Question

Given the performance of 3 mutual funds and S&P500 over the past 15 years in table below:

Return and Risk data for 5 equity mutual funds, 15 year period

Mutual Fund	Average Return %	Standard Deviation %	Beta	R 2
1	15.86	22.85	1.46	.64
2	22.09	17.27	1.24	.79
3	18.39	11.82	0.60	.39
S&P 500	16.35	12.44	1.0	1.0

And assuming that expected market return for next year is 16.35 % and current and average of past 15 years risk free rate is 7.96 %, and using a market risk premium of 8.39% (16.35 -7.96) for the 15 year period, estimate:

a. Sharpe ratios of all 3 funds and S&P 500. Which fund has the highest risk adjusted performance according to Sharpe measure? Which of the above funds have beaten the market?
b. Treynor of all 3 funds and S&P 500. Which fund has the highest risk adjusted performance according to Treynor measure? Which of the above funds have beaten the market?
c. Jensen’s alpha for fund 1
d. Which fund is exposed to most nonsystematic risk?

Answer 1

Sharpe ratio of Mutual fund 1 = (Return of Fund 1 - Risk free rate) / Standard Deviation  of Fund 1

Sharpe ratio of Mutual fund 1 = (15.86% - 7.96%) / 12.85%

Sharpe ratio of Mutual fund 1 = 0.6148

Sharpe ratio of Mutual fund 2 = (Return of Fund 2 - Risk free rate) / Standard Deviation of Fund 2

Sharpe ratio of Mutual fund 2 = (22.09% - 7.96%) / 15.27%

Sharpe ratio of Mutual fund 2 = 0.9253

Sharpe ratio of Mutual fund 3 = (Return of Fund 3 - Risk free rate) / Standard Deviation of Fund 3

Sharpe ratio of Mutual fund 3 = (18.39% - 7.96%) / 14.82%

Sharpe ratio of Mutual fund 3 = 0.7038

Sharpe ratio of S&P 500 = (Return of S&P 500 - Risk free rate) / Standard Deviation of S&P 500  

Sharpe ratio of S&P 500 = (16.35% - 7.96%) / 12.44%

Sharpe ratio of S&P 500 = 0.6744

Fund 2 & 3 have beaten the market based on Sharpe ratio

Treynor ratio of Mutual fund 1 = (Return of Fund 1 - Risk free rate) / Beta of Fund 1

Treynor ratio of Mutual fund 1 = (15.86% - 7.96%) / 1.10

Treynor ratio of Mutual fund 1 = 0.0718

Treynor ratio of Mutual fund 2 = (Return of Fund 2 - Risk free rate) / Beta of Fund 2

Treynor ratio of Mutual fund 2 = (22.09% - 7.96%) / 1.42

Treynor ratio of Mutual fund 2 = 0.0995

Treynor ratio of Mutual fund 3 = (Return of Fund 3 - Risk free rate) / Beta of Fund 3

Treynor ratio of Mutual fund 3 = (18.39% - 7.96%) / 0.65

Treynor ratio of Mutual fund 3 = 0.1605

Treynor ratio of S&P 500 = (Return of S&P 500 - Risk free rate) / Beta of S&P 500

Treynor ratio of S&P 500 = (16.35% - 7.96%) / 1

Treynor ratio of S&P 500 = 0.0839

Fund 2 & 3 have beaten the market based on Treynor ratio

Jensen Alpha for fund 1 = Return of fund 1 - (Risk free rate + Beta * Market risk premium)

Jensen Alpha for fund 1 = 15.86% - (7.96% + 1.1 * 8.39%)

Jensen Alpha for fund 1 = -1.329% or -1.33%

Non systematic risk for Fund 1 = (Standard Deviation  of Fund 1)² - (Beta of Fund 1 * Standard Deviation  of S&P 500)²

Non systematic risk for Fund 1 = (12.85%)² - (1.1 * 12.44%)²

Non systematic risk for Fund 1 = -0.002213

i = -1

Non systematic risk for Fund 1 = 4.70i%

Non systematic risk for Fund 2 = (Standard Deviation  of Fund 2)² - (Beta of Fund 2 * Standard Deviation  of S&P 500)²

Non systematic risk for Fund 2 = (15.27%)² - (1.42 * 12.44%)²

Non systematic risk for Fund 2 = -0.007887

i = -1

Non systematic risk for Fund 2 = 8.88i%

Non systematic risk for Fund 3 = (Standard Deviation  of Fund 1)² - (Beta of Fund 1 * Standard Deviation  of S&P 500)²

Non systematic risk for Fund 3 = (14.82%)² - (0.65 * 12.44%)²

Non systematic risk for Fund 3 = 0.0154249

Non systematic risk for Fund 3 = 12.42%

Fund 3 has the most amount of Non systematic risk