Applications of the Normal Distribution

Safety-first rules focus on shortfall risk, the risk that portfolio value will fall below some minimum acceptable level over some time horizon.

safety-first ratio(SFRatio) = [E(RP) – RL]/σ_P

Value at Risk (VaR) is a money measure of the minimum value of losses expected over a specified time period at a given level of probability.

eg. 95% one-day VaR is $5 million

[Practice Problems] A client holding a £2,000,000 portfolio wants to withdraw £90,000 in one year without invading the principal. According to Roy’s safety-first criterion, which of the following portfolio allocations is optimal?

A.  Allocation A

B.  Allocation B

C.  Allocation C

[Solutions] B

Allocation B has the highest SF ratio. The threshold return level RL is £90,000/£2,000,000 = 4.5%. To compute the allocation that is safety-first optimal, select the alternative with the highest ratio:

Allocation A = (6.5 – 4.5) / 8.35 = 0.240

Allocation B = (7.5 – 4.5) / 10.21 = 0.294

Allocation C = (8.5 – 4.5) / 14.34 = 0.279