The following returns are predicted for XYZ, Inc. stock under three projected ec

Question

The following returns are predicted for XYZ, Inc. stock under three projected economic scenarios.
Pi = Probability of economic scenario i. Ri = Return of ABC stock in economic scenario i.

                                Pi            Ri
Boom                    0.25        12
Normal                 0.5          6
Recession            0.25        -2

What is the standard deviations of returns? If the returns are normally distributed, what is the expected range of returns assuming a 95% level of confidence. If the price of stock is currently $50, what is the expected range of prices using a 95% level of confidence.

Explanation / Answer

Expected return is calculable by formula:

Expected Return on Stock E(R) = 3% + 3% - 0.5% = 5.5%

Probability (Pi)

Return (Ri)

Pi * Ri

Sqrd Deviation

[E(R) – Ri]2

Sqrd Deviation * Pi

0.25

12%

3.00%

0.004225

0.00105625

0.5

6%

3.00%

0.000025

0.0000125

0.25

-2%

-0.50%

0.005625

0.00140625

(Standard Deviation)2 = (0.00105625 + 0.0000125 + 0.00140625) = 0.002475

Standard Deviation = 4.97%

Based on the statistics rule for normal distribution, for a normal distribution, 95% of confidence interval = Mean +/- 1.96(Standard Deviation)

Now, for our question, expected return is equivalent to mean.

Hence, 95% confidence interval range = [5.5% - (1.96 * 4.97%)] till [5.5% + (1.96 * 4.97%)] = -4.25% till 15.25%.

So, if stock price if $50, range of prices could be: $50* (1 – 4.25%) till $50 * (1 + 15.25%) = 47.875 till 57.625

Probability (Pi)

Return (Ri)

Pi * Ri

Sqrd Deviation

[E(R) – Ri]2

Sqrd Deviation * Pi

0.25

12%

3.00%

0.004225

0.00105625

0.5

6%

3.00%

0.000025

0.0000125

0.25

-2%

-0.50%

0.005625

0.00140625

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