Phoenix Company common stock is currently selling for $20 per share. Security analysts at Smith Blarney have assigned the following probability distribution to the price of (and rate of return on) Phoenix stock one year from now: $$\begin{array} { l l l }
\text { Price } & \text { Rate of Return } & \text { Probability } \
\$ 16 & - 20 \% & 0.25 \
\$ 20 & 0 \% & 0.30 \
\$ 24 & + 20 \% & 0.25 \
\$ 28 & + 40 \% & 0.20
\end{array}$$ Assuming that Phoenix is not expected to pay any dividends during the coming year, determine the standard deviation of possible rates of return on Phoenix stock (to the nearest tenth of a percent).
A) 45.6%
B) 20.9%
C) 2.2%
D) 21.4%

Question

Phoenix Company common stock is currently selling for $20 per share. Security analysts at Smith Blarney have assigned the following probability distribution to the price of (and rate of return on) Phoenix stock one year from now: $$\begin{array} { l l l } 
\text { Price } & \text { Rate of Return } & \text { Probability } \
\$ 16 & - 20 \% & 0.25 \
\$ 20 & 0 \% & 0.30 \
\$ 24 & + 20 \% & 0.25 \
\$ 28 & + 40 \% & 0.20
\end{array}$$ Assuming that Phoenix is not expected to pay any dividends during the coming year, determine the standard deviation of possible rates of return on Phoenix stock (to the nearest tenth of a percent).
A) 45.6%
B) 20.9%
C) 2.2%
D) 21.4%

Quizplus · Accepted Answer

The standard deviation of the rates of return can be calculated by first finding the expected rate of return and then using it to calculate the variance and standard deviation. The expected rate of return (E(R)) is calculated as the sum of each possible rate of return multiplied by its probability: E(R) = (-20% * 0.25) + (0% * 0.30) + (20% * 0.25) + (40% * 0.20) = -5% + 0% + 5% + 8% = 8%. The variance (σ^2) is the sum of the squared differences between each rate of return and the expected rate of return, each weighted by its probability: σ^2 = [(0.25 * (-20% - 8%)^2) + (0.30 * (0% - 8%)^2) + (0.25 * (20% - 8%)^2) + (0.20 * (40% - 8%)^2)] = 0.25 * (-28%)^2 + 0.30 * (-8%)^2 + 0.25 * (12%)^2 + 0.20 * (32%)^2 = 0.25 * 0.0784 + 0.30 * 0.0064 + 0.25 * 0.0144 + 0.20 * 0.1024 = 0.0196 + 0.00192 + 0.0036 + 0.02048 = 0.0456. The standard deviation (σ) is the square root of the variance: σ = √0.0456 ≈ 21.4%.

Phoenix Company Common Stock Is Currently Selling for $20 Per