Question

Fund A has a sample mean of 0.13 and fund B has a sample mean of 0.18, with the riskier fund B having double the beta at 2.0 as fund A. The respective standard deviations for fund A and B are 15% and 19%. The mean return for your market index is 0.12 with a standard deviation of 8%, while the risk-free rate on the bond market is 8%. (a) Compute the Jensen Index for each of the funds. What does it indicate to you? (b) Compute the Treynor Index for the funds and the market. Interpret the results. (c) Compute the Sharpe Index for the funds and the market. (d) Which fund would you deem most appropriate to invest all of a clients wealth into?

Answer 1

a) Beta of Fund B = 2 , Risk free rate = 8%, Mean return of market = 0.12

Market risk premium = Mean return of market - risk free rate = 0.12 - 8% = 12% - 8% = 4%

According to CAPM, Required return of Fund B = Risk free rate + beta of fund B x market risk premium = 8% + 2 x 4% = 8% + 8% = 16%

Jensen index for fund B = Mean return of Fund B - Required return of Fund B according to CAPM = 0.18 - 16% = 18% - 16% = 2%

Beta of Fund A = Beta of Fund B / 2 = 2 / 2 = 1

According to CAPM , Required return of Fund A = Risk free rate + beta of fund A x market risk premium = 8% + 1 x 4% = 8% + 4% = 12%

Jensen index for fund A = Mean return of Fund A - Required return of Fund B according to CAPM = 0.13 - 12% = 13% - 12% = 1%

Jensen's Index tells us about the excess return a fund has generated in comparison to the return expected according to CAPM. It tell us the excess return of a a fund after taking into consideration the systematic risk of the fund. Higher value of Jensen Index is considered to be better because fund manager has outperformed the market by delivering better return in comparison to return that should be expected considering the systematic risk of fund. It can be used to compare risk adjusted performance of funds

Fund B has higher Jensen Index than Fund A, hence Fund B has delivered better risk adjusted return than Fund A

b) Treynor index for Fund A = (Mean return of Fund A - Risk free rate) / Beta of Fund A = (0.13 - 8%) / 1 = (13% - 8%) = 5%

Treynor index for Fund B = (Mean return of Fund B - Risk free rate) / Beta of Fund B = (0.18 - 8%) / 2 = (18% - 8%) / 2 = 10% / 2 = 5%

We know that Beta of Market = 1

Treynor index for Market = (Mean return of Market - Risk free rate) / Beta of Market = (0.12 - 8%) / 1 = 12% - 8% = 4%

Treynor index gives excess return of fund over risk free rate for per unit systematic risk undertaken. It gives us risk premium per unit systematic risk.Higher value of this index is considered to be better. It is risk adjusted measure for evaluating performance of funds after taking into consideration systematic risk. Here both funds have generated same excess return over risk free rate for per unit systematic risk taken. and both funds have performed better than the market on the basis of treynor index.

c)

Sharpe index for Fund A = (Mean return of Fund A - Risk free rate) / Standard deviation of Fund A = (0.13 - 8%) / 15% = (13% - 8%) / 15% = 5% / 15% = 0.3333

Sharpe index for Fund B = (Mean return of Fund B - Risk free rate) / Standard deviation of Fund B = (0.18 - 8%) / 19% = (18% - 8%) / 19% = 10% / 19% = 0.5263

Sharpe index for Market = (Mean return of Market - Risk free rate) / Standard deviation of Market = (0.12 - 8%) / 8% = (12% - 8%) / 8% = 4% / 8% = 0.50

d) We should invest invest all clients wealth in Fund B because

i) Fund B has higher Jensen Index than Fund A, hence has delivered higher excess return than the return according to CAPM in comparison to Fund A. Fund B has delivered better performance than Fund A after adjusting for the systematic risk of funds.

ii) Fund B has higher Sharpe index than Fund A. Sharpe index gives the excess return of fund over risk free rate for per unit total risk undertaken. It gives us risk premium per unit total risk. Higher sharpe index is considered to better. Hence fund B has higher risk premium than Fund A after taking into consideration their total risks. Hence Fund B gives better risk adjusted performance.