qm 5: portfolio mathematics

formula for expected return on a portfolio

$$E\left(R_{p}\right)=\sum_{i}^{}w_{i}E\left(R_{i}\right)$$

sum of weighted expected returns

portfolio variance

$$\sigma^2(R_{p})=E[(R_{p}-E(R_{p}))^2]$$

portfolio covariance

$$\sigma^2\left(R_{p}\right)=\sum_{i}^{}\sum_{j}^{}w_{j}w_{j}COV\left(R_{i},R_{j}\right)$$

what do positive, negative, and zero covariance mean?

positive: returns on both assets are on same side of expected values

negative: return on one asset above expected value, return on other above expected value

zero: assets are unrelated

correlation formula

$$\rho\left(R_{i,}R_{j}\right)=\frac{COV\left(R_{i,}R_{j}\right)}{\sigma\left(R_{i}\right)\sigma\left(R_{j}\right)}$$

often rearranged for covarainc

correlation interpretation

close to one: strong positive linear relationship (diversification benefits decrease)

close to -1 : strong negative linear relationship

close to 0: weak linear relationship

independence formulas for probability and expectation

$$P\left(X,Y\right)=P\left(X\right)P\left(Y\right)$$

$$E\left(XY\right)=E\left(X\right)E\left(Y\right)$$

what is mean variance analysis

portfolio analysis using expected means, variances, and covariances of asset returns

when does mean variance analysis hold?

holds when investors are risk-averse and either:

1. returns are normally distributed → fully described by mean and variance.

2. investors have quadratic utility functions → a mathematical form of utility where only mean and variance matter.

what is shortfall risk?

the risk that portfolio value or portfolio return will fall below some minimum acceptable level over some time horizon

what are safety-first rules

rules for portfolio selection that focus shortfall risk

what is roy’s safety first criterion

the optimal portfolio minimises P(R_p < R_L)

$R_{p}$ = portfolio return

$R_L$ = minimum acceptable elvel

safety first ratio formula

$$SF=\frac{E\left(R_{p})-R_{L}\right.}{\sigma_{p}}$$

how to choose safety first optimal portfolio?

1. calc each SF ratio

2. choose highest

Sharpe Ratio

subsitiute $R_L$ with $R_F$ the risk free rate

i.e minimise probability returns are less than risk free rate

How do you calculate the probability of a shortfall for a normally distributed portfolio?

P\left(R_{P}<R_{L}\right.) = Normal(–SFRatio) = 1 − N(SF Ratio)

using the standard normal cumulative distribution function.