Portfolio Maths

Topics

Introduction
Portfolio Expected Return and Variance of Return
- Covariance
- Correlation
Forecasting Correlation of Returns: Covariance Given a Joint Probability Function
Portfolio Risk Measures: Applications of the Normal Distribution
References

Table of Contents

Introduction

Calculating the expected return and variance of a portfolio
Calculating covariance and correlation of portfolio returns using a joint probability function
Portfolio risk measures: Roy’s safety-first ratio

Portfolio Expected Return and Variance of Return

Expected Return
A portfolio’s expected return can be calculated as:

E (R_{P}) = \sum_{i = 1}^{n} w_{i} E (R_{i})

where:

$w_{n}$ is portfolio weight of nth security
$R_{n}$ is expected return of nth security
n is number of securities

Example

0.4 in Stock A and 0.6 in Stock B.

| | P(Scenario) | Returns of A | Returns of B |
| --------- | ----------- | ------------ | ------------ |
| Recession | 0.25 | 2% | 4% |
| Normal | 0.5 | 8% | 10% |
| Boom | 0.25 | 12% | 16% |
A → 7.5 and B → 10. Therefore, portfolio at 9.

Covariance

Covariance tells us how movements in a random variable vary with movements in another random variable, whereas variance tells us how a random variable varies with itself.

Assume there are two random variables X and Y. The covariance between X and Y (used to measure how they move together) is given by:

C o v (X, Y) = E ((X - E X) (Y - E Y))

where:

EX is expected return for variable X
EY is expected return for variable Y

Example

The covariance of returns

0.25 (2 - 7.5) (4 - 10) + 0.5 (8 - 7.5) (10 - 10) + 0.25 (12 - 7.5) (16 - 10)

is 0.0015.

Negative if, when the return on one asset is above its expected value, the return on the other asset tends to be below its expected value.
0 if the returns on the assets are unrelated.
Positive when the returns on both assets tend to be on the same side (above or below) their expected values at the same time.

Correlation

The problem with covariance is that it can vary from $\pm \infty$ which makes it difficult to interpret. To address this problem, we use another measure called correlation.

Correlation is a standardized measure of the linear relationship between two variables with values ranging between $\pm 1$ .

0 (uncorrelated variables) indicates an absence of any linear (straight-line) relationship between the variables.
+1 indicates a perfect positive relationship.
-1 indicates a perfect negative relationship.

It is computed as:

ρ (X, Y) = \frac{C o v (X, Y)}{σ (X) σ (Y)}

Example

Covariance of returns is 0.0015. Standard deviation of A is 0.0357 and the standard deviation of B is 0.0424. The correlation is 0.99.

Variance of Returns

Once we know the covariance, we can calculate the variance of a portfolio using this formula:

σ^{2} (R_{P}) = w_{1}^{2} σ_{1}^{2} (R_{1}) + w_{2}^{2} σ_{2}^{2} (R_{2}) + 2 w_{1} w_{2} Cov (R_{1}, R_{2})

Example

Variance of the portfolio:

{0.4}^{2} ({0.0357}^{2}) + {0.6}^{2} ({0.0424}^{2}) + 2 (0.4) (0.6) (0.0015) = 0.00157

For a 3 asset portfolio:

σ^{2} (R_{P}) = w_{1}^{2} σ_{1}^{2} (R_{1}) + w_{2}^{2} σ_{2}^{2} (R_{2}) + w_{3}^{2} σ_{3}^{2} (R_{3}) + 2 w_{1} w_{2} Cov (R_{1}, R_{2}) + 2 w_{2} w_{3} Cov (R_{2}, R_{3}) + 2 w_{1} w_{3} Cov (R_{1}, R_{3})

Variance Covariance Matrix

A variance covariance matrix is defined as a square matrix where the diagonal elements represent the variance and off-diagonal elements represent the covariance.

	Asset 1	Asset 2	Asset 3
Weight	0.2	0.3	0.5
Return	5	6	7

	Asset 1	Asset 2	Asset 3
Asset 1	196	105	140
Asset 2	105	225	150
Asset 3	140	150	400
In the above matrix, the variance of Asset 1, Asset 2 and Asset 3 are 196, 225 and 400 respectively. The covariance between Asset 1 and Asset 2 is 105. The covariance between Asset 1 and Asset 3 is 140. The covariance between Asset 2 and Asset 3 is 150.

Tip

Expected Return: 6.3%
Portfolio Var: 213.69%
Standard Deviation: 14.62%

Forecasting Correlation of Returns: Covariance Given a Joint Probability Function

The same information we saw in section 2 can also be presented in the form of a joint probability function.

Example

0.4 in Stock A and 0.6 in Stock B.

| | P(Scenario) | Returns of A | Returns of B |
| --------- | ----------- | ------------ | ------------ |
| Recession | 0.25 | 2% | 4% |
| Normal | 0.5 | 8% | 10% |
| Boom | 0.25 | 12% | 16% |

	R = 4	R = 10	R = 16
R = 2	0.25
R = 8		0.5
R = 12			0.25
Row 1 and Column 1 represent the returns of A and B respectively. The other cells contain probabilities.

Expected return of A is: 0.25 x 2 + 0.50 x 8 + 0.25 x 12 = 7.5%.
Expected return of B is: 0.25 x 4 + 0.50 x 10 + 0.25 x 16 = 10%.
Covariance of returns = 0.25(2% – 7.5%) (4% – 10%) + 0.5(8% – 7.5%) (10% – 10%) + 0.25(12% – 7.5%) (16% – 10%) = 0.000825 + 0 + 0.000675 = 0.0015

Portfolio Risk Measures: Applications of The Normal Distribution

Shortfall risk
Risk that portfolio’s return will fall below a specified minimum level of return over a given period of time.

Safety first ratio
Used to measure shortfall risk. It is calculated as:

SF Ratio = \frac{R_{P} - R_{L}}{σ_{P}} $ $ w h e r e : - $ R_{P} $ i s e x p e c t e d p o r t f o l i o r e t u r n - $ R_{L} $ i s t h r e s h o l d l e v e l - $ σ_{P} $ i s s t a n d a r d d e v i a t i o n o f p o r t f o l i o r e t u r n s > [! T i p] > T h e p o r t f o l i o w i t h t h e h i g h e s t S F - R a t i o i s p r e f e r r e d, a s i t h a s t h e l o w e s t p r o b a b i l i t y o f f a l l i n g b e l o w t h e t a r g e t r e t u r n . * * R o y^{'} s s a f e t y - f i r s t c r i t e r i a * * I t s t a t e s t h a t a n o p t i m a l p o r t f o l i o m i n i m i z e s t h e p r o b a b i l i t y t h a t t h e a c t u a l p o r t f o l i o r e t u r n w i l l f a l l b e l o w t h e t a r g e t r e t u r n . > [! E x a m p l e] > A n i n v e s t o r i s c o n s i d e r i n g t w o p o r t f o l i o s A a n d B . P o r t f o l i o A h a s a n e x p e c t e d r e t u r n o f 10 * * S o l u t i o n : * * $ $ S F_{A} = \frac{10 - 8}{2} = 1 $ $ $ $ S F_{B} = \frac{15 - 8}{10} = 0.7 $ $ S i n c e A h a s a h i g h e r s a f e t y - f i r s t r a t i o, t h e i n v e s t o r s h o u l d s e l e c t p o r t f o l i o A . > [! T i p] I f t h e r i s k - f r e e r a t e i s s e t a s t h e t h r e s h o l d l e v e l R L, t h e s a f e t y - f i r s t r a t i o b e c o m e s t h e S h a r p e r a t i o . > $ $ Sharpe Ratio = \frac{R_{P} - R_{f}}{σ_{P}}