Question

1. Mr. Malone wants to change the overall risk of his portfolio. Currently, his portfolio is a combination of risky assets with a beta of 1.25 and an expected return of 14%. He will add a risk-free asset (U.S. Treasury bill) to his portfolio. If he wants a beta of 1.0, what percent of his wealth should be in the risky portfolio and what percentage should be in the risk-free asset? If he wants a beta of 0.75? If he wants a beta of 0.50? If he wants a beta of 0.25? Is there a pattern here?

PLEASE GIVE ME A FULL EXPLANATION WITH ALL THE FORMULAS, THANK YOU

Answer 1

Answer-

Existion Beta = 1.25, Expected return= 14%

1.Calculation of risk free asset to be purchased to reduce existing beta to 1.00

Let assume that weight of portfolio be 'w' and weight of risk free asset be '1-w'

Hence,

portfolio beta= weighted avg. beta

1.00 = w*1.25+(1-w)*0

1.00 = 1.25w+0

1.00 = 1.25w

w = 1.00/1.25

weight of risky asset = 0.80

weight of risk free asset = 1-0.80

= 0.20

2. Calculation of risk free asset to be purchased to reduce existing beta to 0.75

Let assume that weight of portfolio be 'w' and weight of risk free asset be '1-w'

Hence,

portfolio beta= weighted avg. beta

0.75 = w*1.25+(1-w)*0

0.75 = 1.25w+0

0.75 = 1.25w

w = 0.75/1.25

weight of risky asset = 0.60

weight of risk free asset = 1-0.60

= 0.40

3.Calculation of risk free asset to be purchased to reduce existing beta to 0.50

Let assume that weight of portfolio be 'w' and weight of risk free asset be '1-w'

Hence,

portfolio beta= weighted avg. beta

0.50 = w*1.25+(1-w)*0

0.50 = 1.25w+0

0.50 = 1.25w

w = 0.50/1.25

weight of risky asset = 0.40

weight of risk free asset = 1 - 0.40

= 0.60

4.Calculation of risk free asset to be purchased to reduce existing beta to 0.25

Let assume that weight of portfolio be 'w' and weight of risk free asset be '1-w'

Hence,

portfolio beta= weighted avg. beta

0.25 = w*1.25+(1-w)*0

0.25 = 1.25w+0

0.25 = 1.25w

w = 0.25/1.25

weight of risky asset = 0.20

weight of risk free asset = 1 - 0.20

= 0.80

SUMMARY

Sr. No	Expected Beta	Weight of Risk free asset	Weight of Risky asset
1	1.00	0.20	0.80
2	0.75	0.40	0.60
3	0.50	0.60	0.40
4	0.20	0.80	0.20