Consider a portfolio choice problem in a world with risk free rate rf and two ri

Question

Consider a portfolio choice problem in a world with risk free rate rf and two risky assets i = 1, 2. Assume that one asset has both a higher expected return and volatility than the other, so that Mu1 > Mu2 and sigma1 > sigma2, and that the returns to each are uncorrelated P12 = 0. Let lambda denote the portfolio weight on stock 1. (a) Find the weight lambda that yields the minimum variance portfolio. (b) Let lambda * be the weight that defines the tangency portfolio. Write the condition that defines lambda* and, without solving for it, determine how it compares to A. (c) We now wish to consider how the composition of the tangency portfolio shifts as the risk free rate increases. Without solving the optimization problem, how do you expect to change lambda * to change as rf increases? (d) Using the envelope theorem, determine how the slope of the tangent line changes as the risk free rate increases. Once again, do not solve the optimization problem.

Explanation / Answer

a) sum of weights=1 =>1+ 2=1 => 2=1- 1

Variance of portfolio= 12*12+ 22*22 +2* 1* 2*1,2* 1* 2

1,2=0 => Variance of portfolio= 12*12+ 22*22 +2* 1* 2*0* 1* 2

=> Variance of portfolio= 12*12+ 22*22

Put value of 2=1- 1 => Variance of portfolio=V= 12*12+ (1- 1)2*22

Differentiating wrt 1 and set to 0 to get minimum

dV/d 1 = d(12*12+ (1- 1)2*22)/d 1=0

2* 1*12 – 2*(1- 1) *22 =0 => 1 = 22/( 12+ 22)

Double differentiate, d2V/d 12=2*12 + 2* *22 >0 thus we obtained a minimum

b)Tangency Portfolio,

Sharpe ratio= (11+ (1- 1) 2- rf)/ 12*12+ (1- 1)2*22

At tangency portfolio Sharpe ratio is maximum,

d Sharpe ratio /d 1 = d((11+ (1- 1) 2- rf)/( 12*12+ (1- 1)2*22)/d 1=0

( 12*12+ (1- 1)2*22)*( 1- 2) - (11+ (1- 1) 2- rf)* (2* 1*12 – 2*(1- 1) *22)=0 ..(apply quotient rule)

-2* (2* 1*12 – 2*(1- 1) *22) + ( 1- 2)*( 12*12+ (1- 1)2*22-2* 12*12 + 2* 1 (1- 1) *22)+ rf (2* 1*12 – 2*(1- 1) *22)=0

-2* (2* 1*12 – 2*(1- 1) *22) + ( 1- 2)*( (1- 12)*22 - 12*12 ) )+ rf (2* 1*12 – 2*(1- 1) *22)=0

12 * (-2* 1* 2– 12*( 1- 2) +2* 1* rf) + 22*( (1- 12)* ( 1- 2)+ 2* 2 *(1- 1) -2* rf (1- 1) ) =0

12 * (2* 1* 2+ 12*( 1- 2)- 2* 1* rf) - 22*( (1- 12)* ( 1- 2)+ 2* 2 *(1- 1) -2* rf (1- 1)   )=0

12*[( 12 + 22)* ( 1- 2)] + 2* 1 [ 2 *(12 * + 22)- 12* rf – 22* rf] - 22 * ( 1+ 2-2* rf)=0

*2*[( 12 + 22)* ( 1- 2)]   + 2* * [ 2 *(12 * + 22)- 12* rf – 22* rf] - 22 * ( 1+ 2-2* rf)=0

is the required condition for tangency portfolio.

The minimum variance portfolio has the lower Sharpe ratio than the tangency portfolio, also minimum variance portfolio has the lower expected return than the tangency portfolio expected return which directly depends on portfolio weight on stock 1 ,1(1- 2)+ 2- rf thus higher expected return higher the value of portfolio weight on stock 1 1 this means tangency portfolio which has higher expected return has higher portfolio weight on stock 1 * than minimum variance portfolio which has the lower expected return thus lower portfolio weight on stock 1 .

c) Differentiating the above condition for tangency portfolio w.r.t rf

(d * /drf) [2* [( 12 + 22)* ( 1- 2)] + 2 [ 2 *(12 * + 22)- 12* rf – 22* rf]] + 222 =0

(d * /drf) [* [( 12 + 22)* ( 1- 2)] + [ (2- rf) *(12 * + 22)]] + 22 =0

(d * /drf) [* [( 12 + 22)* ( 1- 2)] + [ (2- rf) *(12 * + 22)]] =- 22

(d * /drf)= - 22/[* [( 12 + 22)* ( 1- 2)] + [ (2- rf) *(12 * + 22)]] <0 for all values.

As above * is a decreasing function of rf as slope is less than 0.

Therefore as rf increases the * decreases in value and composition of the second stock 2 increases while the weight of stock 1 decreases as rf increases.

d) Hold * constant and calculate d Sharpe ratio /d rf to find,

= d((*1+ (1- *) 2- rf)/( *2*12+ (1- *)2*22)/d rf

= -( *2*12+ (1- *)2*22)- (*1+ (1- *) 2- rf)<0 for all values.

Thus as above Sharpe ratio is a decreasing function of rf as slope is less than 0 The slope of tangent line decreases as risk free rate increases.

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