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Finance Questions:Solutions Author: Instructor: Institution:

Problem 1 You have $30,000 in your margin account, and you want to invest in BMO stock. The minimum margin requirement for BMO is 30%. You just got a quote on BMO as follows: Bid: 55.25 Ask: 55.26 The interest rate on the margin loan is 6% per annum. If you want to buy BMO in margin, what is the maximum number of shares can you buy? Suppose you want to buy 1200 shares of BMO in margin. Answer the following questions: What is the initial margin ratio? Suppose you are going to hold the shares for one year. At what price at the end of next year will your investment break even? (assuming no margin calls in the year) How far could the stock price fall before getting a margin call? If the stock price falls to $40, you would get a margin call. If this happens, how much new fund would you need to add to your account to respond the margin call?

Solution The shares will be bought at the ask price, as it is higher than the bid price. So, cost of buying 1 share is 55.26 Amount for 1200 shares = no. of shares*price per share = 1200*55.26 = 66,312 Amount in margin account = 30,000. Thus amount required on margin = Total purchase price amount in margin account. = 66,312 - 30,000 = 36,312 Initial margin ratio = amount in margin account/investment purchased on margin = 30,000/36,312 = 0.83

Interest rate = 6%. Interest amount for 1 year = interest rate*amount of loan = 6% of 36,312 = 2,179 Total investment on 1,200 shares = amount paid for purchase + loan interest = 66,312+2,179 = 68,491 Break even price = total investment/no. of shares = 68,491/1,200 = 57.08 (break even price) Let the price be (after falling) "x". Total price = 1,200x 30% of margin is required. Balance after margin = 70% 70% of 1,200 x = 840x So 840x = 30,000 (amount of initial margin) or x = 35.71. If the stock price falls to 35.71 or lower amount, you will get a margin call. at $40, value of investment = 40*1200 = 48,000 70% of 48,000 = 33,600 So new fund needed = Current margin value - amount in margin account = 33,600 - 30,000 = 3,600

Problem 2 Assume you sell short 100 shares of common stock at $70 per share, with initial margin at 55%. The minimum margin requirement is 30%. The stock will pay no dividends during the period, and you will not remove any money from the account before making the offsetting transaction. At what price would you face a margin call? If the price is $86 at the end of the period, what is your margin at that point? What would be your profit if you repurchase the stock at $63/share?

Solution Initial Margin =55%=70*0.55=38.5

Money borrowed=70-38.5=31.5 Maintenance Margin=30%=70*0.3=21 Maximum allowable borrowed funds=100%-30%=70%=0.7*70=49 There will be margin call when the borrowed money reaches 49 or loss=49-31.5=17.5 A loss of 17.5 will occur when underlying rises by 17.5 as we are short on the underlying or , underlying closes at 70+17.5= 87.5 Thus at 87.5 we will face margin call At 86, the margin = initial margin +profit/loss= (38.5-(86-70))*100=$2250 Profit If I repurchase the stock at 63=(70-63)*100=$700

Problem 3 Use the following expectations on stocks X and Y to answer the questions below: Bear Market

Normal Market

Bull Market

Probability

0.2

0.5

0.3

Stock X

-20%

18%

50%

Stock Y

-15%

20%

10%

The correlation between stock X and Y is 0.4. a.

What is the expected return for each stock?

b.

What is the standard deviation for each stock?

c.

Assume you invest your $100,000 in a portfolio with $90,000 in stock X and $10,000 in stock Y. What are the expected return and standard deviation of your portfolio?

Solution

Problem 4 You have $800,000 invested in a complete portfolio that consists of a portfolio of risky assets (P) and T-Bills. The information below refers to these assets. E(rp)=12.00% ?p =7.00% T-Bill rate=3.6% Proportion of T-Bill in the complete portfolio: 20% Proportion of risky portfolio P in the complete portfolio: 80% Composition of P: Stock A

40%

Stock B

25%

Stock C

35%

Total

100%

What is the expected return on your complete portfolio? What is the standard deviation of your complete portfolio? What are the dollar amounts of Stocks A, B, and C, respectively, in your complete portfolio? If your degree of risk aversion is A=4, is your complete portfolio optimal? (assuming P is the optimal risky portfolio)

Solution

Total investment = 800,000 Amount in t-bills = 20% of 800,000 = 160,000 Amount in risky assets = 800,000 - 160,000 = 640,000 Interest on t-bills = 3.6% of 160,000 = 5,760 Return on portfolio = 7% (given) = .07*640,000 = 44,800 Thus, expected return = interest on t-bills+return on stocks = 5,760+44,800 = 50,560 rate of return = 50,560/800,000 = 6% Dollar amount of stocks: Stock A = 40% . P amount = 640,000. Thus A = .4*640,000 = 256,000 Stock B = .25*640,000 = 160,000 Stock C = .35*640,000 = 224,000 Degree of risk aversion = 4 (beta) Using CAPM = risk free rate+beta*(market return - risk free rate) 3.6%+4(7-3.6) = 3.6+13.6 = 17.2% As, return on portfolio is less than return per CAPM, it is not an optimal portfolio

Problem 5 SPY and XIU are ETFs tracking the S&P 500 and S&P/TSX 60 index, which are often used as proxies for the US and Canadian stock markets, respectively. From a set of their historical data, the annual expected returns and standard deviations of those two ETFs and their covariance are estimated as follows: SPY: E(r) = 0.36

0.26

XIU: E(r) = 0.44 0.28

Covariance between= 0.0568 Suppose that you have $5 million to invest for one year and you want to invest this money into SPY, XIU and the Canadian one-year T-bill. Assume that the interest rate of the one-year T-Bill is 6% per annum. Suppose that you have the following utility function: U=E(r) – σ2 Answer following questions using EXCEL: 1 Draw the opportunity set offered by these two securities (with an increment of 0.01 in weight). 2 What is the optimal portfolio of SPY and XIU? 3 Determine your optimal asset allocation among SPY, XIU, and T-Bill, in percentage and in dollar amounts.

Solution

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Problem 1 You have $30,000 in your margin account, and you want to invest in BMO stock. The minimum margin requirement for BMO is 30%. You just got a quote on BMO as follows: Bid: 55.25 Ask: 55.26 The interest rate on the margin loan is 6% per annum. If you want to buy BMO in margin, what is the maximum number of shares can you buy? Suppose you want to buy 1200 shares of BMO in margin. Answer the following questions: What is the initial margin ratio? Suppose you are going to hold the shares for one year. At what price at the end of next year will your investment break even? (assuming no margin calls in the year) How far could the stock price fall before getting a margin call? If the stock price falls to $40, you would get a margin call. If this happens, how much new fund would you need to add to your account to respond the margin call?

Solution The shares will be bought at the ask price, as it is higher than the bid price. So, cost of buying 1 share is 55.26 Amount for 1200 shares = no. of shares*price per share = 1200*55.26 = 66,312 Amount in margin account = 30,000. Thus amount required on margin = Total purchase price amount in margin account. = 66,312 - 30,000 = 36,312 Initial margin ratio = amount in margin account/investment purchased on margin = 30,000/36,312 = 0.83

Interest rate = 6%. Interest amount for 1 year = interest rate*amount of loan = 6% of 36,312 = 2,179 Total investment on 1,200 shares = amount paid for purchase + loan interest = 66,312+2,179 = 68,491 Break even price = total investment/no. of shares = 68,491/1,200 = 57.08 (break even price) Let the price be (after falling) "x". Total price = 1,200x 30% of margin is required. Balance after margin = 70% 70% of 1,200 x = 840x So 840x = 30,000 (amount of initial margin) or x = 35.71. If the stock price falls to 35.71 or lower amount, you will get a margin call. at $40, value of investment = 40*1200 = 48,000 70% of 48,000 = 33,600 So new fund needed = Current margin value - amount in margin account = 33,600 - 30,000 = 3,600

Problem 2 Assume you sell short 100 shares of common stock at $70 per share, with initial margin at 55%. The minimum margin requirement is 30%. The stock will pay no dividends during the period, and you will not remove any money from the account before making the offsetting transaction. At what price would you face a margin call? If the price is $86 at the end of the period, what is your margin at that point? What would be your profit if you repurchase the stock at $63/share?

Solution Initial Margin =55%=70*0.55=38.5

Money borrowed=70-38.5=31.5 Maintenance Margin=30%=70*0.3=21 Maximum allowable borrowed funds=100%-30%=70%=0.7*70=49 There will be margin call when the borrowed money reaches 49 or loss=49-31.5=17.5 A loss of 17.5 will occur when underlying rises by 17.5 as we are short on the underlying or , underlying closes at 70+17.5= 87.5 Thus at 87.5 we will face margin call At 86, the margin = initial margin +profit/loss= (38.5-(86-70))*100=$2250 Profit If I repurchase the stock at 63=(70-63)*100=$700

Problem 3 Use the following expectations on stocks X and Y to answer the questions below: Bear Market

Normal Market

Bull Market

Probability

0.2

0.5

0.3

Stock X

-20%

18%

50%

Stock Y

-15%

20%

10%

The correlation between stock X and Y is 0.4. a.

What is the expected return for each stock?

b.

What is the standard deviation for each stock?

c.

Assume you invest your $100,000 in a portfolio with $90,000 in stock X and $10,000 in stock Y. What are the expected return and standard deviation of your portfolio?

Solution

Problem 4 You have $800,000 invested in a complete portfolio that consists of a portfolio of risky assets (P) and T-Bills. The information below refers to these assets. E(rp)=12.00% ?p =7.00% T-Bill rate=3.6% Proportion of T-Bill in the complete portfolio: 20% Proportion of risky portfolio P in the complete portfolio: 80% Composition of P: Stock A

40%

Stock B

25%

Stock C

35%

Total

100%

What is the expected return on your complete portfolio? What is the standard deviation of your complete portfolio? What are the dollar amounts of Stocks A, B, and C, respectively, in your complete portfolio? If your degree of risk aversion is A=4, is your complete portfolio optimal? (assuming P is the optimal risky portfolio)

Solution

Total investment = 800,000 Amount in t-bills = 20% of 800,000 = 160,000 Amount in risky assets = 800,000 - 160,000 = 640,000 Interest on t-bills = 3.6% of 160,000 = 5,760 Return on portfolio = 7% (given) = .07*640,000 = 44,800 Thus, expected return = interest on t-bills+return on stocks = 5,760+44,800 = 50,560 rate of return = 50,560/800,000 = 6% Dollar amount of stocks: Stock A = 40% . P amount = 640,000. Thus A = .4*640,000 = 256,000 Stock B = .25*640,000 = 160,000 Stock C = .35*640,000 = 224,000 Degree of risk aversion = 4 (beta) Using CAPM = risk free rate+beta*(market return - risk free rate) 3.6%+4(7-3.6) = 3.6+13.6 = 17.2% As, return on portfolio is less than return per CAPM, it is not an optimal portfolio

Problem 5 SPY and XIU are ETFs tracking the S&P 500 and S&P/TSX 60 index, which are often used as proxies for the US and Canadian stock markets, respectively. From a set of their historical data, the annual expected returns and standard deviations of those two ETFs and their covariance are estimated as follows: SPY: E(r) = 0.36

0.26

XIU: E(r) = 0.44 0.28

Covariance between= 0.0568 Suppose that you have $5 million to invest for one year and you want to invest this money into SPY, XIU and the Canadian one-year T-bill. Assume that the interest rate of the one-year T-Bill is 6% per annum. Suppose that you have the following utility function: U=E(r) – σ2 Answer following questions using EXCEL: 1 Draw the opportunity set offered by these two securities (with an increment of 0.01 in weight). 2 What is the optimal portfolio of SPY and XIU? 3 Determine your optimal asset allocation among SPY, XIU, and T-Bill, in percentage and in dollar amounts.

Solution

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