Practice Set #1 and Solutions.

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1 Bo Sjö Practice Set #1 and Solutions. What to do with this practice set? Practice sets are handed out to help students master the material of the course and prepare for the final exam. These sets contain worked-out problems in corporate finance and, investments and portfolio management, with a bias toward quantitative problems. These sets are not graded, and there is no need to hand-in the solutions. Students are strongly encouraged to solve them, discuss the solutions with other course participants, and discuss any problems with their teacher. Some questions in the final exam might resemble the problems given here. Question 1 Why is the value of 10 today not the same as receiving 100 in future? Question 2 What is meant by cash flow (or cash flows), and in what way is the concept different from cash flows in accounting? Question 3 What is the discount rate composed of, in general) and why can you not use the same discount rate for all types cash flows? Hint there are two components. Question 4 What is the present value (and the present value factor) of the following amounts (use discrete not continuous discounting): a) Present value of USD155 to be received six years from now, if the discount rate is 3%? b) Present value of USD1,000 to be received twenty years from now, if the discount rate is 8%? c) Present value of USD100,000 to be received seventy years from now, if the discount rate is 8%? Question 5 a) What is the future value of $343, received 3 years from now, invested at 9% discrete interest per annum? b) What is the future value of $1, received 40 years from now, and invested at 8% per annum? c) What is the future value of $1, received 70 years from now, and invested at 8% per annum? 1

2 Question 6 Suppose you invest $1,000 at 6% simple annual rate, what is the future value if a) Interest is paid at the end of year one? b) If interest is paid every six months during one year? c) If interest is paid every quarter? d) If interest rate is paid every month? e) If interest is paid every day over a whole year? f) If interest is paid only during trading days (trading days are the days when the market is open during a year, not including holidays, add a couple of religious and national holidays and find an approximation)? g) If interest is compounded (paid) at the end of each month over three years? h) If interest is compounded (paid) at the end of each six-month period over three years? i) What are the effective yearly interest rates in questions a) to h)? Question 7 Calculate the present values of the following cash flows (given the discrete discount rates) and use the correct annuity formula, annuity, annuity due and growing annuity: a) A yearly cash payment of $30 received during a period of years, starting at the end of the first year, if the discount rate is 6%? b) A semi-annual cash payment of $30 received at the end of every year during a period of 10 years, if the discount rate is 6% p.a.? c) A semi-annual cash payment of $15, received at the end of every year, during a period of years, if the discount rate is 6% p.a.? d) A yearly cash payment of $30 received over a period of years starting today? e) A yearly cash payment of $30 received during a period of years, starting at the end of the first year, and grows with 2% each year, if the discount rate is 6%? Question 8 a) What is the future value of USD155 three years from now if you know that the future yearly interest rates will be r 1 = 3%, r 2 = 5%, r 3 = 6%? b) What is the present value of USD0 received 3 years from now, given the information above? c) What is the 'average' interest rate over the three years in Question a) above? (Hint: Notice that it must be a geometric average interest rate, not an arithmetic one, since we are dealing with multiplicative sums. Compare with the calculations in questions above) d) Try to explain the simplification made in the examples in Questions 1 to 4? Question 9 Suppose you want to donate money to finance a full professor in corporate finance, at the University of Botswana. The estimated cost of the chair is USD110,000. We expect the cost to raise with 2.5% a year forever. How much money must you set aside today in order to cover the costs if the expected rate of return is 5.5%? 2

3 Question 10 a) An investment project has the following cash flows (starting from period 0): $ (-1543, 705, 860, 505). The opportunity cost is 7%. Should the project be accepted? b) An investment project has the following cash flows (starting from period 0): USD (2100, -400, -600, -800). The hurdle rate is 15%. Should the project be accepted? c) Try to find the point were the investors is indifferent to making the project or not. (Hint NPV=0) Question 11 a) What is the YTM of a $1,000 bond, paying 8% coupon, for 25 periods (=years for simplicity), if the price is $1,000? b) What is the YTM if the price is $900? Question 12 Treasury bonds paying an 8% coupon rate with semi-annual payments currently sell at par value. a) What coupon rate would need in order to sell at par if they paid their coupons annually? Question 13 Consider a newly issued bond that pays its coupon once annually, and whose coupon rate is 5%; the maturity is years, and yield to maturity is 8%. a) Find the holding period return for a one-year investment period if the bond is selling at a yield to maturity of 7% by the end of the year. Question 14 What is the value of a 10-year, risk-free pure discount bond, with face value 1,000 at maturity if the interest rate is a) 5% b) 10% c) 15% Question 15 Nic Generators has issued the following bond, Nominal value: 1,000 Time to maturity: years Coupon is 8 % compounded semi-annually, with semi-annual coupon payments. What is the price of the bond if the YTM:s for this class of bonds are 3

4 a) 8% b) 10% c) 6% Question 16 If you have coupon-paying bond, with (annual) coupon rate 2%, Face value 1,000, YTM = 1.5%, and 2 years to maturity. If you buy the bond today and reinvest coupons at 1.5% what is your expected HPR? Question 17 If you are planning to buy a US Treasury bond with a call provision why should you be think before buying the bond on the secondary market? Question 18 In March 12 a US Treasury bond with 6% coupon of 07 offered a semi-annual yield of 2.62 %. (Face value is always 1,000 on US, Bill, Notes and Bonds). What is the price of this bond? Question 19 Briefly explain the following concepts a) Bond indenture and covenants b) Derivative, instruments? Name and explain the common 4 types. c) Zero coupon bonds d) Certificate of deposits e) Commercial Paper f) TBill g) Repurchase agreement (Repo) h) YTM i) Yield curve j) Callable Bond k) Puttable Bond l) Convertible Bond Question What is mean by risk-free interest rates and risk-free bonds? And are they really risk-free? Question 21 a) The Fisher equation for interest rates says that the interest rate consists of two parts, which are they? b) What is the difference between interest rates and yields (yield to maturity)? c) Suppose that firms expect that their orders will go up and they need to invest to meet an increasing future demand. How will that affect the demand/supply for bonds and the interest rate? d) Suppose the public thinks that there is a recession coming in two years, how will they change their consumption and savings behaviour and what is the effect on interest rates in general and the yield curve? 4

5 Solutions to Practice Set #1 Solution 1 The answer is that in a world with limited resources, money has a time value, or an alternative use. This gives rise to interest rates (and yield and returns etc.) that expresses the alternative use of money and thus resources over time. The simple outcome of this limited resource problem is that 100 can be put in a bank account, or invested otherwise, to an earn interest, which in over time builds up to more than receiving 100 at a future date. Solution 2 Cash flow refers to actual money (cash) in minus actual money out during a specified period. In accounting are measured from accounting statements and they are not always identical to real cash flows. Accounting standards, when looking at historical data will to some degree distorted measured cash flows. Solution 3 The discount rate should reflect the alternative use of the money, or to be more exact the opportunity cost of creating the same type of future cash flows on the financial markets. The discount rate is therefore equal to Discount rate = risk-free interest rate + risk premium(s) Risk premiums for bonds are first of all default risk and liquidity risk (explain these) Solution 4 Use the relevant formulas to get the present values: a) USD155 / (1.03) 6 = USD155 x Present value factor (3%, 6) = USD Pvf = b) USD1,000 / (1.08) = USD1,000 x Present value factor (8%, ) = USD Pvf = c) USD100,000 / (1.08) 70 = USD100,000 x Present value factor (8%, 70) = USD Solution 5 a) $343 (1.09) 3 = $ b) $1 x (1.08) 40 = $21.72 c) $1 x (1 + r) 60 x (1 + r) 10 = $

6 Solution 6 The compounding formula for investing C 0 m times a year over T years is m T r FV = C0 1+ apply it to all questions: m Sorry some of the answers are a bit mixed up at the moment. a) USD1,000 (1.06) = $1060 b) USD1,000 [1+ (0.6/2)] 2 = $ c) Answer: USD1, d) Answer: USD1, e) Answer: USD1, f) Answer: USD1,061.83, it is almost the same. The exact number of trading days depend on the market. There are 52 weeks, with Saturdays and Sundays, add a number of religious and national holidays make 250 days on average per year a good approximation. g) Answer: USD 1, h) Answer USD 1, i) Använd formeln för att beräkna EAR: EAR = (1+APR/m) m -1 +a) 6.00%, b) 6.09%, c) 6.14% d)?, e) 6.18%, f) 6.18% g) 6.17%, h) 6.09% Solution 7 Use the annuity formula to get the annuity factor, which is multiplied with the cash flow. Notice that there will be round errors in the annuity formulas depending on how many decimals you use. a) $30 x Annuity factor(6%, ) = $30 x b) $30 x Annuity factor(3%, ) = $30 x A.03 = $ c) $15 x Annuity factor(3%, 40) = $15 x A.06 = $30 x = $ A.03 = d) A yearly cash payment of $30 received over a period of years starting today? This is annuity due, the payments starts directly, not at the end of the periods. The annuity factor becomes 1 1 AD. 06 = (1.06) (1.06), and the answer is $30 x AD.06 = $ e) A yearly cash payment of $30 received during a period of years, starting at the end of the first year, and grows with 2% each year, if the discount rate is 6%? This is a growing annuity: 6

7 1 (1.02) AG.06 = (1.06) = 25 1 [ ] = And the answer is $30 x = $ (1.02) ( ) (1.06) Solution 8 a) USD155(1+r 1) (1+r 2) (1+r 3) = USD155(1.03) (1.05) (1.06) = USD155 (1.1464) = USD b) The question is solved by calculating the present value year by year. From year 3 to year 2 the value is USD0/(1.06) = USD From year 2 to year 1, we get USD179.69, the final answer is that the present value (T=0) is USD c) If you compare with the calculations done in the questions above you will see that the following must hold between the average yearly interest rate and the three yearly rates: (1 + r) 3 = (1+r 1) (1+r 2) (1+r 3). The average yearly rate should correspond to discrete compounding of the (average) single rate over three years, which must be the same as the multiplicative sum of the three yearly rates. The solution is found by solving for r in the expression above. (1 + r) 3 = (1.03) (1.05) (1.06) r = [(1.03) (1.05) (1.06)] 1/3-1 r = Verify that the result using r = 4.66% leads to same answer as in question a. d) When we use a single discount rate to calculate future and present values over several periods, we are implicitly assuming an (geometric) average interest rate over all periods. We assume what is called a flat yield curve. Is there a problem here? The answer is both yes and no. If we could, in every situation find the correct discount rate for each period we would use it, of course. But, that is quite difficult in the real world, as we move into various calculations of risky cash flows. However, calculations show that the error from assuming a flat yield curve is quite small. Thus, the simplification is not costing us that much, in particular compared to the costs of finding the correct ri for each period, and for each future cash flow. Solution 9 In this situation use the PV formula for a growing perpetuity a) PV = C/(r-g) = USD110,000 /( ) = USD3,666,667 Solution 10 a) NPV = -$ $705/ $860/(1.07) 2 + $505/(1.07) 3 = -$ $ = $

8 The answer is yes, since the NPV is greater than zero. (If it is zero you would be indifferent to the project, if NPV is negative the project is rejected because it is destroying resources and makes us/you poorer). b) NPV = USD USD400/(1.15) - USD600/(1.15) 2 - USD800/(1.15) 3 = USD USD c) Here you end up with a third degree polynomial, which requires a financial calculator to solve. However, you can apply trial and error, and come quite close. Set the NPV expression to zero, try to find a discount rate that gives a negative NPV value. Reduce the discount rate to find a positive NPV value. Next raise the discount rate, step by step you are closing in on the discount rate that gives NPV = 0. Solution 11 When the bond is selling at par value, YTM is 8%. The $80 coupon payment is 8% of the face value, corresponding to 8 % interest rate paid on a deposit. If the investor withdraws the interest each year, she is left with the face value after 25 years. This would be identical to a bank account paying $80 in interest a year, were 1000 is deposited, and the $80 in interest is withdrawn just after it has been paid out. In the end, at time T, withdraw the face value of $1000 from the bank account. Thus, the interest rate on this account would be 8%. Thus, for bonds selling at par value, price equals face vale, YTM is equal to the coupon rate. We know that YTM = 8% if the price is $1,000. Now, the price is lower, P 0 = $900. By trial and error we can then find an approximation to YTM. We know that if the price of the bond goes down, YTM must go up. If the price goes up, YTM goes down. Since the price is lower compared with the face value and YTM =8%, we should first try with a higher rate than 8%. Assume a YTM of 10% and find P = $818.46, which is too low. The YTM must be lower that 10% to give the higher price of $900. Next assume a YTM of 9% and find P = $ A bit too high, but closer. If we continue the trial and error process we end up with 9.02%. Using a financial calculator, the more exact result is YTM=9.0197%. Next, find the YTM if the price is $800. Solution 12 The effective annual yield on the semi-annual coupon bonds is 8.16% = (1+8%/2) 2-1. If the annual coupon bonds are to sell at par, then they must offer the same yield, which will require an annual coupon rate of 8.16%. Solution 13 a) Use relevant annuity and other relevant formulas to compute the following: The initial price is: P0 = $705.46, for [T = ; Coupon = 50; FV = 1000; y = 8] The next year s price is: P 1 = $793.29, for [n = 19; Coupon = 50; FV = 1000; y = 7] Thus, the holding period return (HPR) is given by: HPR = [$50 + ($ $705.46)]/$ => HPR = = 19.5% 8

9 Solution 14 The present value of any pure discount bond is its face value discounted back to the present. a) PV = F / (1+r) 10 = $1,000 / (1.05) 10 = $ b) PV = $1,000 / (1.10) 10 = $ c) PV = $1,000 / (1.15) 10 = $ Solution 15 Start by finding the amount of the semi-annual coupon payment. a) Since the stated annual yield is compounded semi-annually, simply divide this rate by two in order to calculate the semi-annual YTM. Semi-annual Interest Rate = 0.08 / 2 = 0.04 The bond has 40 coupon payments, years x 2 = 40, each coupon payment is (0.08/2) x 1,000 = 40. Apply the annuity formula to calculate the PV of the 40 coupon payments. In addition, the $1,000 payment at maturity must be discounted back 40 periods. Actually, for a semi-annual coupon bond, the YTM is like an Internal Rate of Return. The price is, using the annuity formula for the coupon payments A ( ) = 40 P = C A F / (1+(y/2)) = $40 A.04 + $1,000 / (1.04) 40 = $1,000 The price of the bond is $1,000. Notice that whenever the coupon rate and the yield to maturity are the same, the bond is priced at par, at the price is equal to the nominal value (face value) of the bond. That is, its market value is equal to its face value. b) Semi-annual Interest Rate = 0.10 / 2 = 0.05 P 0 = $40 A $1,000 / (1.05) 40 = $ The price of the bond is $ Notice that whenever the coupon rate is below the yield, the bond is priced below par. (Or if the yield goes above the coupon rate, the price falls below the nominal value. c) Semi-annual Interest Rate = 0.06 / 2 =

10 P 0 = $40 A $1,000 / (1.03) 40 = $1, The price of the bond is $1, Whenever the coupon rate is above the market rate, the bond is priced above par. Yield goes down under the coupon rate the price goes above the nominal value. Solution 16 The holding period return is defined as P1 P0 + D1 HPR1 = where P 1 and P 0 is the present and previous price respectively. D 1 is cash P0 payment received during the period, in this case Today the price of the bond is 1 1 FVT P0 + T (1 ) r r + r (1 + r) = ( ) ,000 = (1.015) = T 2 [ $ ] = = You pay $ today for one bond. The first coupon will over one grow to x(1.015) =.3 The second and final coupon is. At maturity you collect 1, = The HPR is (in steps) 1. ( ) / = To get to the HPR and the annual value: (1.0305) 0.5 = (close to 1.5%) The HPR formula for one period is HPR Over a two year period HPR becomes 1 P1 P0 + D1 = P 0 2 P2 P0 + D2 P2 P0 + D2 (1 + HPR 1) = => HPR = 1, where in this case D 2 is the final coupon P0 P0 and the previous coupon at its future value x (1.015). Solution 17 Treasury securities cannot be redeemed before maturity and do not have call provisions. However up to around 1986 they did have call provisions. A call provision gives the issuer the wright to buy pack the bond at stipulated date 8or up ta specific date) at a fixed price. If the interest rates goes down the issuer might want to use this call provision: the right but not the obligation to buy back the bond at a specific date

11 Solution 18 In March 12 a US Treasury bond with 6.2% coupon of 29 offered a semi-annual yield of 2.62 %. (Face value is always 1,000 on US, Bill, Notes and Bonds). What is the price of this bond? The semi-annual coupon payments are $1,000 x 0.062/2 = $31 The number of compounding periods is number of years to maturity x 2 = 17 x 2 = 34 The semi-annual yield is per 6m = 2.62/2 = 1.31%. Plug the information into the formula for the bond value/price 2T C FVT 1 1 P0 = + = $31 2 t T t= 1 y y ( ) 0.031(1.0131) = $ $1,000 = $ = $31x $ = $ $ = $ $1,000 + (1.0131) [ ] 34 + $ Solution 19 For solutions see Textbook and/or the web. Solution Risk-free bonds refer to default free bonds. If you buy this type of bond and hold it to maturity there is a nearly 100% certainty that you will receive the nominal value of that bond. However, the bond is not risk-free, since the value will go up and down with the interest rate until the maturity. Finally, the nominal value of the bond is determined in nominal terms, so the real value (and the purchasing power of the nominal value) will change with inflation. Solution 21 a) Fisher say s that the nominal interest rates consist of a real interest rate and the expected rate of inflation. Write r for the nominal rate, rr for the real rate and we have: e rt = rrt + p b) The yield is the internal rate of return of a bond. The concept includes the yield of all sorts of bonds including both zero coupon bonds and coupon paying bonds. The interest rate is the yield of a zero coupon bond over a specified period, say 3-months, or one-year. c) To solve this question draw a supply and demand diagram, according to Figure 3 in the handout on interest rates. If firms want to invest more they need to borrow more. Thus they demand loanable funds and supply more bonds to the market. The supply curve for bonds shifts to the right, the price of bonds will fall and the interest rate will go up. Since firms are likely to borrow 11

12 on longer terms the effect will be more clear on long-term rates rather than on money market rates. d) If households, and the public, think s that the future is grim they will save today to have a higher income (from savings) when times gets worse in the future. Thus, they supply more loanable funds, they buy bonds with a maturity corresponding to the expected bad times. The demand for bond increases, the demand curve shifts to the right. Bond prices go up and the interest rate goes down. In particular we expect a stronger effect on two-year bonds and interest rates. This is likely to show in the changing slope and bending of the yield curve. 12

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