 # STD of portfolio, with risk free asset question....

An investor’s portfolio currently consists of 100% of stocks that have a mean return of 16.5% and an expected variance of 0.0324. The investor plans to diversify slightly by replacing 20% of her portfolio with U.S. Treasury bills that earn 4.75%. Assuming the investor diversifies, what are the expected return and expected standard deviation of the portfolio? ERPortfolio óPortfolio A) 10.63% 2.59% B) 10.63% 14.40% C) 14.15% 14.40% D) 14.15% 2.59%

D?

the correct answer is C Since Treasury bills (T-bills) are considered risk-free, we know that the standard deviation of this asset and the correlation between T-bills and the other stocks is 0. Thus, we can calculate the portfolio expected return and standard deviation. Step 1: Calculate the expected return Expected ReturnPortfolio = (wT-bills * ERT-bills) + (wStocks * ERStocks ) = (0.20) * (0.0475) + (1.00-0.20) * (0.165) = 0.1415, or 14.15%. Step 2: Calculate the expected standard deviation When combining a risk-free asset and a risky asset (or portfolio or risky assets), the equation for the standard deviation of ó1,2 = [(w12)( ó12) + (w22)( ó22) + 2w1w2 ó1 ó2ñ1,2]1/2 reduces to: ó1,2 = [(wStocks)( óStocks)] = 0.80 * 0.03241/2 = 0.14400, or 14.40%. (Remember to convert variance to standard deviation).

Well i seriously doubt this calculation > > Step 2: Calculate the expected standard deviation > When combining a risk-free asset and a risky asset > (or portfolio or risky assets), the equation for > the standard deviation of > ó1,2 = [(w12)( ó12) + (w22)( ó22) + 2w1w2 ó1 > ó2ñ1,2]1/2 why did you divide it by 2?? > reduces to: ó1,2 = [(wStocks)( óStocks)] = 0.80 * > 0.03241/2 = 0.14400, or 14.40%. even with this calculation im getting the answeras 1.29% >(Remember to > convert variance to standard deviation).

reema, I think the essence of this question is to get us to think of how to compute the std of a portfolio that includes a risk free asset. you compute the stock part of the formula the exact same way as you normally would: (w1)2 x (std)2 = (in this case they give us the variance, so we do not need to square the std) (.8)2 x (.0324) = .020736 – this equals the variance of the portfolio note: we do not add the weight of the risk free asset, because its std = 0 take the square root of .020736 – .144, and this equals the std of the port that includes risk and risk free assets. hope this helps.

To calculate the standard deviation of the new portfolio : 1. First calculate the STD for old portfolio - SQRT (0.0324) = 0.18 2. STD with the risk free asset is - 0.8 * 0.18 = 0.144 = 14.4% So the answer is C.

yancey Wrote: ------------------------------------------------------- > reema, > > I think the essence of this question is to get us > to think of how to compute the std of a portfolio > that includes a risk free asset. > > you compute the stock part of the formula the > exact same way as you normally would: > > (w1)2 x (std)2 = (in this case they give us the > variance, so we do not need to square the std) > > (.8)2 x (.0324) = .020736 – this equals the > variance of the portfolio > > note: we do not add the weight of the risk free > asset, because its std = 0 > > take the square root of .020736 – .144, and this > equals the std of the port that includes risk and > risk free assets. > > hope this helps. thanks yancey. got your point

in the case of addition of a risk free asset, the formula just boils down to std portfolio = std non risk free * weight non-risk free (when you take the sqrt) so 0.8 * .18 = .144 = 14.4%