Assignment 1

Instructions

Assignment 1 should be submitted after you have completed Unit 2. This assignment is worth 15 percent of your final grade.

Assignment 1 contains five problems. The maximum mark for each problem is noted at the beginning of the problem. This assignment has a total of 100 marks.

Read the requirements for each problem and plan your responses carefully. Although your responses should be concise, ensure that you answer each of the required components as completely as possible. If supporting calculations are required, present them in good form.

When you receive your graded assignment, carefully review the comments the marker has made. This review component is an important step in your learning process. If you have any questions or concerns about the evaluation, please contact the Student Support Centre.

Problem 1 (15 marks)

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?

Problem 2 (15 marks)

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?

Problem 3 (15 marks)

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.

• What is the expected return for each stock? What is the standard deviation for each stock?
• 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?

Problem 4 (15 marks)

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(r_p).00%

?_p .00%

T-Bill .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%

Problem 5 (40 marks)

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:

(r) ? ?²

Answer following questions using EXCEL:

Draw the opportunity set offered by these two securities (with an increment of 0.01 in weight).

What is the optimal portfolio of SPY and XIU?

Determine your optimal asset allocation among SPY, XIU, and T-Bill, in percentage and in dollar amounts.

Assignment 1 Instructions Assignment 1 should be submitted after you have completed Unit 2. This assignment is worth 15 percent of your final grade. Assignment 1 contains five problems. The maximum mark for each problem is noted at the beginning of the problem. This assignment has a total of 100 marks. Read the requirements for each problem and plan your responses carefully. Although your responses should be concise, ensure that you answer each of the required components as completely as possible. If supporting calculations are required, present them in good form. When you receive your graded assignment, carefully review the comments the marker has made. This review component is an important step in your learning process. If you have any questions or concerns about the evaluation, please contact the Student Support Centre. Problem 1 (15 marks) 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. 1) If you want to buy BMO in margin, what is the maximum number of shares can you buy? 2) Suppose you want to buy 1200 shares of BMO in margin. Answer the following questions: a. What is the initial margin ratio? b. 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) c. How far could the stock price fall before getting a margin call? d. 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? FNCE 401v6 Assignment 1 Oct 9/2013 Problem 2 (15 marks) 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. 1) At what price would you face a margin call? 2) If the price is $86 at the end of the period, what is your margin at that point? 3) What would be your profit if you repurchase the stock at $63/share? Problem 3 (15 marks) Use the following expectations on stocks X and Y to answer the questions below: Probability Stock X Stock Y Bear Market 0.2 -20% -15% Normal Market 0.5 18% 20% Bull Market 0.3 50% 10% The correlation between stock X and Y is 0.4. 1) What is the expected return for each stock? 2) What is the standard deviation for each stock? 3) 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? Problem 4 (15 marks) 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% FNCE 401v6 Assignment 1 Oct 9/2013 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 Stock B Stock C Total 40% 25% 35% 100% 1) What is the expected return on your complete portfolio? 2) What is the standard deviation of your complete portfolio? 3) What are the dollar amounts of Stocks A, B, and C, respectively, in your complete portfolio? 4) If your degree of risk aversion is A=4, is your complete portfolio optimal? (assuming P is the optimal risky portfolio) FNCE 401v6 Assignment 1 Oct 9/2013 Problem 5 (40 marks) 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) - 2 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. FNCE 401v6 Assignment 1 Oct 9/2013 FNCE 401 Equations and Formulas Bond Equivalent Yield 1000 p 365 rBEY p n Where p is the bond price, n is the maturity of the bill in days. Bank Discount Yield 1000 p 360 rBDY 1000 n Where p is the bond price, n is the maturity of the bill in days. Bond Price Calculated from Bond Equivalent Yield 1000 p n 1 rBEY 365 Where rBEY is the bond equivalent yield, n is the maturity of the bill in days. Margin Ratio Margin Ratio Equity value Market value of assets - Loan Market value of assets Market value of assets Margin ratio related to short sales Margin Ratio Market value of assets Value of stock owed Net Asset Value Net asset value Market value of assets minus liabilites Shares outstandin g 1 Real interest rate, nominal interest rate, and inflation rate 1 + R = (1 + r)(1 + h) Where R = nominal rate: r = real rate: h = inflation rate. Holding-Period Return (HPR) HPR Ending price of a security - Beginning price Cash dividend (or Cash income) Beginning price Expected Return E(r) p ( s ) r ( s ) s Where p(s) is the probability of each scenario and r(s) is the holding-period return in each scenario. Variance of Return 2 p ( s )[r ( s ) E (r )] 2 s Where p(s) is the probability of each scenario, r(s) is the holding-period return in each scenario, and E (r ) is the expected return. Covariance between two-security returns Cov(ra , rb ) p( s)[ra ( s) E (ra )] [rb ( s) E (rb )] s Where p(s) is the probability of each scenario, r(s) is the holding-period return in each scenario, E(ra) is the expected return of security a, and E(rb) is the expected return of security b. Correlation Coefficient (ra , rb ) Cov (ra , rb ) a b 2 Utility function used in the textbook U E(r) - 1 A 2 2 Where U is the utility value and A is an index of the investor's aversion to taking on risk. Portfolio's Expected Return and Variance n E(rp ) wi E (ri ) i 1 n n n i 1 i 1 j1 i j P2 wi2 i2 wi w j Cov(ri , rj ) Where n is the total number of securities in the portfolio, and wi is the weight in the ith security. Capital Asset Pricing Model (CAPM) E(ri ) r f Cov(r i , rM ) M2 [ E(rM ) rf ] i [E(rM ) rf ] Where rf is the risk free rate, and rM is the rate of return on market portfolio. Bond Price Formula Bond price = PVCoupons + PVFace amount 1 1 1 Bond Price coupon (1 ) Par value T r (1 r ) (1 r ) T Where r is the yield to maturity and T is the time to maturity. Forward Rate and spot rate under rational expectations (1 y n ) n (1 f n 1 ) (1 y n 1 ) n 1 Where yn is the spot rate for pure discount bond with n periods to maturity, and fn+1 is the forward rate for period n. 3 Macaulay's duration formula T D t wt t 1 Where D is the duration value, wt Cash flow at time t 1 t bond price . (1 yield to maturity) Modified duration D* is equal to D / (1+y), where y is the yield to maturity. Duration and Bond Price Change (1 y) P -D D * y P 1 y Convexity Convexity at yield y 1 P(1 y) 2 n Cash flow at time t t 1 (1 y) t Then P 1 2 D * y Convexity y P 2 Dividend Discount Model V0 = (D1 + P1) / (1 + k) Constant growth: P0 = D1 / (k - g) Required return: k = (D1 / P0) + g Growth opportunities: P0 = (EPS / k) + PVGO g ROE b P 1 b E k g 4 (t 2 t ) ROE and its decomposition Debt ROE (1 TaxRate ) ROA ( ROA InterestRa te) Equity and ROE Net Pr ofit Pr etax Pr ofit EBIT Sales Assets Pr etax Pr ofit EBIT Sales Assets Equity Trin Statistic Trin Volume declining/ number declining Volume advancing/ number advancing Put-call parity theorem C P S 0 X (1 r f ) T Black-Scholes pricing formula C S 0 N (d 1 ) Xe rT N (d 2 ) Where d1 ln(S / X ) (r 2 / 2)T T . d 2 d1 T Spot futures price parity F0 S 0 (1 r f ) D S 0 (1 r f d ) Where F0 is the futures price, S0 is the current stock price, rf is the risk free rate, and d =D/S0 is the dividend yield. Commodity Futures Price F0 P0 (1 r f c) T Where c is the carrying cost. Commodity futures price when commodities are not stored. 5 1 rf F0 E ( PT ) 1 k T Interest rate parity 1 rhom e F0 E 0 1 r foreign T Where E0 is the current exchange rate in terms of number of home currency per unit of foreign currency, F0 is the forward price expressed in number of home currency per unit of foreign currency, and rhome and rforeign are respectively the home and foreign interest rate. Performance Measures Sharpe' s Measure : (rP rf ) Treynor' s Measure : P (rP rf ) P Jensen' s Measure : P rP [rf P (rM - rf )] Appraisal Ratio : P (e P ) M 2 Measure : rp * rM Note: Notation for the formulas follows that in the textbook of Bodie et al. Therefore, symbols in formulas are not all explained. Students are encouraged to familiarize themselves with the formula notation. It would be appreciated if errors could be communicated to Eric Wang at ericw@athabascau.ca. 6 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 investmento. 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=7038.5=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=4931.5=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 TBills. The information below refers to these assets. E(rp)=12.00% ?p =7.00% TBill rate=3.6% Proportion of TBill 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 tbills = 20% of 800,000 = 160,000 Amount in risky assets = 800,000 160,000 = 640,000 Interest on tbills = 3.6% of 160,000 = 5,760 Return on portfolio = 7% (given) = .07*640,000 = 44,800 Thus, expected return = interest on tbills+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(73.6) = 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) - 2 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