Question

NB:

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Assume that you recently graduated with a major in finance, and you just landed a job in the trust department of a large regional bank. Your first assignment is to invest KES 10 Million from an estate for which the bank is a trustee. Because the estate is expected to be distributed to the heirs in about one year, you have been instructed to plan for a one-year holding period. Further, your boss has restricted you to the following investment alternatives, shown with their probabilities and associated outcomes.  

		Returns on alternatives investments
		Estimated rate of return
State of economy	Prob.	T-Bills	High-Tech Corporation	Excel collections Ltd	Ribbon manufacturing	Market portfolio
Recession	0.1	8%	-22%	28%	10%	-13%
Below average	0.2	8%	-2%	14.7%	-10%	1%
Average	0.4	8%	20%	15%	7%	15%
Above average	0.2	8%	35%	-10%	45%	29%
Boom	0.1	8%	50%	-20%	30%	43%

High-Tech Corporation is an electronics firm; Excel Collections Ltd. Collects past-due debts; and Ribbon Manufacturing Ltd. manufactures tyres and other rubber and plastics products. The bank also maintains an “Index fund” which owns a market –weighted fraction of all publicly traded stocks; you can invest in that fund and thus obtain average stock market results.

a) Calculate the expected rate of return on each alternative.

Expected Rate of Return of T - Bills

{(0.1*8%)+(0.2*8%)+(0.4*8%)+(0.2*8%)+(0.1*8%)} = 8.00%

Expected Rate of Return of High - Tech Corporation

{(0.1*-22%)+(0.2*-2%)+(0.4*20%)+(0.2*35%)+(0.1*50%)} = 17.40%

Expected Rate of Return of Excel Collections Ltd

{(0.1*28%)+(0.2*14.7%)+(0.4*15%)+(0.2*-10%)+(0.1*-20%)} = 7.74%

Expected Rate of Return of Ribbon Manufacturing

{(0.1*10%)+(0.2*-10%)+(0.4*7%)+(0.2*45%)+(0.1*30%)} = 13.80%

Expected Rate of Return of Market Portfolio

{(0.1*-13%)+(0.2*1%)+(0.4*15%)+(0.2*29%)+(0.1*43%)} = 15.00%

b) Basing a decision solely on expected returns is only appropriate for risk neutral individuals. Because the beneficiaries of the trust, like most people are risk averse, and the riskiness of each alternative is an important aspect of the decision. Calculate the standard deviation for each alternative.

Standard Deviation is square root of {(Pr*(Rr-ERR)2)+(Pba*(Rba-ERR)2)+(Pa*(Ra-ERR)2)+(Pav*(Rav-ERR)2)+(Pb*(Rb-ERR)2)}

Standard Deviation of T - Bills

(0.1*(8%-8%) )+(0.2*(8%-8%) )+(0.4*(8%-8%) )+(0.2*(8%-8%) )+(0.1*(8%-8%) )

= 0+0+0+0+0

= 0.00%

Standard Deviation of High - Tech Corporation

(0.1*(-22%-17.4%) )+(0.2*(-2%-17.4%) )+(0.4*(20%-17.4%) )+(0.2*(35%-17.4%) )+(0.1*(50%-17.4%) )

= 155.236+75.272+2.704+61.952+106.276      = 401.44

= 20.04%

Standard Deviation of Excel Collections Ltd

(0.1*(28%-7.74%) )+(0.2*(14.7%-7.74%) )+(0.4*(15%-7.74%) )+(0.2*(-10%-7.74%) )+(0.1*(-20%-7.74%) )

= 41.04676+9.68832+21.08304+62.94152+76.95076      = 211.7104

= 14.55%

Standard Deviation of Ribbon Manufacturing

(0.1*(10%-13.8%) )+(0.2*(-10%-13.8%) )+(0.4*(7%-13.8*%) )+(0.2*(45%-13.8%) )+(0.1*(30%-13.8%) )

= 1.444+113.288+18.496+194.688+26.244         = 354.16

= 18.82%

Standard Deviation of Market Portfolio

(0.1*(-13%-15%) )+(0.2*(1%-15%) )+(0.4*(15%-15%) )+(0.2*(29%-15%) )+(0.1*(43%-15%) )

= 78.4+39.2+0+39.2+78.4      = 235.2

= 15.34%

c) Calculate the coefficient of variation (CV) for the different securities. Does the CV produce the same rankings as the standard deviation?

d) Suppose you create a two-stock portfolio by investing KES 4 million in High-Tech Corporation and KES 6 million in Excel Collections Ltd.

i. Calculate the expected return, the standard deviation and the coefficient of variation for this

portfolio.

ii. How does the riskiness of this two-stock portfolio compare to the riskiness of the individual stocks if they are held in isolation?

Answer 1

The first two parts are correctly calculated by you

c) The Coefficient of Variation (CV) is given by = Standard Deviation / Average or mean return

So, CV for T- Bills = Std. Dev/ mean return = 0/0.08 = 0

CV for High Tech Corporation = 20.04%/17.4% = 1.1515

CV for Excel Collections = 14.55%/7.74% = 1.8799

CV for Ribbon Mfg = 18.82%/ 13.80% = 1.3637

CV for Market portfolio =15.33%/15% = 1.0224

As per CV, the alternative investments in increasing order of risk are :

T.Bills, Market Portfolio, High Tech Corp, Ribbon Mfg and the most risky is Excel Collections.,

whereas if we take standard deviation alone, the alternative investments in increasing order of risk are

T.Blls, Excel Collections, Market portfolio, Ribbon manufacturing and then High Tech Corporation.

Hence, CV produces different rankings than Standard Deviation. (CV is better)

d) Out of KES 10 million, KES 4 million are invested in High Tech Corp and KES 6 million in Excel Collections , Hence the weight of investments is 40% and 60% respectively in High Tech Corp and Excel Collections

The portfolio so made will have returns in each state of economy calculated as under :

Portfolio Return = Weight in High Tech Corp * return of High Tech Corp + weight in Excel * Return in Excel Solutions

So for Recession, Portolio return = 0.4 * (-22%) + 0.6 * 28% = 8% & so on.

State of Economy	Prob	High Tech Corp	Excel Collections	Portfolio
Recession	0.1	-22%	28%	8.00%
Below Average	0.2	-2%	14.70%	8.02%
Average	0.4	20%	15%	17.00%
Above Average	0.2	35%	-10%	8.00%
Boom	0.1	50%	-20%	8.00%

i) The Expected portfolio return and standard Deviation is calculated now as

Rp= Pi* R i=1

Ι=ΙΛ εί21 - H): 43 -

R_p = 0.1*0.08+0.2*0.0802+0.4*0.17+0.2*0.08+0.1*0.08 = 0.1160 = 11.6%

$\sigma _{p}^{2}$ = 0.1*(8%-11.6%)² +0.2*(8.02%-11.6%)² +0.4*(17%-11.6%)² + 0.2*(8%-11.6%)² +0.1*(8%-11.6%)²

=19.41%

$\sigma _{p}$ = 4.41%

CV_p = 4.41%/11.6% = 0.38

ii) The portfolio is the less riskier (Least CV) than any of the earlier investments considered and gives a reasonable rate of return. In particular, we can say that investing in this portfolio is better than investing in Excel solutions alone in terms of both risk and return. and this portfolio is also better than High Tech Corp. in terms of risk.