PS4. Bonds (Solutions) PDF

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FINANCIAL ECONOMICS 2018/19 SEMINAR 4: PRICING OF BONDS

1. Final Exam-December 2009, question 1. Assume an economy in which there are 3 risk free bonds. The first bond, A, has a price of € 997.5 and has an annual coupon of 5%, a maturity of 1 year and a nominal of € 1,000; the second bond, B, pays a coupon of 8% over a nominal of also € 1,000, matures in two years and its price is € 1,026.4; and the third bond, C, matures in three years, costs € 947.2 today and pays an annual coupon of 6%, also over a nominal of € 1,000. a) Suppose that a fourth bond appears in the market, which we’ll call D, with a nominal of € 1,000, a maturity of 3 years, and which pays an annual coupon of 3%. Construct a portfolio with identical payoffs to those of bond D using bonds A, B and C. b) Prove that the no-arbitrage price of bond D is € 868.6. If bond D trades at a market price of € 880, would there exist an arbitrage opportunity in this market? If so, describe a strategy that would allow you to take advantage of such an arbitrage opportunity. c) Using the information about bonds A, B and C, calculate the no-arbitrage price of the basic bonds that mature in 1, 2 and 3 years and prove that the spot interest rates for those three maturities are 5.26%, 6.6% and 8.17%. Hint: Use the fundamental asset pricing equation for coupon bonds that uses the basic bond prices.

a) The information on the prices and payments of the three bonds of the statement (A, B, C), together with that of the bond to be replicated (D) is shown in the following table:

PRICE BOND A B C D

997.5 1026.4 947.2 PD

t=1 1050 80 60 30

PAYOFFS t=2 0 1080 60 30

t=3 0 0 1060 1030

To replicate the payments of the D bond, the units of A bonds (zA), B bonds (zB) and C bonds (zC) MUST solve the following system of equations:

30 = 1050 zA + 80zB + 60zC 30 = 1080 zB + 60zC 1.030 = 1060zC

Solving the system we get the solution:

zA = -0.02496 zB = -0.02621 zC = 0.97170

b) The price of asset D must coincide, to avoid arbitrage opportunities, with the cost of its replicating portfolio; that is to say: PD = (-0.02496)997.5 + (-0.02621)1.026.4 + 0.97170947.2 = 868.6 If the price is 880 euros, the Law of One price is violated and there is an arbitrage opportunity. The arbitrage will consist of buying 1 unit of the replicating portfolio and short sell 1 unit of the D bond. This strategy has a profit of (880-868.6) = 11.4 euros.

c) The prices of the three basic bonds can be obtained from the pricing equation applied to each bond:

997.5 = 1050 b1 1026.4 = 80 b1 + 1080b2 947.2 = 60b1 + 60b2 + 1060b3 The solution is:

b1 = 0.95 b2 = 0.88 b3 = 0.79 The interest rates are therefore:

r1 = 1/b1-1 = 5.26% r2 = (1/b2)1/2 -1 = 6.6% r3 = (1/b3)1/3 -1 = 8.17%

2. September Exam 2010, question 1. Three basic bonds are traded in the market, which mature in 1, 2 and 3 years and whose prices are 0,976; 0,9426 and 0,8941 euro, respectively. a) Assume two new bonds start trading in this market: a basic bond with a maturity of 4 years and a coupon bond (called bond A) with a nominal of 1000 euro and an annual coupon rate of 3% which matures in 4 years. If the price of bond A is 965,031, prove, under the assumption of absence of arbitrage, that the price of the basic bond maturing in 4 years is 0,855. b) Under the same assumptions of the previous section, obtain the spot interest rates for the four maturities (t=1, 2, 3 and 4), and show that the forward rate for an investment that occurs in t=3 and matures in t=4, f 4 , is 4,6%. c) Under the same assumptions of the previous sections, now suppose someone is willing to lend or borrow 1000 euro in t=3 to be repaid in t=4 at the rate of 6,6%, instead of 4,6% as we calculated in the previous section. Describe a strategy that would allow you to take advantage of this arbitrage opportunity. d) Now assume that a fifth basic bond is introduced that matures in t=5 and has a price of 0,875. Is there an arbitrage opportunity using only the basic bonds? e) Finally, given the spot interest rates, indicate which are the expectations about the spot interest rate that will exist between t=1 and t=2 for borrowing and lending,E1 r2 , under the Pure Expectations theory and the Preference for liquidity theory. For the latter assume that the liquidity premium, 1 L2 is 0,50%.

a) New bonds:

A b.b. 4

t=0 PA=965,031

t=1 30

b4 = ?

PA = 965.031 = 30 b1 + 30 b2 +30 b3 + 1030 b4 Solving the equation for b4, we get b4 = 0.855

t=2 30

t=3 30

t=4 1030

b) Using bt = (1+rt)-t , we get: r1 0.0246

r2 0.0300

r3 0.0380

r4 0.0399

Using (1+r4)4 = (1+r3)3 (1+ f4) , we get f4 = 0.045730994

c)

Lend over 3yrs PV of 1000 Lend through forward 1000 Borrow over 4 years PV of 1066 Total

t=0 -894.1

t=3 1000 -1000

911.43 17.33

0

t=4 1066 -1066 0

d) No, there is no arbitrage opportunity. (However, if we were sure that interest rates will never be negative, we could buy the 4year basic bond, short the 5-year basic bond, an reinvest the payoff of the 4-year basic bond (1 euro) from year 4 to year 5 at a non-negative rate, so the final payoff in year 5 is nonnegative – an example of sequential arbitrage.) e) We know that f2 =1.032 /1.0246 =3.54% PEH  E[1r2]= f2 =3.54% PLH  E[1r2]=f2 - 1L2 = 3.54% - 0.5% =3.04%

3. Final Exam-December 2010, question 1. Consider an economy in which there are two riskless coupon bonds, Bond A and Bond B. The nominal value of Bond A is €100, the coupon rate is 12% and it matures in 1 year. The nominal value of Bond B is €100, the coupon rate is cB and it matures in 2 years. The price of Bond A is €106.67, while the price of Bond B is €120.05. a) Knowing that the price of the 2 year basic bond (b2) is 0.842, compute: the coupon rate of Bond B (cB); the 1 year basic bond price; and the annualized spot interest rates for 1 and 2 years. b) What is the implicit forward rate f2.? What is the internal rate of return (IRR or i) of Bond B? c) Consider a new riskless bond (Bond C) with a maturity of two years, a nominal value of €500 and a coupon of 18%. How much should it cost for there not to be arbitrage opportunities?

d) Suppose that the market price of Bond C is €550. Is there an arbitrage opportunity? If so, exploit the arbitrage opportunity. Specify your positions in Bonds A, B and C. Also, indicate the net payoffs of the strategy in t=0, t=1 and t=2, presenting the information in a table. e) Interpret the term structure of interest rates introduced in point a). Restrict yourself to the liquidity preference theory. Assume that the liquidity premium is 2% (1 L2  2.0% ).

Bl B2

t=0 106.67 120.05

t=l 112 c * l00

t=2 (l + c) * l00

106.67 = 112 * b1 b1 = 0.952 bt = (1+rt)-t  r1 = 0.0499 = 0.05 120.05 = b1 * 100 * c + b2 * l00 * (l + c)  c = 0.1998 = 19.98% r2 = (1/0.842)0.5-1 = 0.08979 = 0.09

b) l + f2 = 1.092 / 1.05 = 1.1315

IRR: Using the formula in the help, i = 0.0865.

c) Pb = 500 * 0.18 * b1 + 500 * (1, l 8) * b2 = 500 * 0. l 8 * 0.952 + 500 * (1, l 8) * 0.842 = 582.46

d)

There would be an arbitrage opportunity, since the market price is lower than the price found in section c). Strategy: long position in the bond and short in the replicating portfolio. To find the replicating portfolio, we solve: 90 = 112 · z1 + 19.98 · z2 590 = 120 · z2 solving: z1 = -0.0737; z2 = 4.9175 Cost of replicating portfolio = 582.48 Therefore:  We buy the bond  We buy 0.0737 units of bond 1  We short 4.9175 units of bond 2

e) Under the preference for liquidity hypothesis, investors have a preference for investing in short-term securities. The theory implies that the slope of the interest rate curve is influenced not only by the expected changes in interest rates, but also by the liquidity premium that investors demand from long-term bonds.

In particular, in this case the theory assumes: f2 = E[1r2] + 1L2. Since 1L2 = 0.02 and f2 = 0.1315, then E[1r2] = 0.1115.. 4. Final Exam-June 2011, question 1. There are three bonds (A, B and C) being traded in a financial market, all of them with a nominal value of 1000 Euros. The first one, bond A, has a price of 1000.69 Euros, a maturity of 2 years and pays an annual coupon of 5%. Bond B pays a coupon of 4%, has a maturity of 2 years and its Price is 981.958 Euros. Bond C pays a coupon of 3%, has a maturity of 3 years and its Price is 971.86. a) Find the prices of the basic bonds for 1,2, and 3 years consistent with the absence of arbitrage opportunities in this market and check that these prices are 0.9662, 0.907, y 0.889, respectively. (Hint: you can compute the prices of the basic bonds 1 and 2 using exclusively the information about bonds A and B). b) Compute the risk-free interest rates for 1, 2 and 3 years in this economy. c) Answer if there is sequential arbitrage in this economy and explain why. d) Assume that a new bond starts being traded at 900 Euros, a bond that we are going to call bond D, that does not pay coupons, that has a nominal value of 1000 Euros and a maturity of 2 years. Form a portfolio composed of bonds A and B that replicates the payoffs of bond D. Prove that there exists an

arbitrage opportunity and describe precisely an arbitrage opportunity, that is indicate: - the positions that you should take (number of units of each asset and if they are long or short). The strategy has to include only positions in bonds A, B and D. - the payoffs or cash flows of each strategy in each date. e) Using your answer for part c), compute the forward interest rates implied by the term structure of interest rates in this economy. Assuming that the pure expectation hypothesis/theory is satisfied, which are the market’s expectations with respect to the interest rates for investing a period of one year, one and two years from now? f)

Assume that a new zero-coupon bond, E, starts being traded, that has a maturity of 3 years and a nominal value of 1000 Euros. Compute its price, its internal rate of return and its duration. If the interest rates increased by 10 basis points (0.10%), using the duration, by how much would the price of bond E decrease in relative terms with respect to its current price (by what percentage)?

a) 1000.69 = 50b1 + 1050b2 981.958 = 40b1 + 1040b2 Solution is: {[b1 = 0.96617, b2 = 0.90703]} 971.86 = 30b1 + 30b2 + 1030b3 971.86 = 30*0.9662 + 30*0.907 + 1030b3. Solution is: {[b3 = 0.8889941748]} b) 0.9662 = 1/ (1+r1). Solution is {[r1 = 3.498240530 * 10-2]} 0.907 = 1/ (1+r2)2. Solution is {[r2 = 5.001706292 * 10-2]} 0.889 = 1/ (1+r3)3. Solution is {[r3 = 3.999858006 * 10-2]} c) b1 > b2 > b3. Therefore, there is no sequential arbitrage d) 100 * b2 = 907 > 900 Arbitrage strategy: buy D and sell portfolio that replicates D. Replicating portfolio: 0 = 50zA + 40zB 1000 = 1050zA + 1040zB Solution is: {[zA = -4.0, zB = 5.0]} Cost: -4*1000.69 + 5*981.958 = 907.03 Strategy: Buy 1D, Buy 4A, Sell 5B

Payments: t=0: -900 - 4*1000.69 + 5*981.958 = 7.03 t=1: 4*50 - 5*40 = 0.0 t=2: 1000 + 4*1050 – 5*1040 = 0.0 e) (1 + 0.035) (1 + f2) = (1 + 0.05)2. Solution is: {[f2 = 0.065]} (1 + 0.05)2 (1 + f3) = (1 + 0.04)3. Solution is: {[f2 = 0.0203]} Term rates match the expected rates under the PEH. f) PE =

100

= 888.9963587

(1+0.04)3

As it is a zero-coupon bond: D=3 TIR = 4% dB/B = -(3/(1.04))*0.0010 = -0.00288  The relative price drop would be 0.29% 5. Final Exam-June 2012, question 1. Consider the following default-free bonds with nominal €100: - Bond A: One–year zero coupon bond, price today = €95.24. - Bond B: Two-year 5.5%-coupon bond, price today = €99.13. a) Compute the prices of basic bonds with maturities of 1 and 2 years. b) Find the 1- and 2-year spot interest rates ( r1 and r2), and the forward rate f2 for a loan between t=1 and t=2. c) What is the yield to maturity (or internal rate of return) of bond B? d) Suppose that a third bond, bond C, is issued. It is a two-year zero-coupon bond with a face (or nominal) value of €1000 and a market price of €900. Is there an arbitrage opportunity? If yes, define an arbitrage strategy and calculate the arbitrage profit (you do not need to construct the detailed table of payoffs). e) Calculate the duration of bond B. Use it to approximate the impact on its price of a uniform decrease of 2% in interest rates. Compare it with the exact increase in the bond price and explain why there is a difference.

a) From 100 b1 = 95.24 5.5 b1 + 105.5 b2 = 99.13 we can calculate b1 = 0.9524 and b2 = 0.8900. b) From b1 = 1 / (1 + r1) and b2 = 1 / (1 + r2) 2 we calculate r1 = 5% and r2 = 6%.

From (1 + r1) (1 + f2) = (1 + r2) 2 we obtain f2 = 7%. c) Solving 99.13 = 5.5 / (1 + i) + 105.5 / (1 + i)2, we obtain i=5.97%. d) The price of C must be equal to 1000 b2 = 890 EUR in the absence of arbitrage. An arbitrage strategy: Short 1 unit of C long 9.479 units of B Short 0,521 units of A Profit = 900 + 0.521 * 95.24 -9.479 * 99.13 = 10 EUR e) Present value of cash flows using the IRR: 5,190 and 93,948. Therefore, the weights are 0.0523 for t=1 and 0.9477 for t=2. It follows that Dur= 1*0.0523 + 2*0.9477 = 1.948 DM = DUR/(1+y) = 1,838. Therefore, if dy=-0.02, dB/B = -DM*dy = 0.0368 or 3.68% The price of the bond calculated with the new interest rates is 102.8869, that is, a real increase of 3.78%. The difference is due to the convexity of the relationship between the price of the bond and the discount rates. 6. Final Exam-September 2012, question 1. Assume that there are 2 risk-free coupon bonds. Their face values are both equal to €100 and their maturities are one and two years. Their coupons are 6% and 10% respectively, and their market prices are €100.95 and €101.605. a) Compute the prices of the basic bonds 1 and 2 and the spot interest rates for one and two years. b) What is the yield to maturity (or internal rate of return) of the two-year bond? Prove that the implied forward rate f2 of a loan which delivers the principal in one year time and maturity of two years is 13.78%? Hint: The solution for an equation of the form ax2 bx c 0 2 b b 4ac 2a. is x

c) Now assume that the prevailing forward rate in the market is 15% instead of 13.78%. If there is an arbitrage opportunity, take advantage of it using this forward rate and the spot rates of a). Remark: In order to design an arbitrage strategy suppose that you borrow €1000 and invest the same amount in t=0.

d) Consider a new bond with maturity of two years, a face value of €1000 and a coupon of 15%. How much should it cost in order to avoid arbitrage opportunities? e) Exploit an arbitrage opportunity if the market price of the previous bond was €1000. Specify your positions in the two initial coupon bonds and the last coupon bond in d).

a) Calculate the prices of the basic bonds 1 and 2, and the spot interest rates for 1 and 2 years. (Precio) t=0 100,95 101,605

Bono 1 Bono 2

(t = 1) 106,00 10,00

(t = 2)

110,00

Bond 1 = 106.00 units of bb1

 100,95  106,00  b1  b1  r1 

100,95  0,9524 106,00

1  1 0,05  5% 0,9524

Bond 2 = 10 units of bb1 and 110 of bb2  101,605  10  b1  110  b 2  b2 

101,605  10  b1 101,605  10  0,9524  0,8371  110 110

1

 1 2 r2     1  0,093  9,3%  0,8371 

b) 101,605 

10 110  i=9.1%  1  i  1  i 2

1 f 2  1  r2   1,093 1  r1  1,05 2

c) There is arbitrage:

2

 1,1378  f 2  13,78%

1  f 2   1 r2  1  r1 

2

 1,15 

1,0932  1,15  1,1378 1,05

Arbitrage strategy: Borrow € 1000 for two years at 9.3% Lend € 1000 for one year at a spot rate r1=5% Lend the procceds of the loan you provided for one year at a forward rate f2=15%:

b.b.2 b.b.1 f2

t=0 +1000 -1000 0

t=1 + 1050 -1050

t=2 -1194.65 +1207.5

Arbitrage profit at t=2 is 12.85 euros

d) Bond 3: N = € 1000, coupon = 15%, T = 2. Bond 3 = portfolio of 150 bb1 and 1150 bb2 Price (Bond 3) = 150 𝑏1 + 1150 𝑏2 = 150(0.9524) + 1150(0.8371) = € 1105,53 e) If the market price of the bond were € 1000 there would be an arbitrage opportunity since the price of the bond does not coincide with the no-arbitrage price, € 1105.53. The arbitrage strategy will consist of a long position in bond 3 and a short position in the replicating portfolio consisting of bonds 1 and 2 We must solve the following system of equations:

150  106  z 1  10  z 2  1150  0  z 1  110  z 2 Solution: z1  0, 43; z 2  10,45. We denote the new portfolio as Z.

7. July 2018 Assume that the pure expectation hypothesis is true. There is a one-year basic bond with a price 0.99 and a two-year bond paying a 5% annual coupon that trades at 104.9 €. Compute the expected interest rate for a loan between year 1 and year 2.

5 ∗ 0.99 + 105 ∗ b2 = 104.9, Solution is: {[b2 = 0.951 904 8]} 1/0.951 = (1/0.99) ∗ (1 + f2), Solution is: f 2= 4% Under the PEH: E[1r2] = f2 = 4%

8. June 2018 Consider the following information on two default-free bonds:

Bond X Bond Y

Coupon rate 0% 5%

Nominal 100 100

Maturity 2 years 2 years

Price 96.4 106.17

Suppose that two basic bonds maturing in one and two years, respectively, start to trade in the market. If there are no arbitrage opportunities, what are the prices of these bonds?

Solving the system of fundamental pricing equations for the two bonds: b2 = 0.964 b1 = ( 106.17-105*0.964)/5 =

0.99

9. June 2018 Given the data from the previous question. If the pure liquidity preference hypothesis is true and the liquidity premium for the period between year 1 and year 2 , 1L2, is 1%, compute the expected interest for a loan that is initiated in one year with maturity two years later? r1=1/b1-1 = r2 = (1/b2)0.5-1

0.010 0.0185

f2 = E(1r2)= f2-1L2

0.026971 0.01697...