Question

Variance and standard deviation

Hull​ Consultants, a famous think tank in the​ Midwest, has provided probability estimates for the four potential economic states for the coming year. The probability of a boom economy is 13%​, the probability of a stable growth economy is 20​%, the probability of a stagnant economy is 54​%, and the probability of a recession is 13​%.

Calculate the variance and the standard deviation of the three​ investments: stock, corporate​ bond, and government bond. If the estimates for both the probabilities of the economy and the returns in each state of the economy are​ correct, which investment would you​ choose, considering both risk and​ return?  

Investment	Forecasted Returns for Each Economy
	Boom		Stable Growth		Stagnant		Recession
Stock	23​%		10​%		6​%		−10​%
Corporate bond	9%		7%		6​%		4%
Government bond	8​%		6​%		5​%		3​%

​Hint: Make sure to round all intermediate calculations to at least seven​ (7) decimal places. The input​ instructions, phrases in parenthesis after each answer​ box, only apply for the answers you will type.

What is the variance of the stock​ investment?

%

​(Round to six decimal​ places.)

What is the standard deviation of the stock​ investment?

%

​(Round to two decimal​ places.)

What is the variance of the corporate bond​ investment?

%

​(Round to six decimal​ places.)

What is the standard deviation of the corporate bond​ investment?

%

​(Round to two decimal​ places.)

What is the variance of the government bond​ investment?

​%

​(Round to six decimal​ places.)

What is the standard deviation of the government bond​ investment?

%

​(Round to two decimal​ places.)

If the estimates for both the probabilities of the economy and the returns in each state of the economy are​ correct, which investment would you​ choose, considering both risk and​ return?  ​(Select the best​ response.)

A.

The corporate bond would be the best choice because it has the highest expected return and the lowest risk.

B.

The stock investment would be the best choice because it has the highest volatility and therefore the best chance of a high return.

C.

The government bond would be the best choice because it has the lowest risk.

D.

There is not enough information to make this decision.

Answer 1

Variance of stock:

N = 4

Expected return from stock = 0.13x23 + 0.2x10 + 0.54x6 - 0.13x10 = 6.93%

Variance = 0.13x(23-6.93)² + 0.2x(10-6.93)² + 0.54x(6. 93-6)² + 0.13x(10-6.93)² = 37.1491

Standard Deviation = sq. Root of variance = 6.095006

For Corporate bond:

Expected return from corporate bond = 0.13x9 + 0.2x7 + 0.54x6 + 0.13x4 = 6.33

Variance = 0.13x(9-6.33)² + 0.2x(7-6.33)² + 0.54x(6.33-6)²+ 0.13x(6.33-4)² = 1.7811

Standard Deviation = square root of variance = 1.334578

For government bond:

Expected return from government bond = 0.13x8 + 0.2x6 + 0.54x5 + 0.13x3 = 5.33

Variance = 0.13x(8-5.33)² + 0.2x(6-5.33)² + 0.54x(5.33-5)² + 0.13x(5.33-3)² = 1.7811

Standard Deviation = 1.334578

For selecting which is the best investment bet, we need to look at the risk to reward ratio.

Coefficient of Variance (CoV) = Risk to reward ratio = Standard Deviation (risk) / Expected return (reward)

Cov stocks = 6.095006/6.93 = 0.87951

CoV Corporate bond = 1.334578/6.33 = 0.2108339

CoV of government bond = 1.334578/5.33 = 0.250389

So the best option is to pick corporate bond as it has the best risk to reward ration.