You have been given this probability distribution for return for Cheese, Inc. stock:

State of Economy Probability Return
Boom 0.20 20%
Normal 0.45 10%
Recession 0.35 2%

Assuming that the expected return on Cheese's stock is 9.2%, what is the standard deviation of these returns?

Question

You have been given this probability distribution for return for Cheese, Inc. stock:

State of Economy	Probability	Return
Boom	0.20	20%
Normal	0.45	10%
Recession	0.35	2%

Assuming that the expected return on Cheese's stock is 9.2%, what is the standard deviation of these returns?

BlackTom AI · Accepted Answer

To analyze this problem, I’ll first restate what we’re given and what we’re asked to compute, then I’ll walk through each relevant calculation step.

We’re given a probability distribution for Cheese, Inc. stock returns with three states of the economy:
- Boom: probability 0.20, return 20%
- Normal: probability 0.45, return 10%
- Recession: probability 0.35, return 2%
The problem also states that the expected return is 9.2%, which should match the weighted average of the returns:
E[R] = 0.20*(0.20) + 0.45*(0.10) + 0.35*(0.02) = 0.04 + 0.045 + 0.007 = 0.092 = 9.2%.

Next, we compute the variance of the returns, using the formula Var(R) = Σ p_i (R_i − E[R])^2.
- For Boom: R_i = 0.20, E[R] = 0.092, difference = 0.20 − 0.092 = 0.108. Square = 0.011664. Weighted by p = 0.20 gives 0.20 * 0.011664 = 0.0023328.
- For Normal: R_i = 0.10, difference = 0.10 − 0.092 = 0.008. Square = 0.000064. Weighted by p = 0.45 gives 0.45 * 0.000064 = 0.0000288.
- For Recession: R_i = 0.02, difference = 0.02 − 0.092 = −0.072. Square = 0.005184. Weighted by p = 0.35 gives 0.35 * 0.005184 = 0.0018144.

Sum these contributions to obtain Var(R):
Var(R) = 0.0023328 + 0.0000288 + 0.0018144 = 0.004176.

The standard deviation is the square root of the variance:
SD(R) = sqrt(0.004176) ≈ 0.0646, or 6.46%.

Therefore, the standard deviation of these returns is approximately 6.46%.

Note on the answer options: since a single option is provided, 6.46% is the value you should select. If there were multiple options, I would similarly verify each candidate by computing its deviation from the mean and comparing to the correct calculation, identifying any misapplications of the variance formula or arithmetic errors.

FINANCE 261 Quiz 1

You have been given this probability distribution for return for Cheese, Inc. stock: State of Economy Probability Return Boom 0.20 20% Normal 0.45 10% Recession 0.35 2% Assuming that the expected return on Cheese's stock is 9.2%, what is the standard deviation of these returns?

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