Question

Question: The stock of Business Adventures sells for $40 a share. Its likely dividend payout and end-of-year price depend on the state of the economy by the end of the year as follows:

a. Calculate the expected holding-period return and standard deviation of the holding period return. All three scenarios are equally likely.

b. Calculate the expected return and standard deviation of a portfolio invested half in Business Adventures and half in Treasury bills. The return on bills is 4%.

Short answer

a. E(HPR) = 8.75%,σ= 17.88%

b. E(r) = 6.375%, σ= 8.94%

Step-by-step solution

Given information

The selling price of Business Adventure stocks = $40 per share.

In boom scenario, dividend= $2.00 and Stock price =$50

In normal scenario, dividend = 1.00 and Stock Price = 43

In recession scenario, dividend = .50 and Stock Price = 34

All the above three scenarios are equally likely.

Solution for part (a)

The formula for finding holding period returns is given below:

HPR=( ((end of period value – original value) + income))/ Original value)*100%

The holding period returns for the three scenarios are:

Boom: (50 – 40 + 2)/40 = 0.30 = 30.00%

Normal: (43 – 40 + 1)/40 = 0.10 = 10.00%

Recession: (34 – 40 + 0.50)/40 = –0.1375 = –13.75%

Since all the scenarios are equally likely, expected holding period return or

Expected return =[Probability (Scenario) x Return in Scenario]

E (HPR) = [(1/3) x 30%] + [(1/3) x 10%] + [(1/3) x (–13.75%)] = 8.75%

and Standard deviation of holding period return =

(Standard Deviation)² =[Probability (Scenario x Deviation from mean in Scenario)]

σ^2*(HPR) = [(1/3) x (30 – 8.75)²] + [(1/3) x (10 – 8.75)²] + [(1/3) x (–13.75 – 8.75)²]

= 319.79

σ= Ö319.79 = 17.88%

Solution for part (b)

Expected return when investment divided by half:

Expected return =[Probability (Scenario) x Return in Scenario]

E(r) = (0.5 x 8.75%) + (0.5 x 4%) = 6.375%

and Standard deviation when investment divided by half:

σ = 0.5 x 17.88% = 8.94%