Define shortfall risk, calculate the safety-first ratio, and select an optimal portfolio using Roy's safety-first criterion.
Roy's safety-first criterion is an application of the normal distribution. If returns are normally distributed, the optimal portfolio maximizes the safety-first ratio (denoted by SFRatio), calculated as
SFRatio = (mean return – shortfall level)/standard deviation of returns
= [E(RP) – RL]/σP.
When there are several portfolios to choose from and the returns are normally distributed, we calculate each portfolio's SFRatio and choose the one with the highest ratio.
Explain the relationship between normal and lognormal distributions, and why the lognormal distribution is used to model asset prices.
Distinguish between discretely and continuously compounded rates of return. Calculate and interpret a continuously compounded rate of return, given a specific holding period return.
Explain the Monte Carlo simulation and historical simulation. Describe their major applications and limitations.
The Monte Carlo simulation tries to represent the operation of a complex financial system. Using a computer, we simulate the generation of a large number of random samples from specified probability distributions. We thereby represent the operation of risk. We use the Monte Carlo simulation in
financial risk management, and
valuing complex securities.
The Monte Carlo simulation only provides statistical estimates, not exact results.
An alternative to the Monte Carlo simulation is the historical simulation, in which we use repeated sampling from historical data-series. This method is based on actual data, while the Monte Carlo simulation is based on statistical data. Historical simulation can only reflect risks that are represented in the sample historical data. Unlike the Monte Carlo simulation, the historical simulation does not reflect “what-if“ analyses.