Understanding Investment Returns, Margin, and Portfolio Optimization

Posted on Sep 5, 2024 in Mathematics

Question 2

Consider the three stocks in the following table. P_t represents the price in time t, and Q_t represents shares outstanding at time t. Stock C splits two for one in the last period.

a. Calculate the rate of return on a price-weighted index of the three stocks for the first period (t = 0 to t = 1).

b. What must happen to the divisor for the price-weighted index in year 2?

c. Calculate the rate of return for the second period (t = 1 to t = 2).

d. Calculate the first-period rates of return on the following indexes of the three stocks:

1. A market-value-weighted index.
2. An equally weighted index.

Answer

a. At t = 0, the value of the index is: (120 + 60 + 135)/3 = 105

At t = 1, the value of the index is: (135 + 52.5 + 150)/3 = 112.5

The rate of return is: (112.5/105) – 1 = 7.14%

b. In the absence of a split, Stock C would sell for 150, so the value of the index would be: 337.5/3 = 112.5

After the split, Stock C sells for 75. Therefore, we need to find the divisor (d) such that:

112.5 = (135 + 52.5 + 75)/d d = 2.33

c. The return is zero. The index remains unchanged because the return for each stock separately equals zero.

d.

1. Total market value at t = 0 is: ($48,000 + $48,000 + $108,000) = $204,000
2. Total market value at t = 1 is: ($52,000 + $42,000 + $120,000) = $216,000. Rate of return = ($216,000/$204,000) – 1 = 5.88%

An alternative solution:

The return on each stock is as follows:

r_A = (135/120) – 1 = 0.125 r_B = (52.5/60) – 1 = –0.125 r_C = (150/135) – 1 = 0.111

The value-weighted average is: [(48/204)(.125) + (48/204)(–0.125) + (120/204)(.111)] = 0.0588 = 5.88%

(ii) The equally-weighted average is: [(1/3)(.125) + (1/3)(–0.125) + (1/3)(.111)] = 0.037 = 3.70%

Question 3

In less than 150 words explain forward and futures contracts and the difference between them.

Answer

A forward contract is an agreement between two parties where one party agrees to deliver an asset on a specific date at an agreed-upon price, and the second party agrees to receive the asset. A futures contract is similar to a forward contract, except there’s a daily settlement where the buyer’s (long position) and seller’s (short position) accounts are settled daily (debited or credited), and margin calls are in place if needed. Futures contracts are usually more standardized and exchange-traded.

Question 4

Nicole Trader opens a brokerage account and purchases 1,000 shares of ABC at $100 per share. She borrows $50,000 from her broker to help pay for the purchase. The interest rate on the loan is 10%.

a. What is the margin in Nicole’s account when she first purchases the stock?

b. If the price falls to $80 per share by the end of the year, what is the remaining margin in her account? If the maintenance margin requirement is 30%, will she receive a margin call?

c. How far does the stock price have to fall for Nicole to get a margin call?

d. What is the rate of return on her investment?

Answer

a. The stock is purchased for: (1,000 x $100) = $100,000

The amount borrowed is $50,000. Therefore, the investor put up equity, or margin, of $50,000. This will make her account margin = $50,000/$100,000 = 50%

b. If the share price falls to $80, then the value of the stock falls to $80,000. By the end of the year, the amount of the loan owed to the broker grows to:

($50,000 x 1.10) = $55,000.

Therefore, the remaining margin in the investor’s account is:

($80,000 – $55,000) = $25,000.

The percentage margin is now: ($25,000/$80,000) = 0.3125 = 31.25%.

Therefore, the investor will not receive a margin call.

c. The value of the 1,000 shares is: 1,000P. Equity is: (1,000P – $50,000)

You will receive a margin call when: (1,000P – $50,000) / 1,000P = 0.30 when P = $71.43 or lower.

d. The rate of return on the investment over the year is:

(Ending equity in the account – Initial equity) / Initial equity = ($25,000 – $50,000) / $50,000 = -0.50 = -50.00%

Question 5

Casey Ambitious opened an account to short sell 40,000 shares of ABC from the previous problem. The initial margin requirement was 50%. (The margin account pays 5% interest.) A year later, the price of ABC has risen from $100 to $110, and the stock has paid a dividend of $3 per share.

a. What is the remaining margin in the account?

b. If the maintenance margin requirement is 30%, will Casey receive a margin call?

c. How high can the price of the stock go before she gets a margin call?

d. What is the rate of return on the investment?

Answer

a. The initial margin was: (0.50 x 40,000 x $100) = $2,000,000

As a result of the increase in the stock price, Casey loses:

($10 x 40,000) = $400,000. Therefore, the margin decreases by $400,000. Moreover,

Casey must pay the dividend of $3 per share to the lender of the shares, so that the margin in the account decreases by an additional $120,000. Therefore, the remaining margin is: [$2,000,000(1.05) – $400,000 – $120,000] = $1,580,000.

b. The percentage margin is: ($1,580,000/$4,400,000) = 0.3591 = 35.91%

Therefore, there will not be a margin call.

c. $2,000,000(1.05) – $40,000P / 40,000P = 0.30 when P = $115.38 or higher

d. The rate of return on the investment is:

(Ending equity in the account – Initial equity) / Initial equity = ($1,580,000 – $2,000,000) / $2,000,000 = -0.21 = -21.0%

Question 6

Your client needs your help regarding the following investment opportunities.

If the only information you have about your client is that s/he prefers more to less, what can you say about their desirability of these projects for your client, using stochastic dominance criteria? i.e., which investment is better? Show your work.

Answer

If the only information we have about our client is that s/he prefers more to less, then we can only use the First Degree Stochastic Dominance (FSD). To do that, we must compare the cumulative probability distribution of both options as follows:

Because the two cumulative probability distributions cross, none of the two distributions stochastically dominate the other, and thus we cannot choose between the two.

Question 7

The expected returns for ABC and XYZ are 10% and 20%, respectively. Their standard deviations are also 10% and 20%, respectively. If the correlation between returns of the two companies is -1, what combination of the two stocks will provide a minimum risk (standard deviation) portfolio? What is this minimum risk? Carefully show your work.

Answer

Let us consider ABC as security 1 and XYZ as security 2. The general formula for the variance of the portfolio is:

σ²_p = W²₁(σ²₁) + W²₂(σ²₂) + 2W₁W₂Corr(r₁, r₂). Since Correlation = -1, this will change to σ²_p = W²₁(σ²₁) + W²₂(σ²₂) – 2W₁W₂σ₁σ₂ = (W₁σ₁ – W₂σ₂)² = [W₁σ₁ – (1 – W₁)σ₂]²

Taking the derivative of σ²_p with respect to W₁ and equating it to zero will result in [W₁σ₁ – (1 – W₁)σ₂] = 0, which will be reduced to W₁(σ₁ + σ₂) = σ₂. Solving for W₁ will result in W₁ = σ₂ / (σ₁ + σ₂) = 20/(10+20) = 2/3.

This means W₂ = 1/3.

Note that one could reach the same result by simply setting σ²_p = [W₁σ₁ – (1 – W₁)σ₂]² to zero. To calculate the minimum risk, we need to replace σ₁ and σ₂ with their values (10 and 20) resulting in

σ²_p = [(2/3)(10) – (1/3)(20)]² = 0.

Question 8

Assume that the average variance of return for an individual security is 150 and the average co-variance is 30. What is the expected variance of an equally weighted portfolio of 5, 10, 20, 50, and 100 securities? Show your work.

Answer

As shown in chapter 6 of your text, the formula for the variance of an equally weighted portfolio (where W_i = 1/N for i = 1, …, N securities) is: σ²_P = 1/N[σ²_j + (N-1)σ_kj], where σ²_j is the average variance across all securities, σ_kj is the average covariance across all pairs of securities, and N is the number of securities. We can calculate the expected variance for various portfolios by replacing the appropriate values in this equation; for example, for N = 5, we obtain

σ²_P = (1/5)[150 + (5-1)30] = 54. Similarly, we can replace 10, 20, 50, and 100 for N in the formula with σ²_j = 150 and σ_kj = 30 to obtain the values in the following table:

Question 9

The expected returns for the bond portfolio (portfolio 1) and stock portfolio (portfolio 2) are 10% and 25%, respectively. Their standard deviations are also 10% and 25%, respectively. Assume the correlation between returns of the two portfolios is 0.3. Furthermore, assume that the risk-free lending and borrowing rate is 5%.

If you are a mean-variance optimizer and have a $100 million dollar to invest, how would you allocate this money amongst the risk-free asset, bond portfolio, and stock portfolio? Show your work.

Answer

To obtain the Markowitz portfolio, we need to solve the following system of equations:

z₁ = [R₁ – R_f / σ²₁] – [z₂(σ₁₂ / σ²₁)]

z₂ = [R₂ – R_f / σ²₂] – [z₁(σ₁₂ / σ²₂)]

Replacing all appropriate values we obtain

100z₁ = 75z₂ + 5

75z₁ = 625z₂ + 20

Solving will result in Z₁ = 1/35 and Z₂ = 1/35.

Therefore, W₁ = [(1/35)/(1/35 + 1/35)] = 1/2 and W₂ = [(1/35)/(1/35 + 1/35)] = 1/2.

The expected return of this portfolio of risky securities is:

(1/2)(10) + (1/2)(25) = 17.5%,

and the standard deviation of this portfolio of risky security is:

[(1/2)²(10)² + (1/2)²(25)² + 2(1/2)(1/2)(.30)(10)(25)]^1/2 = 14.79%

If you want to have a portfolio with a 10% return, you need to put (10/14.79) = 67.612% of your money ($67.612 million) on the risky portfolio. This money will be equally divided between bond and stock portfolios ($33.806 million each). The rest ($32,388) will be invested in the risk-free asset.

Question 10

:

Suppose you are buying an asset that you expect to provide you with $1,000/year perpetually but you are not sure about the risk of the asset. The risk‐free interest rate is 6 percent and the expected rate of return on the market portfolio is 16 percent. If you think the beta of the firm is 0.5, when indeed the beta is actually 1, how much more will you offer for the asset than it is truly worth?

Answer:

Assume that the $1,000 is a perpetuity. If beta is .5, the cash flow should be discounted at the rate

6 + .5  (16 – 6) = 11%

PV = 1000/.11 = $9,090.91

If, however, beta is equal to 1, the investment should yield 16%, and the price paid for the asset should be:

PV = 1000/.16 = $6,250

The difference, $2,840.91, is the amount you will overpay if you erroneously assumed that beta is .5 rather than 1.