Consider the following information for a mutual fund, the market index, and the
ID: 2651097 • Letter: C
Question
Consider the following information for a mutual fund, the market index, and the risk-free rate. You also know that the return correlation between the fund and the market is .97.
Year Fund Market Risk-Free
2008 –21.20 % –40.5 % 2 %
2009 25.1 21.1 4
2010 14.0 14.2 2
2011 6.2 8.8 4
2012 –2.16 –5.2 3
What are the Sharpe and Treynor ratios for the fund? (Do not round intermediate calculations. Round your answers to 4 decimal places.)
Sharpe ratio
Treynor ratio
Explanation / Answer
WN. (‘1)- Calculation of average rate of return
Years
Fund
Market
Risk Free
2008
-21.2
-40.5
2
2009
25.10
21.1
4
2010
14.00
14.2
2
2011
6.20
8.8
4
2012
-2.16
-5.2
3
Average Rate of Return ( Sum of returns/ 5)
4.39
-0.32
3
WN (‘2) Calculation of beta of Fund
Let’s denote fund return is x and market return is y
Fund (x)
Market (y)
x*y
Y2
-21.2
-40.5
858.60
1640.25
25.10
21.1
529.61
445.21
14.00
14.2
198.80
201.64
6.20
8.8
54.56
77.44
-2.16
-5.2
11.23
27.04
21.94
-1.6
1652.8
2391.58
Beta of fund= [Sum (xy)- n x (xy)]/ Sum (y2) – n x Y2
Beta= (1652.8 – 5 x 4.39 x -0.32 )/ (2391.58- 5 x -0.32 x -0.32)
Beta of fund= 0.694
It is to be noted that beta of market is always 1.
WN(‘3)- Standard Deviation of Fund
D= yearly return- average return
Fund (x)
D
D2
P
P x D2
-21.2
-25.59
654.85
0.2
130.97
25.10
20.71
428.9
0.2
85.78
14.00
9.61
92.35
0.2
18.47
6.20
1.81
3.28
0.2
0.66
-2.16
-6.55
42.9
0.2
8.58
244.46
Standard Deviation = (244.46)0.5 = 15.64
WN(‘3)- Standard Deviation of Market
Market
D
D2
P
P x D2
-40.5
-40.18
1614.43
0.2
322.89
21.1
21.42
458.82
0.2
91.76
14.2
14.52
210.83
0.2
42.17
8.8
9.12
83.17
0.2
16.63
-5.2
-4.88
23.81
0.2
4.76
478.21
Standard deviation of market= (478.21)0.5 = 21.87
Sharpe Ratio=( Return – Risk Free return)/ Standard Deviation
Hence Sharpe for Mutual Fund= (21.94-3)/ 15.64
Sharpe ( Mutual Fund)= 1.21
Sharpe ( Market)= (-1.6- 3)/21.87
Sharpe (Market)= -0.21
Treynor Ratio= (Return- Risk Free Return)/ Beta
Treynor ( Fund)= (21.94-3)/0.694= 27.29
Treynor ( Market)= (-1.6-3)/1= -4.6
Years
Fund
Market
Risk Free
2008
-21.2
-40.5
2
2009
25.10
21.1
4
2010
14.00
14.2
2
2011
6.20
8.8
4
2012
-2.16
-5.2
3
Average Rate of Return ( Sum of returns/ 5)
4.39
-0.32
3
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