Many stocks in the security market

at
Categories
Tags

 

I. Suppose that there are many stocks in the security market and that the characteristics of stocks A and B are given as follows:
Stock Expected Return Standard Deviation
A 12% 6%
B 15% 8%
Correlation : -1

Suppose that it is possible to borrow at the risk-free rate Rf. What must be the value of the risk-free rate ?
(Hint: Think about constructing a risk-free portfolio from stocks A and B)

Please include your calculations for each question.

II. Suppose you have a project that has a 70% chance of doubling your investment in a year and a 30% change of halving your investment in a year.
What is the standard deviation of the rate of return on this investment?

Please include your calculations for each question.

III. Consider a risky portfolio. The end-of-year cash flow derived from the portfolio will be either 70000 USD or 190000 USD with equal probabilities of 50%. The alternative risk-free investment in T bills pays 5% per year.
Please include your calculations for each question.
III.a
If you require a risk premium of 8%, how much will you be willing to pay for the portfolio?
III.b
Suppose that the portfolio can be purchased for the amount you found in III.a. What will be the expected rate of return on the portfolio?
III.c
Now suppose that you require a risk premium of 12%. What is the price that you will be willing to pay?
III.d
Comparing your answers to III.a and III.c, what do you conclude about the relationship between the required risk premium on a portfolio and the price at which the portfolio will sell?

 

 

Sample Solution

I. Risk-Free Rate Calculation

We can construct a risk-free portfolio using Stocks A and B because their correlation is -1 (perfect negative correlation). This means when one stock goes up, the other goes down by the same amount, effectively canceling out the risk.

Steps to find the risk-free rate (Rf):

  1. Set the weights for the portfolio. Since we want a risk-free portfolio, we need to choose weights that completely offset the risk. Let weight (wA) be for Stock A and weight (wB) be for Stock B. As their returns are negatively correlated, we can choose wA to be positive and wB to be negative in a 1:1 ratio (e.g., wA = 1 and wB = -1). This ensures when A goes up, B goes down by the same proportion, eliminating risk.

  2. Calculate the expected return of the portfolio.

Expected Return (Rp) of portfolio = wA * Expected Return (A) + wB * Expected Return (B)

Rp = (1) * (12%) + (-1) * (15%) = -3%

Note: A negative expected return might seem counterintuitive, but it’s because this portfolio is designed to be risk-free, and potentially sacrificing some return is the trade-off.

  1. Since the portfolio is risk-free, its expected return must equal the risk-free rate (Rf).

Therefore, Rf = -3%.

Important Note: In reality, constructing a perfectly risk-free portfolio is challenging due to factors like transaction costs and potential imperfections in negative correlations. This is a theoretical example to illustrate the concept.

II. Standard Deviation of Project Return

This problem uses the concept of expected value and variance.

  1. Define outcomes and probabilities.
  • Outcome 1: Investment doubles (100% return) with a 70% chance (probability = 0.7).
  • Outcome 2: Investment halves ( -50% return) with a 30% chance (probability = 0.3).
  1. Calculate the expected return (E).

E = (Return in Outcome 1 * Probability 1) + (Return in Outcome 2 * Probability 2)

E = (100% * 0.7) + (-50% * 0.3) = 35%

  1. Calculate the squared deviations from the mean for each outcome.

  • (Outcome 1 return – Expected return)^2 = (100% – 35%)^2 = 42.25%

  • (Outcome 2 return – Expected return)^2 = (-50% – 35%)^2 = 72.25%

  1. Calculate the variance (σ^2).

σ^2 = [(Outcome 1 squared deviation * Probability 1) + (Outcome 2 squared deviation * Probability 2)]

σ^2 = [(42.25% * 0.7) + (72.25% * 0.3)] = 50.75%

  1. Calculate the standard deviation (σ).

σ = √σ^2 = √(50.75%) ≈ 7.1%

Therefore, the standard deviation of the rate of return on this investment is approximately 7.1%.

III. Risky Portfolio Valuation

III.a. Price with 8% Risk Premium

  1. Calculate the expected return (E) of the portfolio.

E = (Probability of high return * High return) + (Probability of low return * Low return)

E = (0.5 * $190,000) + (0.5 * $70,000) = $130,000

  1. Calculate the required return (Rr) considering the risk premium.

Risk-free rate (Rf) is not given, but we know the required risk premium (Rp) is 8%.

Rr = Rf + Rp (Assuming we know the risk-free rate)

Since we don’t have the actual Rf, let’s represent it as ‘x’. So, Rr = x + 8%.

  1. ** equate the expected return with the risk-free rate plus the risk premium.**

E = Rr

$130,000 = x + (8% of the unknown price)

Note: We cannot solve for the exact price without knowing the risk-free rate (x). However, we can conclude that the investor will be willing to pay an amount such that the expected return (considering the risk premium) equals the portfolio’s average return.

This question has been answered.

Get Answer