Cov}(R_A, R_B)}{\sigma_A\,\sigma_B}

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)

Independence Use it

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

Quick example

P(X = 1) = 0.40 and P(Y = 1) = 0.25, with X and Y independent. P(X = 1 and Y = 1) = 0.40 × 0.25 = 0.10.

On your calculator

0.4×0.25= 0.1

0.4ENTER0.25× 0.1

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)

Shortfall risk and Roy's safety-first ratio

Known

MemorizeRoy's safety-first in three lines

Step	Rule
Threshold	$R_L$ = payout ÷ portfolio value
Ratio	$\mathrm{SFRatio} = [E(R_p) - R_L] \div \sigma_p$
Choice	highest ratio = lowest $P(R_p < R_L)$
Probability	$N(-\mathrm{SFRatio})$

Memory hook The ratio counts how many standard deviations of cushion sit between the expected return and the threshold. More cushion, smaller left tail.

Know it cold Replace $R_L$ with the risk-free rate and the ratio becomes the Sharpe ratio. When the question gives a threshold, use it, not the risk-free rate.

Key terms

shortfall risk Memorize
threshold return Memorize
Roy's safety-first criterion Memorize
safety-first ratio Memorize
value at risk Know the idea
stress testing Know the idea

Choosing a portfolio with Roy's criterion

Roy's safety-first criterionminimize P(return below R_L)

1. Set the threshold R_L Use it payout ÷ portfolio value, or the minimum return you can accept
2. Compute each SFRatio Memorize (E − R_L) ÷ σ, one per portfolio
3. Choose the highest ratio Memorize with normal returns it has the lowest shortfall probability
4. Shortfall probability Memorize N(−SFRatio), read from the z-table

Ways to measure the risk of a bad outcome

Portfolio risk measuresstandard deviation treats gains and losses alike; these look at the bad side

Standard deviation Memorize spread both above and below the mean
Shortfall risk Memorize chance of finishing below a threshold return R_L
- Safety-first ratio standard deviations between the mean and R_L
Value at risk Know the idea money loss exceeded only with a small stated probability
Stress testing Know the idea loss under a specified extreme event

Formulas

Threshold return Use it

R_L = \frac{\text{money that must be earned}}{\text{portfolio value}}

Quick example

A 4,000,000 portfolio must fund a 160,000 payout without touching capital. 160,000 ÷ 4,000,000 = 4.0%.

On your calculator

160000÷4000000= 0.04

160000ENTER4000000÷ 0.04

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

Roy's safety-first ratio Memorize

\mathrm{SFRatio} = \frac{E(R_p) - R_L}{\sigma_p}

Quick example

Expected return 9%, threshold 3%, standard deviation 15%. (9 − 3) ÷ 15 = 0.40.

On your calculator

9−3=÷15= 0.4

9ENTER3−15÷ 0.4

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

Shortfall probability (normal returns) Memorize

P(R_p < R_L) = N(-\mathrm{SFRatio})

Quick example

Ratio 0.40. N(−0.40) = 1 − N(0.40) = 1 − 0.6554 = 34.5% chance of finishing below the threshold.

On your calculator

9−3=÷15= 0.4
+|- -0.4, then read N(-0.40) in the z-table: 0.3446

+|- change sign

9ENTER3−15÷ 0.4
CHS -0.4, then read N(-0.40) in the z-table: 0.3446

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

Cheat sheet · PDF · 4 A4 pages

Portfolio Mathematics on 4 pages

What to memorize, the concept maps, every formula with a worked example and keystrokes for the TI BA II Plus and HP 12C, how each idea is asked, and the traps. Print it, or keep it on your phone.

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