Present Value and the Risk/Return Trade-Off

Question 1 

Formula for perpetuity: 

Principal 

Principal 

We need to make a funding of $ 1,000,000 to achieve a $ 50,000 perpetuity. 

Formula for continuous compounding: 

A 

A = final amount 

P = initial amount 

R = interest rate 

T = time 

A 

In 10 years, the value of this bank account will be 

Question 2 

Computing the present value: 

PV 

PV = present value 

CF= cash flow per year 

n = number of years 

r = interest rate 

computing the future value: 

FV  

FV = future value 

r = interest rate 

n = number of years 

for 8% interest rate the future and present value are: 

annual interest rate:	8%	Year	payment	present factor	present value	future value
		1	5000	0.925925926	$ 4,629.63	$ 6,802.44
		2	6000	0.85733882	$ 5,144.03	$ 7,558.27
		3	7000	0.793832241	$ 5,556.83	$ 8,164.80
		4	8000	0.735029853	$ 5,880.24	$ 8,640.00
		5	9000	0.680583197	$ 6,125.25	$ 9,000.00
			total		$ 27,335.98	$ 40,165.52

For 10% interest rate the future and present values are: 

annual interest rate:	10%	Year	payment	present factor	present value	future value
		1	5000	0.909090909	$ 4,545.45	$ 7,320.50
		2	6000	0.826446281	$ 4,958.68	$ 7,986.00
		3	7000	0.751314801	$ 5,259.20	$ 8,470.00
		4	8000	0.683013455	$ 5,464.11	$ 8,800.00
		5	9000	0.620921323	$ 5,588.29	$ 9,000.00
			total		$ 25,815.74	$ 41,576.50

Question 3 

Let the return for each asset be the random variable X. Calculating the expected value E ( , multiply the random variable X (returns) with the probability associated with it. The total is the expected value for the assets A, B, C respectively. the expected value for Asset A is 7.4, this means that asset A should expect 7.4 returns. Asset B should expect 17.4 returns and Asset C should expect 5 returns. 

Asset A			Asset B			Asset C
Probability	return	Expected value of Asset A	probability	return	Expected value of Asset B	probability	return	Expected value of Asset B
0.3	5	1.5	0.1	25	2.5	0.1	4	0.4
0.4	8	3.2	0.3	20	6	0.8	5	4
0.3	9	2.7	0.5	15	7.5	0.1	6	0.6
			0.1	14	1.4
	total	7.4		total	17.4		total	5

Obtaining the standard deviation for each asset, taking the returns which is the random variable X, subtract the mean (expected value) of each asset, square the results and multiply with the probability associated with each set of returns and then obtain the square root of the whole process. 

standard deviation Asset A			Standard deviation Asset B			standard deviation Asset B
3.26863886			3.440930107			0.447213595

1.  

CV=SD/Mean asset A			Asset B			Asset C
0.441707954			0.197754604			0.026306682

Standard deviation is the proxy for risk. To compare Asset A, B, C the coefficient of variation. The coefficient of variation explains for one unit of mean how much risk we have. The coefficient of variation for Asset c is the least and hence has the less variation there is the less risk there is. 

Question 4 

Required return = Risk free rate + Risk coefficient (Expected return- Risk free rate) 

1% + 1.2(8%-1%) = 9.4% 

Beta increases by 50%, obtaining the new beta, 

Required new return =1% +1.8(8%-1%) = 13.6% 

Percentage=wise change in required return is calculated by taking the difference between the new returns and the old returns, the difference is divided by the old returns and multiply by 100. That is, 

Percentage-wise change in required return = new return – old return 

= 13.6% - 9.4% = 4.2% 

The percentage-wise change in required return is 4.2% 

Market return increases by 50% the new market return is computed as: 

New required return = 1% +1.2(12% - 1%) = 14.2% 

14.2% - 9.4% = 4.8% 

The percentage=wise change is 4.8%. 

Question 5 

Beta is a measure of the volatility, or systematic risk, of a security or portfolio in comparison to the market as a whole. Investors use beta to gauge how much risk a stock adding to a portfolio. A beta value of 1.0 indicates that its price activity is strongly correlated with the market. It has a systematic risk (Cordis, 2014) . Beta values less than one means that the security is theoretically less volatile than the market. The stock being included in the portfolio makes it less risky than the same portfolio without the stock. Beta values greater than one shows that the security’s price is theoretically more volatile than the market. Trendy Tech will have the highest beta. This is because it has sensitive movements in the market. Oily oil should have the lowest beta because the stock depends just on oil and not on the market movements. Conglomerated should have a beta close to one. It is involved in every industry and is highly diversified so its stock price will depend heavily on the market as a whole. 

Reference 

Cordis, A. (2014). Accounting Ratios and the Cross-section of Expected Stock Returns. Journal of Business Finance & Accounting, 41(9-10), 1157-1192. https://doi.org/10.1111/jbfa.12092 .