Question

Assume that a potential project has a 75% chance of doubling your investment in a year and a 25% chance of halving your investment in a year. What is the standard deviation of the rate of return on this investment?

Answer 1

The standard deviation of the rate of return on the investment with a 75% chance of doubling and a 25% chance of halving is approximately 0.649.

Step-by-step explanation:

The question asks for the standard deviation of the rate of return on an investment with a 75% chance of doubling and a 25% chance of halving. To calculate this, first we need to determine the expected return and then the standard deviation.

The expected rate of return (mean) is calculated as follows:

Expected Rate of Return = 0.75 x 2 + 0.25 x 0.5 = 1.625 or 162.5%

The returns are 2 (for doubling) and 0.5 (for halving), therefore:

• Variance = (2 - 1.625)² x 0.75 + (0.5 - 1.625)² x 0.25
• Variance = (0.375)² x 0.75 + (-1.125)² x 0.25
• Variance = 0.140625 x 0.75 + 1.265625 x 0.25
• Variance = 0.10546875 + 0.31640625
• Variance = 0.421875

The standard deviation is the square root of the variance, which can be calculated using a calculator as it is not a perfect square. If calculated, it gives:

Standard Deviation = √0.421875 ≈ 0.649 (approximately, in terms of the rate of return)

This is the standard deviation of the investment's return after one year.

Answer 2

To calculate the standard deviation of the rate of return on an investment with a 75% chance of doubling and a 25% chance of halving, we find the expected return and then the variance. After computing these values, we ascertain the standard deviation is approximately 64.95%.

Step-by-step explanation:

The student's question presents a scenario where an investment can either double or halve in a year, with corresponding probabilities of 75% and 25%. To compute the standard deviation of the rate of return, we first need to find the expected return and then the variance, before taking the square root of the variance for the standard deviation.

Let us denote the random variable for the rate of return as R. There are two possible outcomes for R:

• It can double, which represents a 100% return or a value of 1 (since rate of return is often expressed as a decimal), with a probability (P) of 0.75.
• It can halve, which represents a -50% return or a value of -0.5, with a probability (P) of 0.25.

The expected return (E(R)) is calculated as follows:
E(R) = (1 × 0.75) + (-0.5 × 0.25) = 0.75 - 0.125 = 0.625

The variance (Var(R)) is calculated with the formula:
Var(R) = ∑ (P × (R - E(R))^2)
= (0.75 × (1 - 0.625)^2) + (0.25 × (-0.5 - 0.625)^2)
= (0.75 × 0.140625) + (0.25 × 1.265625)
= 0.1055 + 0.3164
= 0.4219

Now, we calculate the standard deviation (SD) by taking the square root of the variance:
SD(R) = √Var(R)
= √0.4219 ≈ 0.6495 or 64.95%