EXPECTED RETURNS Suppose you won the lottery and had two options: (1) receiving

Question

EXPECTED RETURNS

Suppose you won the lottery and had two options: (1) receiving $1.8 million or (2) taking a gamble in which, at the flip of a coin, you receive $3.6 million if a head comes up but receive zero if a tail comes up.

a. What is the expected value of the gamble? (Round to two decimal places)

b. Suppose the payoff was actually $1.8 million - that was the only choice. You now face the choice of investing it in a U.S. Treasury bond that will return $1,917,000 at the end of a year or a common stock that has a 50-50 chance of being worthless or worth $4,320,000 at the end of the year.

1. The expected profit on the T-bond investment is $117,000. What is the expected dollar profit on the stock investment? (Round to two decimal places)

2. The expected the rate of return on the T-bond investment is 6.5%. What is the expected rate of return on the stock investment? (Round to two decimal places)

3. Exactly how large would the expected profit (or the expected rate of return) have to be on the stock investment to make you invest in the stock, given the 6.5% return on the bond? (Round to two decimal places)

Explanation / Answer

Part a:

Expected Value of the Gamble = (Probability of Head x Amount received on head coming up) + (Probability of Tail x Amount received on tail coming up)

When you toss a coin, there are only 2 possible events - either head comes up or tail comes up. So the probability of head coming up = probability of tail coming up = 0.50

We'll receive $3.6 million if head comes up, and $0 if tail comes up, so:

Expected Value of the Gamble = (0.50 x $3.6 million) + (0.50 x $0) = $ 1.8 million

Part b:

1. If we invest the $1.8 million in a common stock it has a 50-50 chance (probability =0.50 ) of having a value of $4,320,000 or $0 at the end of the year.

So, we find the expected value at the end = ($4,320,000 x 0.50) + ($0 x 0.50) = $ 2,160,000

Hence, Expected Dollar Profit on stock investment = Expected Ending Value - Amount Invested

= $ 2,160,000 - $ 1,800,000 = $ 360,000

2. Expected Rate of Return on Stock investment :

If the stock becomes worth $4,320,000 at year end, its rate of return = $4,320,000 - 1,800,000 / 1,800,000 = 140%

If the stock becomes worth $0 at year end, its rate of return = 0 - 1,800,000 / 1,800,000 = -100%

The chance of either rate of return is 50-50.

Hence, Expected Rate of Return on Stock = (140% x 0.50) + (-100% x 0.50) = 70% - 50% = 20%

3. To find the minimum expected rate of return on the stock, we can use the capital asset pricing model or capm according to which, required rate of return = risk-free rate + beta of stock x (expected market return – risk free rate).

Here the risk-free rate is the T-bond's rate of return = 6.5%. Expected market return from stock = 20%. However since we don't know the stock's beta, we can't arrive at the final value.

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