Question

Molly and Jack are trying to decide how to allocate their investment portfolio. The risk-free rate is 4%. An advisor has alerted them to 3 funds as follows.

Fund	E(r)	σ
Canadian Bond	6%	8%
U.S. Equity	18%	25%
Canadian Small Cap	15%	18%

a) Is there one of these funds that stands out as being superior to the others? Support your conclusion with numerical analysis.

b) Regardless of your answer in Part A, assume Molly and Jack both decide they are going to allocate their investment portfolio between the risk-free asset and the above U.S. Equity fund. Their advisor asks them to fill out a questionnaire which will arrive at a risk factor for each of them. It turns out Molly has a risk factor of 2, whereas Jack has a risk factor of 5. The advisor tells them he uses this factor to help them determine how to allocate their portfolio between the two funds. He attempts to find the highest value for U in the following equation:

? = ?(?) − ??2

Since A is the risk factor mentioned above, is Molly or Jack more willing to take additional risk?

c) Based on the above equation, what is the optimal allocation of funds (i.e. % of their portfolio) between the U.S. equity fund and the risk-free asset for each of Molly and Jack?

Answer 1

A. Comparing the three funds with respect to the Reward-to-Risk ratio, known as the Sharpe Ratio, we see that the Canadian Small Cap has the highest Sharpe Ratio. Hence, it can be considered as superior to other funds mentioned.

Sharpe Ratio = [ E(R) - Rf ] / σ(R)

Risk-Free Rate	4%
Fund	Expected Return	Standard Deviation	Sharpe Ratio
Canadian Bond	6%	8%	0.25
US Equity	18%	25%	0.56
Canadian Small Cap	15%	18%	0.61

B. The risk factor (A) represents the Risk Aversion of the investor. Higher the value of risk factor, the more risk-averse the investor is. For a given value of expected return, Molly (with a lower risk factor of 2) can take more risk in her investments than Jack (with a higher risk factor of 5).

C. At the optimal allocation level, the Capital Allocation Line (CAL) becomes tangential to the Utility Curve [? = ?(?) − ??2], i.e. Slope of CAL becomes equal to the slope of Utility Curve.

Slope of CAL = Sharpe Ratio of US Equity Fund = [ E(R) - Rf ] / σ(R) = 0.56

Slope of Utility Curve = d E(r) / d σ = 2*A*σ

Equating the two slopes, σ = 0.56/(2*A)

MOLLY:

σ = 0.56/4 = 0.14

σP = w*σUS Equity

0.14 = w*0.25

w = 0.56

i.e. Molly should invest 56% of her fund in US Equity fund, and remaining 44% in Risk free asset.

JACK:

σ = 0.56/10 = 0.056

σP = w*σUS Equity

0.056 = w*0.25

w = 0.224

i.e. Jack should invest 22.4% of his fund in US Equity fund, and remaining 77.6% in Risk free asset.