QM LM5 Portfolio mathematics

Question 1

How do you calculate the expected value of a portfolio?

Answer 1

• Add up the expected value of each asset multiplied by their weighting (proportion of the portfolio)

Question 2

How do we calculate portfolio variance, and what information do we need?

Answer 2

We need:
- The expected returns of each asset
- The covariances between them

• We make a variance covariance matrix. The diagonal will be variances, the others will be covariances. Above the diagonal and below are the same

When the assets have zero correlation the portfolio variance will just be the weighted sum of the variances.

Question 3

What is covariance?

Answer 3

Covariance is the sum of the deviations from expected value of one asset class, multiplied my the same for the second asset class, all over (n - 1)

Question 4

What is variance with respect to variance?

Answer 4

The covariance of an asset with itself

Question 5

How can you reduce porfolio risk?

Answer 5

• Selecting assets with zero or negative covariance
• However, if you had a portfolio with perfect negative correlation it would be zero risk and therefore only earn the risk free rate. You must take on risk to earn returns.
• This is because negative covariance means when one asset goes up the other goes down (so they would cancel out)
• Therefore ZERO correlation is what you’re looking for

Question 6

How do you weight probabilities to find an expected outcome if the sub-outcomes are not independent?

Answer 6

You can calculate joint probability
Probability of A and B occuring = probability of A x probability of B
This gives you the requisite weighting

Question 7

What are safety first rules?

Answer 7

Rules for portfolio management that focus on shortfall risk - the risk a portfolio value (or return) will fall below some minimum acceptable level over some time horizon.

Question 8

What is the safety first ratio?

Answer 8

[E(RsubP) - RsubL] / SigmaP

Where,
- E(RsubP) is the expected return of a given portfolio composition
- RsubL is the minimum level of return required
- SigmaP is the standard deviation of a given portfolio composition

We attempt to maximise this ratio (larger denominator) to find the safest portfolio

Question 9

Why can’t you use decimals for the covariance matrix?

Answer 9

You can - you just have to remember that the numbers you get are percent squared, meaning that you have to divide by 100^2, or 10 000, to get the decimal number