Question

a) “The standard deviation of a portfolio's return cannot be reduced to zero by holding all the securities in the market.” True or false? Explain. (2 mark)

An investor buys 1 share of ABC Ltd at the price of $32 on December 1, 2019. The firm is not expected to pay any dividends. Consider the following three possible scenarios for the share price on December 1, 2020:

• $50 with a probability of 30%
• $35 with a probability of 60%
• $23 with a probability of 10%

b) Calculate the expected return for holding the share for a year. (2 mark)

c) Calculate the variance of return and standard deviation of return.

d) On December 1, 2020, the share is worth $36 and the investor just received a dividend of $2.50. Calculate the total holding period return and capital gains return over the one-year period.

e) Explain the difference between expected return and realised return.

Answer 1

a). Standard deviation measures total risk including both systematic and nonsystematic risk. Only the nonsystematic risk can be offset by the variation of other securities in a portfolio. The systematic (or nondiversifiable) risk always remains so even if all the securities in a market are held in a portfolio, standard deviation will decrease but will never be zero. Thus, the given statement is true.

b). Expected price = sum of [probability*share price]

= (30%*50) + (60%*35) + (10%*23) = 38.3

Expected return = (expected price/purchase price) -1 = (38.3/32) -1 = 19.69%

c). Return is calculated as (price after one year/purchase price) - 1

Variance of return = sum of [probability*(return - expected return)^2]

= 30%*((50/32)% - 19.69%)^2 + 60%*((35/32)% - 19.69%)^2 + 10%*((23/32)% - 19.69%)^2 = 0.0342

Standard deviation of return = variance^0.5 = (0.0342)^0.5 = 18.49%

d). Total holding period return (HPR) = ((price after one year + dividend)/purchase price) -1

= ((36+2.50)/32) -1 = 20.31%

Capital gains return = (price after one year/purchase price) -1 = (36/32) -1 = 12.50%

e). Realised return is the actual return which an asset gives over a period of time whereas expected return is the return expected on an asset over a period of time based on potential outcomes with given probabilities. Risk comes from the fact that realised return may not be the same as the expected return.