Question

Suppose you have $200 with which you can buy shares of stock from two companies: ABC Hot Chocolate Company and XYZ Lemonade. Each company's stock currently sells for $100 per share. ABC hot chocolate company: If the temperature next year is lower than average, the interest rate will be 20%. There is a 50% chance that it will be colder than average next year. But if the temperature is warmer (50% chance), there is no returns from this stock. XYZ lemonade: If the temperature next year is higher than average, the interest rate will be 20%. There is a 50% chance that it will be warmer than average. But if the temperature is colder (50% chance), there is no returns from this stock Choice 1: Only buy 2 stocks from ABC hot chocolate company. Choice 2: Only buy 2 stocks from XYZ lemonade company. Choice 3: buy 1 stock from ABC and buy 1 stock from XYZ. Question: 1. Please calculate the expected return rate for every stock from each company. 2. please calculate the expected return for each choice. 3. Please compare the risk for these three choices and choose which one choice is more riskless. (eg: for choice 1, 50% chance to get 20$, but also 50% chance get nothing).

Answer 1

In the question it is mentioned that the interest rate will be 20% in both the cases, I believe they mean to see that the return from the stock is 20%. i.e. If the temperature is lower than average then return of ABC hot chocolate company is 20% and similarly if the temperature is warmer than average then return of XYZ lemonade company is 20%.

Question 1:

We will use the following formula to calculate the expected return from each stock:

Expected return from stock = $\sum$ probability of event occuring * stock returns if the event occurs

Expected return from ABC Hot Chocolate company = probability (temperature is colder than average) * stock returns if temperature is colder + probability (temperature is warmer than average) * stock returns if temperature is warmer

= 50% * 20% + 50% * 0 = 10%

Expected return from XYZ lemonade company = probability (temperature is warmer than average) * stock returns if temperature is warmer + probability (temperature is colder than average) * stock returns if temperature is colder

= 50% * 20% + 50% * 0 = 10%

Answer: Expected return from each stock is 10%

Question 2:

Calculate expected return from each choice?

Choice 1: Only buy 2 stocks from ABC Hot chocolate company.

It doesn't matter how many stocks we buy from ABC hot chocolate company because the expected return depends upon the stock and not the quantity that we buy. Hence the expected return of choice 1 is 10%

Choice 2: Only buy 2 stocks from XYZ lemonade company.

It doesn't matter how many stocks we buy from XYZ lemonade company because the expected return depends upon the stock and not the quantity that we buy. Hence the expected return of choice 2 is also 10%

Choice 3: Buy one stock from ABC Hot chocolate company and one stock from XYZ lemonade company

Expected return of portfolio = weighted average of expected return of stocks

= 1/2 * Expected return(ABC Hot chocolate) + 1/2 * Expected return (XYZ lemonade company) (weight of each stock is 100*1/(100*1+100*1) = 100/200 = 1/2)

= 1/2*10% + 1/2*10% = 10%

3) Calculate risk for each choice

Formula to calculate risk:

sigma =

Σ(X - E(X) 2)

Risk from Choice 1:

X = (20%, 0%)

E(X) = 10%

sigma =
(20%-10%) + (0%-10%)2
=
(10%)2+(-10%)2

=
(100%)+ (100%)
=
(200%) V2
=10%

choice 2 is similar to choice 1, hence the risk for choice 2 is also 10%

Choice 3:

Choice 3 is a portfolio of two stocks. These two stocks are not at all correlated with each other, because if first stock performs well then other stock has zero returns and vice versa.

weight of each stock in the portfolio = 1/2 = 0.5

variance of portfolio = w²*sigma1² +w2*sigma2² .

= (1/2)² * 10%² + (1/2)² * 10%²

= 1/4*(100%+100%) = 50%

sigma of portfolio = square root of (50%) = 7.0711%

Hence the risk of the portfolio is 7.0711%

Comparing the risk of choice 1 (10%), choice 2 (10%) and choice 3 (7.0711%), choice 3 has less risk than the other two and hence is more riskless. This is because of the diversification effect.