Suppose you estimated a Security Characteristic Line (SCL) for a stock X using t

Question

Suppose you estimated a Security Characteristic Line (SCL) for a stock X using the monthly excess returns on A and a proxy for the market portfolio M for the past 5 years: Some of the results from the estimation are given below: r^ext = ax + b_xr^eMt + e_xt Over the 5-year sample period, the average monthly excess return and standard deviation of the market portfolio proxy were 0.5% and 4%. a. Over the 5-year sample period, what was the average monthly excess return on the stock X? Suppose the standard deviation of the stock X over the 5-year sample period was 9%. Decompose the variance of the stock X (sigma^2_x = 9^2 = 81) into the systematic component and firm-specific component . Suppose you invested your entire wealth in the stock X for the past 5 years. Did you outperform the market? Evaluate based on the appropriate measure of performance.

Explanation / Answer

Question a:

r(xt) = a(x) + b(x) x r(xm) + e(xt)

where r(xt) = excess return of stock x , a(x) = alpha of stock x , r(xm) = excess return of market , e(xt) = non systematic risk or random risk

taking expectations on both sides we get

E[r(xt)] = E[a(x)] + E[b(x) x r(xm)] + E[e(xt)]

Now E[e(xt)]=0

Also b(x) & r(xm) are assumed to be independent and hence E[b(x) x r(xm)] = E[b(x)] x E[r(xm)]

So the equation now becomes

E[(r(xt)] = 0.22% + 1.3 x 0.5% = 0.22% + 0.65% = 0.87%

Question b:

Variance of Stock Return can be broken down into two components viz.

1) Variance attributable to uncertainty of common macroeconomic factors = beta square x variance of market

2) Variance attributable to firm specific uncertainty

Systematic component measure = (1.3)^2 x (4)^2 = 1.69 x 16 = 27 [Variance = square of standard deviation, have rounded the answer, it actually comes out to 27.04]

Standard deviation of stock = 9, Variance = 81

Hence Variance attributable to firm specific uncertainty = 81 - 27 = 54

We can also cross validate our answer as R square = systemic variance / total variance = 27.04 / 81 = 0.3338

Answer c:

Comparative analysis of returns can be very easily conducted using the Sharpe Ratio

Sharpe Ratio = (Asset Return - Risk Free Rate) / Asset Standard Deviation

We estimate Sharpe Ratio for X as well as the overall market to make an evaluation

Sharpe Ratio (x) = 0.87 / 9 = 0.0967

Shrpe Ratio (Market) = 0.50 / 4 = 0.1250

Since sharpe ratio (x) is lower, we can conclude that on a risk adjusted basis, we did not outperform the market.

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