Question

The following table shows the annualized return of S&P total return index from 1999 to 2003:

Year Total Return (%)

1999 21.0

2000 -9.1

2001 -11.9

2002 -22.1

2003 28.7

(a) Calculate the geometric return from 1999 to 2003.

(b) Calculate the mean absolute deviation and the sample standard deviation of the index

return.

(c) Calculate the 45th-percentile of the index return.

(d) Sketch the drawdown curve. What is the maximum drawdown of the index return over this

period?

(e) Assume an annualized risk-free rate of 5%, what is the Sharpe ratio of the index return over

this period?

Answer 1

(a) Geometric return is calculated by solving the following equation:

$Return_{geo} =\sqrt[n]{(1+r_{1}) \times (1+r_{2}) \times (1+r_{3}) \times (1+r_{4}) \times (1+r_{5}) } - 1$

$Return_{geo} =\sqrt[5]{(1+0.21) \times (1-0.091) \times (1-0.119) \times (1-0.221) \times (1+0.287) } - 1$

$Return_{geo} =-0.57669 \%$

(b) Mean absolute deviation is calculated below:

Year	Returns		MAD
1999	21.0%	abs(0.21-0.0132)=	19.68%
2000	-9.1%	abs(-0.091-0.0132)=	10.42%
2001	-11.9%	abs(-0.119-0.0132)=	13.22%
2002	-22.1%	abs(-0.221-0.0132)=	23.42%
2003	28.7%	abs(0.287-0.0132)=	27.38%
Sum	6.6%	Sum	94.1%
Count	5	Count	5
Mean	6.6/5= 1.32%	MAD	94.1/5 = 18.82%

Sample standard deviation is calculated below:

Year	Returns	Squared deviations
1999	21.0%	(0.21-0.0132)^2=	3.87%
2000	-9.1%	(-0.091-0.0132)^2=	1.09%
2001	-11.9%	(-0.119-0.0132)^2=	1.75%
2002	-22.1%	(-0.221-0.0132)^2=	5.48%
2003	28.7%	(0.287-0.0132)^2=	7.50%
Sum	6.6%	Sum	19.7%
Count	5	Count	4
Mean	0.0132	Variance	19.7/4 = 4.92%
Sample standard deviation			4.92^0.5 = 22.19%

• As we need to find the sample standard deviation, we need to divide the sum of the squared deviations by n-1 = 5-1 = 4 to find the variance
• After calculating the variance, we need to take a square-root of it to find the sample standard deviation

(c)

• To calculate the 45th percentile we first need to reorder the returns in ascending order as follows:

Year	Returns
2002	-22.1%
2001	-11.9%
2000	-9.1%
1999	21.0%
2003	28.7%

• Next we multiply the required percentile with the number of observations: 0.45 x 5 = 2.25
• Next we round off 2.25 to the next highest whole number ie 3
• The third observation is the 45th percentile , which is 9.1%

(d) Drawdown is calculated below:

Year	Returns	Drawdown
1999	21.0%	0.731707317
2000	-9.1%	-0.317073171
2001	-11.9%	-0.414634146
2002	-22.1%	-0.770034843
2003	28.7%	1

• Maximum return is 28.7%
• We divide each return with this to find the draw down of the year or what percentage is it of the maximum return
• Then we plot the same as follows:
• 
  Drawdown 20b0 2001 20b2 2003 -0.21999 -0.4 -0.5
  

We can see the lowest fall is in 2002 when the return goes down to -22.1% and the drawdown is -0.770034843 ie the maximum drawdown

(e)

$Sharpe \ ratio = \frac{Return_{geo} - Return_{riskfree}}{standard \ deviation}$

$Sharpe \ ratio = \frac{-0.57669\% - 5\%}{4.92 \% } = -1.133007667$