Problem 1 (10 marks)

Three years ago, you purchased a bond for $974.69. The bond had three years to maturity, a coupon rate of 8% paid annually, and a face value of $1,000. Each year you reinvested all coupon interest at the prevailing reinvestment rate shown in the table below. Today is the bond's maturity date. What is your realized compound yield on the bond?

Time	Prevailing reinvestment rate
0 (purchase date)	6.0%
1	7.2%
2	9.4%
3 (maturity date)

Problem 2 (15 marks)

You will be paying $10,000 a year in education expenses at the end of the next two years. Currently the yield curve is flat at 8%.

• If you want to fully fund and immunize your obligation with a single issue of a zero-coupon bond, what maturity bond must you purchase? What must be the market value and the face value of the zero-coupon bond?
• Instead of using a single zero-coupon bond, you prefer to use a one-year T-Bill and a five-year zero-coupon bond to fund and immunize your obligation. How much of each security will you buy?

Problem 3 (15 marks)

A newly issued bond has the following characteristics:

Par value = $1000

Coupon rate = 8%

Yield to Maturity = 8%

Time to maturity = 15 years

Duration = 10 years

• Calculate modified duration using the information above. If the yield to maturity increases to 8.5%, what will be the change (in dollar amount) in bond price?
• Identify the direction of change in modified duration if:

i. the coupon of the bond is 4%, not 8%.

ii. the maturity of the bond is 7 years, not 15 years.

• How can you construct a portfolio with a duration of 8 years using this bond and a 5 year zero coupon bond?

Problem 4 (10 marks)

You have been provided with the following information zero coupon bonds with $1000 face value.

Maturity - semi -annual periods	semi-annual spot rates
1	4.25
2	4.15
3	3.95
4	3.70
5	3.50
6	3.25
7	3.05
8	2.90

• Compute the forward interest rates. Graph the yield curve.
• Explain the factors that account for the shape of the curve.

Problem 5 (10 marks)

Company HTA had a free cash flow for the firm (FCFF) of $1,500,000 last year. It is expected the FCFF will keep a sustainable growth rate of 5%. The company has 2 million common shares outstanding. In addition, the following information has been gathered:

Capital structure: D/.2:0.8?

Market value of Debt: V_D =$5,000,000;

Required return on equity: k_E %

Cost of debt before tax %

Tax rate: t_c %;

Determine the fair value of HTA stock.

Problem 6 (15 marks)

Company JUK has a ROE of 25% and the company will not pay any dividend for the next 3 years. It is estimated that the company will pay $2 dividend per share after three years and then to level off to 5% per year forever.

The company has a beta of 2. Assume the risk-free interest rate is 4%, and the market risk premium is 8%.

• What is your estimate of the fair price of a share of the stock? If the market price of a share is equal to this intrinsic value, what is the P/E ratio?
• What do you expect its price to be 1 year from now? Is the implied capital gain consistent with your estimate of the dividend yield and the market capitalization rate?

Problem 7 (10 marks)

MicroSense, Inc., paid $2 dividends per share last year. It is estimated that the company?s ROEs will be 12% and 10%, respectively, next two years. The plowback rate in next two years will be 0.6. It is expected that the dividends will grow at a sustainable rate of 3% per year after two years. Assume that the expected return on the market is 8%, the risk-free rate is 4%, and the beta of the stock is 1.4. What is the fair price of the stock?

Problem 8 (15 marks):

An analyst uses the constant growth model to evaluate a company with the following data for a company:

Leverage ratio (asset/equity): 1.8

Total asset turnover: 1.5

Current ratio: 1.8

Net profit margin: 8%

Dividend payout ratio: 40%

Earnings per share in the past year: $0.85

The required rate on equity: 15%

Based on an analysis, the growth rate of the company will drop by 25 percent per year in the next two years and then keep it afterward. Assume that the company will keep its dividend policy unchanged.

• Determine the growth rate of the company for each of next three years. Use the multi-period DDM to estimate the intrinsic value of the company?s stock.
• Suppose after one year, everything else will be unchanged but the required rate on equity will decrease to 14%. What would be your holding period return for the year?

Assignment 3 Instructions Assignment 3 should be submitted after you have completed Unit 5. This assignment is worth 15 percent of your final grade. Assignment 3 contains eight 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. Problem 1 (10 marks) Three years ago, you purchased a bond for $974.69. The bond had three years to maturity, a coupon rate of 8% paid annually, and a face value of $1,000. Each year you reinvested all coupon interest at the prevailing reinvestment rate shown in the table below. Today is the bond's maturity date. What is your realized compound yield on the bond? Time 0 (purchase date) 1 2 3 (maturity date) FNCE 401v6 Assignment 3 Prevailing reinvestment rate 6.0% 7.2% 9.4% Oct 9/2013 Problem 2 (15 marks) You will be paying $10,000 a year in education expenses at the end of the next two years. Currently the yield curve is flat at 8%. 1. If you want to fully fund and immunize your obligation with a single issue of a zerocoupon bond, what maturity bond must you purchase? 2. What must be the market value and the face value of the zero-coupon bond? 3. Instead of using a single zero-coupon bond, you prefer to use a one-year T-Bill and a fiveyear zero-coupon bond to fund and immunize your obligation. How much of each security will you buy? Problem 3 (15 marks) A newly issued bond has the following characteristics: Par value = $1000 Coupon rate = 8% Yield to Maturity = 8% Time to maturity = 15 years Duration = 10 years 1. Calculate modified duration using the information above. 2. If the yield to maturity increases to 8.5%, what will be the change (in dollar amount) in bond price? 3. Identify the direction of change in modified duration if: i. the coupon of the bond is 4%, not 8%. ii. the maturity of the bond is 7 years, not 15 years. 4. How can you construct a portfolio with a duration of 8 years using this bond and a 5 year zero coupon bond? FNCE 401v6 Assignment 3 Oct 9/2013 Problem 4 (10 marks) You have been provided with the following information zero coupon bonds with $1000 face value. Maturity - semi -annual periods semi-annual spot rates 1 4.25 2 4.15 3 3.95 4 3.70 5 3.50 6 3.25 7 3.05 8 2.90 1. Compute the forward interest rates. 2. Graph the yield curve. 3. Explain the factors that account for the shape of the curve. Problem 5 (10 marks) Company HTA had a free cash flow for the firm (FCFF) of $1,500,000 last year. It is expected the FCFF will keep a sustainable growth rate of 5%. The company has 2 million common shares outstanding. In addition, the following information has been gathered: Capital structure: D/E=0.2:0.8 Market value of Debt: VD =$5,000,000; Required return on equity: kE =15% Cost of debt before tax =6% Tax rate: tc =25%; Determine the fair value of HTA stock. FNCE 401v6 Assignment 3 Oct 9/2013 Problem 6 (15 marks) Company JUK has a ROE of 25% and the company will not pay any dividend for the next 3 years. It is estimated that the company will pay $2 dividend per share after three years and then to level off to 5% per year forever. The company has a beta of 2. Assume the risk-free interest rate is 4%, and the market risk premium is 8%. 1. What is your estimate of the fair price of a share of the stock? 2. If the market price of a share is equal to this intrinsic value, what is the P/E ratio? 3. What do you expect its price to be 1 year from now? Is the implied capital gain consistent with your estimate of the dividend yield and the market capitalization rate? Problem 7 (10 marks) MicroSense, Inc., paid $2 dividends per share last year. It is estimated that the company's ROEs will be 12% and 10%, respectively, next two years. The plowback rate in next two years will be 0.6. It is expected that the dividends will grow at a sustainable rate of 3% per year after two years. Assume that the expected return on the market is 8%, the risk-free rate is 4%, and the beta of the stock is 1.4. What is the fair price of the stock? FNCE 401v6 Assignment 3 Oct 9/2013 Problem 8 (15 marks): An analyst uses the constant growth model to evaluate a company with the following data for a company: Leverage ratio (asset/equity): 1.8 Total asset turnover: 1.5 Current ratio: 1.8 Net profit margin: 8% Dividend payout ratio: 40% Earnings per share in the past year: $0.85 The required rate on equity: 15% Based on an analysis, the growth rate of the company will drop by 25 percent per year in the next two years and then keep it afterward. Assume that the company will keep its dividend policy unchanged. 1. Determine the growth rate of the company for each of next three years. 2. Use the multi-period DDM to estimate the intrinsic value of the company's stock. 3. Suppose after one year, everything else will be unchanged but the required rate on equity will decrease to 14%. What would be your holding period return for the year? FNCE 401v6 Assignment 3 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 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