Question

Q1. Portfolio Returns

i. stock has mean of 8% and stdev of 20%;

ii bond has mean of 4% and stdev of 10%;

iii correlation b/w stock and bond of -0.2;

iv. cash return is 1% for lending and borrowing.

Q1b: the mean and stdev of a fully invested yet unleveraged portfolio that assign weights based on the inverse of risk is 5.33% and 8.43%, respectively.

1.if you want to target 10% stdev risk per year, how would you combine Q1b risk parity portfolio with cash in this case?

2.if you want to target 10% stdev risk per year, yet you believe correlation has changed to +0.5. How would you combine Q1b risk parity portfolio with cash in this case?

Answer 1

1. As the fully invested and unleveraged portfolio has a standard deviation of 8.43%, but the target standard deviation risk is 10%, one has to borrow at cash rate of 1% and invest in the portfolio. The weights of the stock is 1/3 and the weight of Bond is 2/3 to get a return of 5.33%

Weight of Risky asset = Target standard deviation/ Standard deviation of portfolio

=0.1/0.0843 = 1.18624

and weight of Cash = 1-1.18624 = -0.18624

So, one must borrow at cash rate of 1% , an amount equal to 18.624% of amount invested in the unleveraged portfolio and invest this borrowed amount also in the unleveraged portfolio

The return of the total portfolio thus will be = 1.18624* 5.33% - 0.18624*1% = 6.13642% with a risk of 10%

2. if correlation coefficient is +0.5, the risk of the unleveraged portfolio is

= (0.3333^20.2^2+ 0.6667^2*0.1^2+2*0.3333*0.6667*0.2*0.1*0.5)^0.5

= 0.11547 or 11.55%

Now, As the fully invested and unleveraged portfolio has a standard deviation of 11.55%, but the target standard deviation risk is 10%, one has to disinvest some amount from the original portfolio and invest  at cash rate of 1%.

Weight of Risky asset = Target standard deviation/ Standard deviation of portfolio

=0.1/0.1155 = 0.8658

and weight of Cash = 1-0.866 = 0.1342

So, one must invest 13.42% of total money in cash rate of 1% , and retain only 86.58% of total amount invested in the unleveraged portfolio

The return of the total portfolio thus will be = 0.8658* 5.33% + 0.1342*1% = 4.75% with a risk of 10%