Unformatted text preview: MGTB09H3 – Principles of Finance Fall 2010 Additional selfstudy exercises topics 2, 3 and 4 (answers are given at the end) First make these exercises on your own, and then check the solutions! Questions Question 1 (Time Value of Money) Khayat & Elkind Financial Planners offered Eileen Chan two different investment plans. Plan A is a perpetuity that pays $4,500 per year. Plan B is a 10 year, $15,000 annual annuity. Both plans will make their first payment two years from today. At what discount rate would Eileen be indifferent between these two plans? Question 2 (TVM) Katherine Hui would like to purchase a new Toyota from Bai & Becz Autos. She will finance the car purchase by borrowing $20,000. The loan contract is in the form of a 30 month annuity due at 12.685% EAR. a) Calculate the monthly interest rate. b) What will her monthly payment be? Question 3 (TVM) Consider an investment that pays $2,000 every third year forever. The annual discount rate is 10%. a) What is the value of the investment when the first cash flow occurs one year from now? b) What would the value be if the first cash flow occurs in two years? c) In four years? Question 4 (TVM) Florence Lee recently bought a house in Scarborough. To finance the purchase she took a mortgage loan of $250,000 from the Christidis & Ribeiro Trust Company for an amortization period of 25 years at a quoted interest rate of 8% per year compounded semiannually. a) What is her monthly payment? Florence makes payments at the end of each month. b) After 10 years of payments how much does she still owe to the Trust Company? c) What is the total amount of interest (in $) that she has paid during these 10 years? Question 5 (TVM) Navpreet Singh is about to enter university and has two options to her. 1. Study Business. If she does this, her undergraduate degree would cost her $12,000 a year for 4 years. Having obtained this, she would need to get two years of practical experience: in the first year she would earn $30,000, and in the second year she would earn $35,000. She 1 then would need to obtain her MBA degree, which will cost her $20,000 per year for 2 years. After that she will be fully qualified and can earn $80,000 per year for 21 years. 2. Study Science. If she does this, her undergraduate degree would cost her $15,000 a year for 4 years and then she would earn $50,000 a year for 25 years. All earnings and costs are paid at the end of the year. What advice would you give her if the applicable rate of return were 10% per year? Question 6 (TVM) Sheila Santos is considering the following three investments offered by Angelo Simoes & Navneet Hans Investments Limited. The investments will pay: 1. $60,000 every four years forever, with the first payment occurring one year from today. 2. $5,000 every quarter for 20 years (80 payments in total), with the first payment occurring two months from today. 3. $10,000 every 6 months for 25 years (50 payments in total) starting 2 years from today. Which of the above 3 investments should she select if each investment costs $140,000 and her required rate of return is 12% per year? Question 7 (TVM) Vivian Tsang has just bought a house in Scarborough. To help finance the purchase she took a mortgage loan of $300,000 at 8% interest rate compounded semiannually. Amortization period for the mortgage loan is 20 years. a) Determine how much she has to pay at the end of every month to pay off the loan in 20 years. b) How much will she owe to the bank after making payments for 15 years? (i.e., what is the mortgage balance after 15 years?) c) Suppose two years after taking the mortgage, she wants to end the mortgage by paying off the remaining balance. However, the bank will charge her a penalty for doing this. The penalty equals the amount of interest she would have paid in the next 3 months. (Hence, the interest she would have to pay in the first three months when there are still 18 years left of maturity.) Determine the amount of interest penalty she has to pay. Question 8 (Bonds) Valesquez & Vinokour Manufacturing has two different bonds currently outstanding. Bond X has a face value of $100,000 and matures in 20 years. The bond makes no payments for the first 4 years, then pays $6,000 semiannually over the subsequent 6 years, after that pays $8,000 semiannually for 5 years, and finally pays $9,000 semiannually over the last 5 years. Bond Y has a face value of $100,000 and a maturity of 10 years; it makes no coupon payments over the life of the bond. If the required rate of return on both bonds is 12% compounded semiannually, what is the current price of Bond X? And of Bond Y? Question 9 (Bonds) Truong & Llewellyn Enterprises issued a new series of bonds on January 1, 1976. The bonds were sold at par ($1,000), had a 12% coupon, and had an original maturity of 30 years. Coupon payments are made annually on December 31 of each year. a) What was the yield to maturity of the bond on January 1, 1976? b) What was the price of the bond on January 1, 1981 (5 years later), assuming that the level of interest rates had fallen to 10%? 2 c) Calculate the bond’s current yield on January 1, 1996 (20 years after the bonds were issued), assuming that the interest rate was 13% at that time. Question 10 (Stocks) Jingshu & Serhad Manufacturing (JSM) is experiencing rapid growth. JSM has just paid a dividend and the stock price is currently equal to $80 per share. Dividends are expected to grow by 40% per year during the next 2 years, 20% during the 3rd year, 10% during the 4th year, and then 5% per year indefinitely. The required rate of return is 10% per year. a) Calculate the dividend that JSM has just paid out. b) What is the projected dividend for the coming year? Question 11 (Stock) You can buy a share of the Mary Mina & Vivian Leung Technologist’s (MLT) stock today for $24. MLT’s last dividend was $1.60. In view of the MLT’s low risk, its required rate of return is only 12%. Dividends are expected to grow at a constant rate, g, in the future. Assume that the required rate of return is expected to remain at 12%. What is MLT’s expected share price 5 years from now? Question 12 (Stock) Mayur Gandhi & Daniel Tsang Computers (MDC) is experiencing a period of rapid growth. Earnings and dividends are expected to grow at a rate of 18% during the next two years, at 15% in the 3rd year, and at a constant rate of 6% thereafter. MDC’s last dividend was $1.15, and the required rate of return on the shares is 12%. a) Calculate the price of a share today. b) Calculate P1 and P2. c) Calculate the dividend yield, capital gains yield, and total return for years 1, 2, and 3. Question 13 (Stocks) Anar & Rebecca Wang Technologies (A&RWT) is a young startup company. No dividends will be paid on the stock for the next 2 years. First dividend will be paid 3 years from today. Dividends are expected to grow at 25% per year during the 4th and the 5th year, 15% during the 6th year, 10% per year during the 7th and 8th year, and then 5% per year indefinitely. A&RWT stock is currently selling for $107.673 per share, and the required rate of return is 10%. a) What dividend will be paid 3 years from today? b) If you purchase the stock 7 years from today and sell it after holding it for one year, what will be the dividend yield, capital gain yield, and the total yield? 3 Solutions Answer 1 (TVM) Find the discount rate for which both plans have the same present value. Both plans have their first payment two years from now, so we can just compare the PVs at time 1 (one period before the first payment). $4,500/k = $15,000((1‐1/(1+k)10)/k) $4,500 = $15,000(1‐1/(1+k)10) 0.30 = 1‐1/(1+k)10 10 0.70 = 1/(1+k) 1/0.70 = (1+k)10 k = (10/7)1/10‐1 = 3.63% Answer 2 (TVM) a) She will make monthly payments, so we first need to calculate the monthly interest rate. EAR = 12.6825% Monthly rate = (1.126825)1/12‐1 = 0.01 b) Next, use the formula for the present value of an annuity due to calculate her monthly payment. 11,218.91. Answer 3 (TVM) a) The cash flows take place every three years, we therefore need to calculate the three‐year effective interest rate. Effective 3 year rate = (1.10)3‐1 = 0.331 = 33.1% We use the formula for the PV of a perpetuity PV=PMT/k. This formula assumes the first payment occurs one period from today. The first cash flow occurs one year from now. Hence, by using this formula, we calculate the PV two years ago (i.e., one three‐year period before the first cash flow). PVtwo years ago = $2,000/0.331 = $6,042.30 Calculate the value today (at time 0): (use formula for FV, with n=2 years and k=0.10 per year) PVtoday = $6,042.30(1.1)2 = $7,311.18 b) If the first cash flow occurs two years from now, the formula for the PV of a perpetuity gives us the PV one year ago (i.e. one three‐year period before the first cash flow). Hence, PVone year ago = $6,042.30. In order to calculate the PV today, calculate FV with n=1 (and k=0.10 per year): PVtoday = $6,042.30(1.1) = $6,646.53 c) If the first cash flow occurs four years from now, the formula for the PV of a perpetuity gives us the PV one year from now (i.e. one three‐year period before the first cash flow). We need to discount this value back by one year (using k=0.10 per year) to calculate the value today: PVtoday = $6,042.3/1.1 = $5,493 4 Answer 4 (TVM) a) To calculate the monthly payment, we first need to calculate the monthly interest rate. We know the quoted rate of 8% which is based on semi‐annual compounding. Hence, first calculate the EAR. EAR = (1+0.08/2)2‐1 = 0.0816 = 8.16% Next, calculate the effective monthly rate: Monthly rate = (1+EAR)1/12‐1 = 1.08161/12 – 1 = 0.006558 = 0.6558% We know that the value of the mortgage today should equal the PV of all future payments. The payments occur at the end of each month and thereby can be seen as an ordinary annuity. The total mortgage Florence takes equals the PV of this annuity. Now, calculate the monthly payment using the formula for the PV of an ordinary annuity. Everything is in months, hence: n=25 years x 12 =300 months. 1
⎡
⎢1 − 1.006558 300
250,000 = PMT × ⎢
⎢ 0.006558
⎢
⎣ ⎤
⎥
⎥ ⇒ PMT = $1,908 ⎥
⎥
⎦ b) The remaining balance of a loan or mortgage at a certain point in time always equals the present value of all future payments. After 10 years of payments, there are still 15 years of payments to go → the remaining number of monthly payments is 15 × 12 = 180. Hence, use the formula for the PV of an ordinary annuity, with n=180, pmt = $1980 and k=0.6558%. 1
⎡
⎢1 − 1.006558180
Loan _ balance = 1908 × ⎢
⎢ 0.006558
⎢
⎣ ⎤
⎥
⎥ = 201,236 ⎥
⎥
⎦ c) Next, we need to calculate the total amount of interest paid over the first 10 years. To this end, we first calculate the total amount of money Florence has paid to the bank over the past 10 years. She has paid $1908 per month, for 120 months. Hence: Total paid to the bank = $1908 × 120 = $228,960 Part of this was used to repay the loan, and part of this was interest. We know that the balance of the loan has reduced from $250,000 to $201,236. Hence: Reduction in the loan = 250,000‐201,236 = $48,764 The rest of the amount of money that Florance has paid to the bank in the first ten years was interest payment: Interest paid = 228,960‐48,764 = $180,196 5 Answer 5 (TVM) We will calculate the present values (today, at time 0) of both options. Navpreet should choose the option with the highest PV. Option 1 First, let’s create a timeline. Cash inflows (e.g. earn salary) are positive and cash outflows (e.g. pay for undergraduate degree) are negative. CF ‐12K ‐12K ‐12K ‐12K 30K 35K ‐20K ‐20K 80K … 80K
time 0 1 2 3 4 5 6 7 8 9 … 29 We need to discount all of these cash flows back to time 0. First, we calculate the PV of the last 21 annual salaries of $80,000, since this is an ordinary annuity. The first $80,000 payment occurs at time 9, hence using the formula for the PV of an ordinary annuity will give us the PV at time 8 (i.e. one period before the first cash flow of the annuity). Hence: 1⎤
⎡
1−
21 ⎥
⎢
PV8 = 80,000 × ⎢ 1.10 ⎥ = $691,895.54 ⎢ 0.10 ⎥
⎣
⎦
Next, discount PV8 and all other cash flows back to time 0 (and recognizing that the first four payments are an ordinary annuity as well): 1
⎡
⎢1 − 1.10 4
PV0 = −12,000 × ⎢
⎢ 0.10
⎢
⎣
PV0 = $303,526.91 ⎤
⎥ 30,000 35,000 20,000 20,000 691,895.54
+
−
−
+
⎥+
5 1.10 6
1.10 7
1.10 8
1.10 8
⎥ 1.10
⎥
⎦ Option 2 Again, make a timeline: CF ‐15K ‐15K ‐15K ‐15K 50K … 50K time 0 1 2 3 4 5 … 29 First calculate the present value of the 25 annual salaries starting at time 5. The formula for the PV of an ordinary annuity will give us the PV at time 4 (one period before the first salary). 1
⎡
⎢1 − 1.10 25
PV4 = 50,000 × ⎢
⎢ 0.10
⎢
⎣ ⎤
⎥
⎥ = $453,852.00 ⎥
⎥
⎦ The PV at time 0 of these salaries equals:
PV0 = 453,852.00
= $309,987.02 1.10 4 Calculate the PV at time zero of the four annual college payments: 6 1
⎡
⎢1 − 1.10 4
PV0 = −15,000 × ⎢
⎢ 0.10
⎢
⎣ ⎤
⎥
⎥ = −$47,547.98 ⎥
⎥
⎦ The total PV at time 0 of the second option equals $309,987.02 ‐ $47,547.98 = $262,439.04 The total PV of option 1 is higher → she should go with the first option. Answer 6 (TVM) Calculate the PV at time 0 of the future cash flows of each of the three investments. Then, whenever an investment has a PV that exceeds the initial cost of $140,000, it generates more money than it costs. Sheila should select the investment with the highest PV. 1. The cash flows occur every four years. Hence, first calculate the effective rate over a four‐year period: 4 yr rate = (1.12)4‐1 = 0.5735 = 57.35% We use the formula for the PV of a perpetuity PV=PMT/k. This formula assumes the first payment occurs one period from today. The first cash flow occurs one year from now. Hence, by using this formula, we calculate the PV three years ago (i.e., one four‐year period before the first cash flow). PV3 years ago = $60,000/0.5735 = $104,620.75 Calculate the value at time 0 by calculating the FV, three years ahead using k=0.12 per year. PV at t=0 = $104,620.75(1.12)3 = $146,984.62 2. Payments occur every quarter and hence, we need to calculate the quarterly rate. Quarterly rate = (1+0.12)1/4‐1 = 0.028737 = 2.8737% The 80 quarterly payments can be seen as an ordinary annuity. When using the formula for the PV of an ordinary annuity, we always calculate the PV one period before the first cash flow. The first payment occurs two months from now. Hence, we calculate the PV one month ago. PV1 _ month _ ago 1
⎡
1−
80
⎢
= $5,000 × ⎢ 1.028737
⎢ 0.028737
⎢
⎣ ⎤
⎥
⎥ = $155,954 ⎥
⎥
⎦ In order to calculate the value today (a future value where n=1 months), we first need to calculate the monthly rate: = 0.009489 = 0.9489% Monthly rate = (1.12)1/12‐1 Now, calculate the PV at time 0 of this investment: PV at t=0 = $155,954 ×1.009489 = $157,434 3. First, calculate the semiannual rate: Semiannual rate = (1.12)1/2‐1 = 7 0.0583 = 5.83% The 50 semiannual payments can be seen as an ordinary annuity, with the first payment two years from now. We first calculate their PV one six‐month period before the first payment, one and a half year from now. 1
⎛
⎞
⎜1−
50 ⎟
(10583) ⎟
.
PV1.5 years from now = $10,000⎜
= $161,437 ⎜ 0.0583 ⎟
⎜
⎟
⎝
⎠ To calculate the value today, we should discount this back by three more six month periods: = $136,200 PV at t=0 = $161,437/(1.0583)3 The first two investments have a PV that exceeds the cost of $140,000. The second alternative has the highest PV and therefore, Sheila should select the second investment. Answer 7 (TVM) a) First, calculate the effective annual rate: EAR =( 1+ 0.08/2)2‐1 = 0.0816 The monthly rate equals: Monthly rate = (1+EAR)1/12‐1 = 1.08161/12 – 1 = 0.006558 = 0.6558% Calculate the monthly payment of the mortgage, using the formula for the PV of an ordinary annuity (n= 12 × 20 = 240 monthly payments): $300,000 = 1
⎛
⎞
⎜1−
240 ⎟
.
⎜ (10065582) ⎟ × C ⎜
0.0065582 ⎟
⎜
⎟
⎝
⎠ $300,000 = 120.7208C C = $2,485.07 Hence, her monthly payment is $2,485.07. b) The balance of a loan or mortgage is the PV ofall future payments. After 15 years, there are still 5 years of monthly payments of $2,485.07 left. Hence, calculate the PV of these remaining 60 monthly payments: Amount owed after 15 years = 1
⎛
⎞
⎜1−
⎟
.
⎜ (10065582)60 ⎟ $2,485.07
⎜
⎟
0.0065582
⎜
⎟
⎝
⎠ = $122,937.10 c) We have to calculate the amount of interest that needs to be paid during three consecutive months starting two years from now. Mortgage payments always consist of repayments of the mortgage and interest payments. During a three‐month period, the total amount of mortgage payments is: Three months payments = $2,485.07 ×3 = $7,455.21 8 Next, we calculate the total repayment of the mortgage during these three months. We do this by comparing the mortgage balance after two years with the mortgage balance after two years and three months. After 2 years, there are still 18 years left = 18 × 12 = 216 monthly payments Amount owed after 2 years = 1
⎛
⎞
⎜1−
⎟
.
⎜ (10065582)216 ⎟ $2,485.07
⎜
⎟
0.0065582
⎜
⎟
⎝
⎠ =$286,593.43 After two years and three months, there are still 216 – 3 = 213 monthly payments left: 1
⎛
⎞
⎜1−
.
(10065582)213 ⎟ = $284,764.82 ⎟
Amount owed after 2 years and 3 months = $2,485.07 ⎜
⎜
⎟
0.0065582
⎜
⎟
⎝
⎠
Hence, during these three months, the mortgage principal has been reduced by: Reduction in Principal = $286,593.43‐284,764.82 = 1,828.61 The interest payments during these three months (which is the penalty) equal the total mortgage payments minus the repayments: Penalty = 3 month interest = $7,455.21‐1,828.61 = $5,626.60 Answer 8 (Bonds) The price of a bond equals the PV of all its future cash flows. Bond X: First, make a timeline (everything in six‐month periods). Cash 6,000 … 6,000 8,000 … 8,000 9,000 … 9,000 flow 100,000
Time 0 1 .. 8 9 … 20 21 … 30 31 … 40 The semi‐annual discount rate equals 0.12 / 2 = 0.06. • Step 1: Calculate the value at t= 8 of the 12 semi‐annual payments of $6,000 that start at t=9. 1
⎡
⎢1 − 1.0612
PV8 = 6000 × ⎢
⎢ 0.06
⎢
⎣
• ⎤
⎥
⎥ = $50,303.06 ⎥
⎥
⎦ Now, calculate the value at time 0: PV0 = $50,303.06 / 1.068 = $31,560.76 Step 2: Calculate the value at t=20 of the 10 semi‐annual payments of $8,000 that start at t=21. 1
⎡
⎢1 − 1.0610
PV20 = 8000 × ⎢
⎢ 0.06
⎢
⎣ ⎤
⎥
⎥ = $58,880.70 ⎥
⎥
⎦
9 • Now, calculate the value at time 0: PV0 = $58,880.70 / 1.0620 = $18,359.28 Step 3: Calculate the value at t=30 of the 10 semi‐annual payments of $9,000 that start at t=31. 1
⎡
⎢1 − 1.0610
PV30 = 9000 × ⎢
⎢ 0.06
⎢
⎣ ⎤
⎥
⎥ = $66,240.87 ⎥
⎥
⎦ Now, calculate the value at time 0: PV0 = $66,240.87 / 1.0630 = $11,533.19 • Step 4: Calculate the PV at t=0 of the face value that is paid at t=40 PV0 = $100,000 / 1.0640 = $9,722.22 • Step 5: Calculate the total PV at time 0 of all of these cash flows: Price of bond X = $31,560.76 + $18,359.28 + $11,533.19 + $9,722.22 = $71,175.45 Bond Y: Just discount the face value back by 20 semi‐annual periods: Price of bond Y = $100,000/(1.06)20 = $31,180.74 Answer 9 (Bonds) a) The bonds are sold at par, therefore YTM =12%. b) The price of the bond is the present value of all future cash flows. On January 1, 1981 there were still 25 annual coupon payments to come and the payment of the face value. The coupon payment is $120. The discount rate is 10%. P1 / 1 / 1981 1
⎡
⎢1 − 1.10 25
= 120 × ⎢
⎢ 0.10
⎢
⎣ ⎤
⎥ 1,000
= $1,181.54 ⎥+
25
⎥ 1.10
⎥
⎦ c) The current yield equals the annual coupon payment divided by the current price. The annual coupon payment equals $120. Hence, in order to calculate the current yield on January 1, 1996, we first need to calculate the bond’s price at that date. The bond has 10 years left to maturity. The discount rate is 13%. P1 / 1 / 1996 1
⎡
⎢1 − 1.1310
= 120 × ⎢
⎢ 0.13
⎢
⎣ ⎤
⎥ 1,000
= $945.74 ⎥+
1.1310
⎥
⎥
⎦ Now, calculate the current yield: Current Yield = 120 / 945.74 = 0.1296 = 12.96% 1 0 Answer 10 (Stocks) a) Create a timeline: The dividend at t=0 is denoted by D0. Cash D0 D0×1.40 D0×1.402 D0×1.402×1.20 D0×1.402×1.20×1.10 D0×1.402×1.20×1.10×1.05 … flow Time 0 1 2 3 4 5 … As of D4 dividends grow at a constant rate of 5% forever (i.e. D4 is the first dividend that will grow by 5% forever). Use the formula for the PV of a growing perpetuity to calculate the stock price at t=3 (one period before the first cash flow that is part of the growing perpetuity). This stock price only includes future dividends and therefore does not include D3. P3 = D × 1.40 2 × 1.20 × 1.10
D4 =0
(k − g )
0.10 − 0.05 The stock price at t=0 equals the PV of all future cash flows: the PV of the dividends at time 1, 2, and 3 and the stock price at time 3 that we just calculated. P0 = D3
P
D1
D2
+
+
+ 33
2
3
1.10 1.10
1.10 1.10 D0 × 1.40 2 × 1.20 × 1.10
D × 1.40 D0 × 1.40 2 D0 × 1.40 2 × 1.20
0.10 − 0.05
P0 = 0
+
+
+
2
3
1.10
1.10
1.10
1.10 3
⎡
1.40 2 × 1.20 × 1.10 ⎤
⎢1.40 1.40 2 1.40 2 × 1.20
⎥
0.10 − 0.05 P0 = D0 × ⎢
+
+
+
⎥
2
1.10 3
1.10 3
⎢1.10 1.10
⎥
⎢
⎥
⎣
⎦
$80 = D0 × 43.536
D0 = $1.8375
b) Now we still need to calculate the expected dividend for the coming year: D1 = D0 × 1.40 = $1.8375 × 1.40 = $ 2.572 Answer 11 (Stocks) The current stock price is the present value of all future dividends. Dividends are expected to grow at a constant rate, hence this is a growing perpetuity. First, calculate the growth rate. The dividend one period from now equals D1 = D0(1+g) = $1.60 ×(1+g) Use the formula for the PV of a growing perpetuity to calculate g: P0 = D1/(k‐g) $24 = $1.60(1+g)/(0.12‐g) Therefore g = 5% P5 = D6 / (k‐g) = 1.60(1.05)6/(0.12‐0.05) = $30.63 1 1 Alternatively, we know that when dividends grow at 5% per year, the stock price also grows at 5% a year. Hence: P5 = $24(1.05)5 = $30.63 Answer 12 (Stocks) The price of a stock is the present value of all future dividends. a) D1 = 1.15(1.18) = $1.357 2 D2 = 1.15(1.18) = $1.6013 = 1.15(1.18)2(1.15) = 1.8415 D3 D4 = 1.15(1.18)2(1.15)(1.06) = $1.9519 2
= $1.357/(1.12) + 1.6013/(1.12) + 1.8415/(1.12)3 + 1.9519/(0.12‐0.06)(1.12)3 P0 = $26.95 = $1.2117 + 1.2766 + 1.3108 + 32.53/(1.12)3 b) P1 = $1.6013/1.12 + 1.8415/1.122 + 32.53/1.122 = $28.83 P2 = $1.8415/1.12 + 32.53/1.12 = $30.69 a) In general, the dividend yield, capital gains yield and total return are given by: Total return from time t to time t+1 = E [Dt +1 ] + E [Pt +1 ] − Pt = E [Dt +1 ] + E [Pt +1 ] − Pt
Pt
P
P 123 14 t 4
4 t4
23 dividend yield
capital gain
In this exercise, the yields can be calculated as follows Year Dividend Yield Capital Gain Yield
Total Return 1 2 3 1357
.
= 5.04% 26.95
16013
.
= 5.55% 28.83
18415
.
= 6% 30.69 28.83 − 26.95
= 6.98% 26.95
30.69 − 28.83
= 6.45% 28.83
32.53 − 30.69
= 6% 30.69 12% 12% 12% Answer 13 (Stocks) a) The stock price equals the PV of all future dividends. Let’s make a time line first. Cash flow D3 D3×1.25 D3×1.252 D3×1.252
×1.15 2 D3×1.25
×1.15×1.10 2 D3×1.25
×1.15×1.102 D3×1.252 ×1.15×1.102×1.05 … Time 0 1 2 3 4 5 6 7 8 9 … As of t=8 the dividend with grow forever at a constant rate of 5%. (D8 is the first dividend that grows at 5%). Hence, the formula for the PV of a growing perpetuity gives us the stock price at t=7, one period before the first cash flow that belongs to the growing perpetuity. P7 = D8
D × 1.25 2 × 1.15 × 1.10 2
=3 0.10 − 0.05
0.10 − 0.05 The price at t=0 equals the present value of the dividends at t=3, 4, 5, 6 and 7 and the price at time 7. 1 2 D3
D5
D6
D7
P
D4
+
+
+
+
+ 77
3
4
5
6
7
1.10 1.10
1.10 1.10 1.10
1.10
2
D3
D × 1.25 D × 1.25
D × 1.25 2 × 1.15 D3 × 1.25 2 × 1.15 × 1.10 +3 4+3
+3
+
P0 =
1.10 3
1.10
1.10 5
1.10 6
1.10 7
D3 × 1.25 2 × 1.15 × 1.10 2
0.10 − 0.05
+
1.10 7
P0 = The current stock price is $107.673. We can now calculate D3: ⎡
1.25 2 × 1.15 × 1.10 2
2
2
2
⎢1
1.25 1.25 1.25 × 1.15 1.25 × 1.15 × 1.10
0.10 − 0.05
+
+
+
+
+
107.673 = D3 × ⎢
3
4
5
6
7
1.10
1.10
1.10
1.10 7
⎢1.10 1.10
⎢
⎣
107.673 = D3 × 26.91821
D3 = $4.00 b) We buy the stock at t=7 and sell it at t=8. Hence, we need to calculate the total yield, dividend yield and capital gains yield over the 8th year. Remember: E [Dt +1 ] + E [Pt +1 ] − Pt
E [Dt +1 ] E [Pt +1 ] − Pt
=
=
+
Total return from time t to time t+1 Pt
Pt
P 123 14 t 4
44
23
dividend yield
capital gain We know that the total yield during year 8 will be 10%, since this is the required rate of return. In order to calculate the dividend yield we need to calculate the dividends that will be paid in year 8 and the price in year 7. D8 = D3 × 1.25 2 × 1.15 × 1.10 2 = 4 × 1.25 2 × 1.15 × 1.10 2 = $8.6969 P7 = D8
8.6969
=
= $173.9375 0.10 − 0.05 0.10 − 0.05 Dividend yield = D8 / P7 = 8.6969 / 173.9375 = 0.05 Capital gain yield = total yield – dividend yield = 0.10 – 0.05 = 0.05 Alternatively, we could calculate the capital gains yield as the % change from P7 to P8. First, calculate the stock price at t=8: P8 = D9 / (0.10‐0.05) = D8 × 1.05 / (0.10‐0.05) = 9.1317 / 0.05 =$182.634 Capital gains yield = (182.634 ‐ 173.9375) / 173.9375 = 0.05 (same answer as before!) 1 3 ⎤
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 Fall '10
 Elaine
 D3 Publisher

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