Assume that Rm is the return of an index that has an annual volatility of 25% an

Question

Assume that Rm is the return of an index that has an annual volatility of 25% and no autocorrelation. Calculate the volatilities of: i. A portfolio that leverages the index 1.4 to 1 ii. A portfolio with a 60% portfolio weight in the index and the remainder in cash iii. The index's three month return volatility b. A forty-month return series has a sample skewness and sample excess kurtosis of 0.41 and 0.38, respectively. Calculate the Jarque-Bera (JB) statistic and test, at a 5% significance level, the hypothesis that this series of returns is normal. The critical value for the JB statistic at the 5% significance level is 5.99.

Explanation / Answer

Soln : a-i) A portfolio that leverages index 1.4 to 1 and autocorrelations is 0.

Volatility of index = 25%, that will give volatility of portfolio = 25% *1.4 = 35%

a-ii) Portfolio weight = 60%, the volatility can be calculated = 0.60 * volatilty of index = 0.60*25% = 15%

a-iii) Index 3 month volatility return = annual volatility *t0.5 , here t = 3/12 = 0.25

Volatility return = 25% *(0.25)0.5 = 25*0.5 = 12.5%

b) As per the Jarque- Bera test , The JB statistic = N/6 *(S2 + (C-3)2/4,

where n = Sample Size here it is = 40 , C-3 = excess kurtosis, S = sample skewness

JB = (40)/6 *(0.41^2 + (0.38^2)/4) = 1.36

1.36 < 5% significance level 5.99, therefore distribution is normal and hypothesis cann't be rejected.

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