1. Suppose the market price is 150 for a security paying 100 a year from now and 90 two years from now. Further suppose that the spot one-year rate is 15% and the forward rate for lending from one year out to two years out is 10%.
The PV of the cash flow is:
As the cash flow is cheap, we should buy the cash flow and finance by spot borrowing and forward borrowing.
time (years from now) | 0 | 1 | 2 |
---|---|---|---|
buy the cash flow | -150 | 100 | 90 |
spot borrowing | 158.10 | -181.82 | 0 |
forward borrowing | 0 | 81.82 | -90 |
total net cash flow | 8.10 | 0 | 0 |
Strategy for constructing arb (match cash flow): We start from the year that has the minimal number of cash flows, which is usually the last year. Then we work back our way to the present by matching the cash flows year by year.
2. Consider the following scenario. A marketed claim is a riskless self-amortizing loan that pays $250 a year from now and $150 two years from now and is priced at $350 in the market. A zero-coupon bond maturing one year from now costs 90¢ per dollar of face value, while a zero-coupon bond maturing two years from now costs 80¢ per dollar of face value.
The PV is $250*0.9+$150*0.8 = $345.
The NPVis $345-$350 = -$5.
As the price of the cash flow is high, according to our "deep theoretical insight", we should sell the cash flow, and buy the zero-coupon bonds.
time (years from now) | 0 | 1 | 2 |
---|---|---|---|
sell the cash flow | 350 | -250 | -150 |
buy 1-year zero | -225 | 250 | 0 |
buy 2-year zero | -120 | 0 | 150 |
the total net cash flow | 5 | 0 | 0 |
The implied forward rate f(0,1) = 1/D(0,1)-1 = 1/0.9-1 = 11.11%.
The implied forward rate f(0,2) = D(0,1)/D(0,2)-1 = 0.9/0.8-1 = 12.5%.
The implied forward rate f(0,2) is lower than the actual borrowing and lending rate one year from now, i.e. 12.5% < 15%. Yes, there is an arbtrage opportunity. We should lend forward.
At the scale of $1, the arb is
time (years from now) | 0 | 1 | 2 |
---|---|---|---|
lend forward | 0 | -1 | 1.15 |
buy 1-year zero | -0.9 | 1 | 0 |
short 2-year zero | 0.92 | 0 | -1.15 |
the total net cash flow | 0.02 | 0 | 0 |
4. Suppose we can borrow and lend forward one year from now at 12%, and that a 10% coupon bond maturing one year from now is priced at $100 and a 10% coupon bond maturing two years from now is priced at $97.1. Coupons are paid annually and this year's coupon payment has already been made.
In order to calculate the implied forward rate f(0,2), we need to compute the discount factors first.
D(0,2) can be computed by solve the following equation:
D(0,2) = ($97.1-$10*D(0,1))/$110 = 0.8.
The implied forward rate f(0,2) = D(0,1)/D(0,2)-1 = 0.9091/0.8-1 = 13.6%.
The implied forward rate f(0,2) is higher than the actual borrowing and lending rate one year from now, i.e. 13.6% > 12%. Yes, there is an arbtrage opportunity. We should borrow forward.
At the scale of $100, the arb is
time (years from now) | 0 | 1 | 2 |
---|---|---|---|
borrow forward | 0 | 100 | -112 |
short 1-year coupon bond | 100.16 | -110.18 | 0 |
buy 2-year coupon | -98.87 | 10.18 | 112 |
the total net cash flow | 1.29 | 0 | 0 |
If coupon bonds cannot be bought and sold fractionally, one can construct an arb as follows.
time (years from now) | 0 | 1 | 2 |
---|---|---|---|
borrow forward | 0 | 98.2 | -110 |
lend spot | -1.64 | 1.8 | 0 |
short 1-year coupon bond | 100 | -110 | 0 |
buy 2-year coupon | -97.1 | 10 | 110 |
the total net cash flow | 1.26 | 0 | 0 |
5. Assume you observe the following three coupon bond prices and remaining cash flows (coupons are paid annually and this year's coupon has already been paid):
In order to compute the discount factors, we can construct portfolios that give us cash flow only at the time that we are interested in. However, we will use a different approach, which is more mechanical, and intuitive. We solve the following system of equations:
110*D(0,1) = 100
10*D(0,1)+110*D(0,2) = 105
10*D(0,1)+10*D(0,2)+110*D(0,3) = 107
This gives us D(0,1) = 0.9091, D(0,2) = 0.8719, D(0,3) = 0.8108.
one year par coupon bond yield:
y_{2} = (1-D(0,2))/(D(0,1)+D(0,2)) = (1-0.8719)/(0.9091+0.8719) = 7.2%;
y_{3} = (1-D(0,3))/(D(0,1)+D(0,2)+D(0,3)) = (1-0.8108)/(0.9091+0.8719+0.8108) = 7.3%
6. Suppose a zero-coupon bond maturing one year from now costs 90¢ per dollar of face value, a zero-coupon bond maturing two years from now costs 80¢ per dollar of face value, and a zero-coupon bond maturing three years from now costs 70¢ per dollar of face value.
Calculate:
Using the approximation formula z(0,t) = (1-D(0,t))/t, the yields are
z(0.2) = (1-D(0,2))/2 = (1-0.8)/2 = 10%;
z(0,3) = (1-D(0,3))/3 = (1-0.7)/3 = 10%.
Using the exact formula z(0,t) = D(0.t)^{-1/t}-1, the yields are
z(0,2) = D(0,2)^{-1/2}-1 = 0.8^{-1/2}-1 = 11.8%;
z(0,3) = D(0,3)^{-1/3}-1 = 0.7^{-1/3}-1 = 12.6%;
f(0,1) = 1/D(0,1)-1 = 1/0.9-1 = 11.11%;
f(0,2) = D(0,1)/D(0,2)-1 = 0.9/0.8-1 = 12.5%;
f(0,3) = D(0,2)/D(0,3)-1 = 0.8/0.7-1 = 14.3%
The yield of a par coupon bond is
7. You are managing the final years of a pension fund. There are three remaining dates at which lump-sum payments will be made to beneficiaries: $320 million 6 months from now, $161 million 12 months from now, and $208 million 18 months from now.
To match the cash flows, we start from the last period. We need to buy two T-bond 3 to match the last period's cash flow. Similarly, we need 1.5 T-bond 2 to match the second to last period's cash flow. Finally we need 3 T-bond 1 to match month 6's cash flow (see the table below).
The market value of the pension liability is the value of the portfolio.
time(months out) | 0 | 6 | 12 | 18 |
---|---|---|---|---|
pension fund cash flow | -653 | -320 | -161 | -208 |
TBond 1 | -100*3 | 103*3 | 0 | 0 |
TBond 2 | -98*1.5 | 2*1.5 | 102*1.5 | 0 |
TBond 3 | -103*2 | 4*2 | 4*2 | 104*2 |
8. Suppose a coupon bond with a face value of 100, maturing 3 years from now, has a coupon rate of 10% (paid annually). What are its price and duration if the market discout factors are 0.9, 0.85, 0.8 for 1-year, 2-year and 3-year bonds.
The price is
Duration is
9. Assuming the sensitivity of discount bond prices to a shock is given by sens(t-s)=(1-exp(-.125(t-s)))/.125 (as we have been assuming), compute the Macauley duration and the effective duration of a bond which pays 1/4 of its value at 30 years out, 1/4 of the value 20 years out and 1/2 of its value at 10 years out. Either use the graph on the lecture note to obtain an approximate value, or use the formulas from the lecture note to perform a more exact computation.
sens(10) = (1-exp(-.125*10))/.125 = 5.7
sens(20) = (1-exp(-.125*20))/.125 = 7.3
sens(30) = (1-exp(-.125*30))/.125 = 7.8
sens of the bond = 1/2*5.7+1/4*7.3+1/4*7.8 = 6.6
The effective duration is -log(1-0.125*sens)/.125 = 14
10. Consider a two-period binomial model in which the short riskless interest rate starts at 30% and moves up or down by 10% each period (i.e., up to 40% or down to 20% at the first change). The artificial probability of each of the two states at any node is 1/2.
The interest tree is
|- 50% |- 40% -| 30% -| |- 30% |- 20% -| |- 10%
and the value of the bond is given by
|- 100 |- 71.43 -| 59.52 -| |- 100 |- 83.33 -| |- 100
and the value of the option is given by
|- 0 3.20 -| |- 8.33
The price of a two-year discount bond is $59.52 and the price of the option is $3.20, as immediate exercise is not profitable.
11. As interest rate might follow a mean-reverting process, the assumption of constant artificial probabilities at each node might not be reasonable. Under the assumption of mean-reverting process, the probability of interest rete going up is a decreasing function of the level of interest rate. Assume the probability of going up p_{u} is 0.6 when interest rate r is 20%, 0.4 when r is 30%, and 0.2 when r is 40%. Solve the previous problem under the new assumption.
The interest tree is
|- 50% |- 40% -| 30% -| |- 30% |- 20% -| |- 10%
and the value of the bond is given by
|- 100 |- 71.43 -| 60.44 -| |- 100 |- 83.33 -| |- 100
and the value of the option is given by
|- 0 3.84 -| |- 8.33
The price of a two-year discount bond is $60.44 and the price of the option is $3.84, as immediate exercise is not profitable.
12. Consider a two-year binomial model. Start with an original model in which the short riskless interest rate starts at 10% and moves up or down by 5% each period (i.e., up to 15% or down to 5% at the first change). The artificial probability of each of the two states at any node is 1/2.
The interest tree is
|- 20% |- 15% -| 10% -| |- 10% |- 5% -| |- 0%
and the value of the two-year bond is given by
|- 100 |- 86.96 -| 82.82 -| |- 100 |- 95.24 -| |- 100
and the value of the one-year bond is given by
|- 100 90.91 -| |- 100
The discount factors computed from the model is
D^{om}(0,1) = 0.9091,
D^{om}(0,2) = 0.8282.
The discount factors from the economy are
D(0,1) = 1/1.11 = 0.9009,
D(0,2) = 1/(1.12)^{2}= 0.7972. Therefore, the fudge factors are f1 = 0.9091/0.9009 = 1.0091,
f2 = (0.8282/0.9091)/(0.7972/0.9009) = 0.9110/0.8849 = 1.0295.
The adjusted interest rates are r_{1}^{*} = (1+r_{1})*f1-1 = (1+0.1)*1.0091-1 = 11%
r_{2}^{*} = (1+r_{2})*f2-1, it is 1.15*1.0295-1 = 18.4% for upstate, and 1.05*1.0295-1 = 8.1% for down state.
The adjusted interest rate process is
|-18.4% 11% -| |- 8.1%
The above formula is just an approximation. We can compute the implied forward rates to verify.
The implied forward rates in the economy:
f(0,1) = 11%,
f(0,2) = D(0,1)/D(0,2)-1 = 0.9009/0.7972-1 = 13%.
The implied forward rates from the model are
f^{om}(0,1) = 10%,
f^{om}(0,2) = D^{om}(0,1)/D^{om}(0,2)-1 = 0.9091/0.8282-1 = 9.8%.
The difference in interest rate in first year is 1%, and 3.2% in the second year.
So the adjusted interest rate process is
|-18.2% 11% -| |- 8.2%Thus we are ensured that the results are indeed quite close.