stochastic INTEREST RATE ANALYSIS
I.SHORT RATES:
The short rate r(t,j) at state (t, j) is the risk-free rate for borrowing/lending for the upcoming period. At time 0, the risk-free rate is 7%. A deposit of 1 at time 0 will yield 1.07 at the end of the period, risk-free. If the short rate goes “up” next period to 9.1%, then a deposit of 1 at time 1 will yield 1.091 at the end of the period, risk-free. If, on the other hand, the short rate went “down” next period to 6.3%, then a deposit of 1 at time 1will yield 1.063 at the end of the period, risk-free.
SHORT RATE PROCESS r[t, j] / u = 1.3 / d = 0.90 / 1 / 2 / 3 / 4 / 5 / 6
0.07 / 0.091 / 0.1183 / 0.15379 / 0.199927 / 0.259905
0.063 / 0.0819 / 0.10647 / 0.138411 / 0.179934
0.0567 / 0.07371 / 0.095823 / 0.124570
0.05103 / 0.066339 / 0.086241
0.045927 / 0.059705
0.041334
II.ELEMENTARY PRICES:
The Elementary Price at node j at time t, EP(t, j), represents the no-arbitrage value at time 0 of receiving exactly one dollar at node j at time t, and zero dollars in all other states. Here, j = 0, 1, 2, … , t, where j = 0 refers to the top node in column t. It can be calculated using discounted expectation, keeping in mind that the interest rate that applies for the period is the known value of the short rate.
ELEMENTARY PRICES EP(t, j) / q = 0.50 / 1 / 2 / 3 / 4 / 5 / 6
1 / 0.46729 / 0.214157 / 0.095751 / 0.041494 / 0.017290 / 0.006862
0.46729 / 0.433954 / 0.296303 / 0.175390 / 0.094323 / 0.046831
0.219798 / 0.304554 / 0.275719 / 0.202837 / 0.130154
0.104002 / 0.191299 / 0.215503 / 0.189381
0.049476 / 0.113351 / 0.152679
0.023652 / 0.064839
0.011357
Examples:
1.EP(1,0) = [0.5(1) + 0.5(0)]/(1.07) = 0.467290. Note EP(1, 0) = EP(0, 1).
2. Calculation for EP(2, 1).
0 / 1 / 2[0.5(0.458295) + 0.5(0.470367)]/1.07 = 0.433954 / [0.5(0) + 0.5(1)]/(1.091) = 0.458295 / 0
[0.5(1) + 0.5(0)]/(1.063) = 0.470367 / 1
0
The Elementary Price matrix can be completed very efficiently if one computes the prices forward in time, as follows:
ELEMENTARY PRICES EP(t, j) / q = 0.50 / 1 / 2 / 3 / 4
1 / 0.46729 / 0.214157 / 0.095751 / 0.041494
0.46729 / 0.433954 / 0.296303 / 0.175390
0.219798 / 0.304554 / 0.275719
0.104002 / 0.191299
0.049476
3.Start with EP(0, 0) = 1, and EP(1,0) = EP(0, 1) = 0.5/[1 + R(0,0)].
4.Now consider, once again, EP(2, 1). Its value at state (t=1, j=0) is 0.458295. We already know that the value of at time 0 of receiving exactly one dollar at state (t=1, j=0) and zero dollars in all other states is 0.467290. Hence:
The value at state (t=1, j=0) of 0.458295 equals (0.458295)(0.467290) = 0.214156at time 0.
Similarly, the value of EP(2, 1) at state (t=1, j=1) is 0.470367. We already know that the value of at time 0 of receiving exactly one dollar at state (t=1, j=1) and zero dollars in all other states is 0.467290. Hence:
The value at state (t=1, j=1) of 0.470367 equals (0.470367)(0.467290) = 0.219798 at time 0.
The total value at time 0 is the sum of these two numbers, and equals 0.214156 + 0.219798 = 0.433954.
5.Consider EP(4,2). Its value at state (t=3, j=2) is 0.5/(1.07371) = 0.465675 since r(3, 2) = 0.07371. From the table, we know that the value of at time 0 of receiving exactly one dollar at state (t=3, j=2) and zero dollars in all other states is 0.304554. Hence:
The value at state (t=3, j=2) of 0.465675 equals (0.465675)(0.304554) = 0.141823 at time 0.
Similarly, the value of EP(4, 2) at state (t=3, j=1) is 0.5/1.10647 = 0.451888 since r(3,1) = 0.10647. From the table, we know that the value of at time 0 of receiving exactly one dollar at state (t=3, j=1) and zero dollars in all other states is 0.296303. Hence:
The value at state (t=3, j=1) of 0.451888 equals (0.451888)(0.296303) = 0.133896 at time 0.
The total value at time 0 is the sum of these two numbers, and equals 0.141823 + 0.133896 = 0.275719.
6.General (Recursive) Formula:
“up” “across”
EP(t, j) = 0.5*EP(t-1, j-1)/(1 + r(t-1, j-1)) + 0.5*EP(t-1, j)/(1 + r(t-1, j)).
where it is understood EP( ) = 0 if the state does not exist.
III.ZERO COUPON BOND PRICES:
The value B(0, t) represents the no-arbitrage price at time 0 of receiving 1 dollar in all states at time t. This zero coupon bond is said to have a maturity of t periods.
The value B(0, t) is easily obtained from the Elementary Prices via the formula:
B(0, t) = EP(t, 0) + EP(t, 1) + EP(t, 2) + … + EP(t, t).
ZERO COUPON BOND PRICES B(0, t)1 / 2 / 3 / 4 / 5 / 6
0.934579 / 0.867908 / 0.800610 / 0.733378 / 0.666957 / 0.602103
1.Consider, for example, B(0, 3):
B(0, 3) = 0.095751 + 0.296303 + 0.304554 + 0.104002 = 0.800610.
2.Zero coupon bond prices can also be expressed as an expectation, as follows. Define the (random) discount process
,t = 1, 2, … , T; D(0) = 1.
Then B(0, t) = E[D(t)]. Note that D(t) is only based on information up to (and including) time t-1.
3.Consider, once again, B(0, 3). At time t = 2 the short rate process r(0), r(1), r(2) is known. The value of 1 at time 3 will be discounted by ( [1 + r(0)][1 + r(1)][1 + r(2)] )-1 . There four possible paths for the short rate process up (and including) time t = 2: (0.07, 0.091, 0.1183), (0.07, 0.091, 0.0819), (0.07, 0.063, 0.0819), (0.07, 0.063, 0.0567). Each path is equally likely with probability equal to 0.25. Now
which equals the value of B(0,3).
IV.SPOT RATES:
The spot rate s(t) is the effective rate of interest one receives when one purchases a zero coupon bond with a maturity of t periods. The collection of spot rates is called the yield curve.
ZERO COUPON BOND PRICES B(0, t)1 / 2 / 3 / 4 / 5 / 6
0.934579 / 0.867908 / 0.800610 / 0.733378 / 0.666957 / 0.602103
SPOT RATES s(t)
1 / 2 / 3 / 4 / 5 / 6
0.070000 / 0.073404 / 0.076944 / 0.080608 / 0.084377 / 0.088232
1.Consider, for example, a 3-period zero coupon bond. It cost 0.800610 to purchase, and will pay off 1 at time t = 3. The appropriate interest rate satisfies the identity
0.800610*(1 + s(3))3 = 1, which implies that s(3) = (0.800610)-1/3 – 1 = 0.076944 or 7.6944%.
In general, s(t) satisfies the identity
B(0, t)*(1 + s(t))t = 1.
2.The definition of spot rate implicitly assumes a compounding convention. The example above assumed a one-period compounding convention (as does the whole table). Typically, the period length is one year, and semi-annual compounding is used. In this case, the appropriate interest rate satisfies the general identity
B(0, t)*[(1 + s(t)/2)2]t = 1.
V.FORWARD RATES
The forward rate f(t, j), t < j, is the rate of interest one can lock in for borrowing/lending money over the period from time t to time j.
FORWARD RATES f(t,j)1 / 2 / 3 / 4 / 5 / 6
1 / 1 / 0.076782 / 0.080433 / 0.084167 / 0.088001 / 0.091915
2 / 1 / 0.084058 / 0.087860 / 0.091756 / 0.095723
3 / 1 / 0.091674 / 0.095624 / 0.099639
4 / 1 / 0.099589 / 0.103643
5 / 1 / 0.107712
1.Construct the following portfolio at time zero: short 1 t-period zero coupon bond, and purchase
B(0, t)/B(0, j) j-period zero coupon bonds. The cost of this portfolio is zero since
B(0, t) – [B(0, t)/B(0, j)] B(0, j). (Keep in mind the price of an s-period zero coupon bond is B(0, s).)
The holder of this portfolio will have to pay 1 at time t to honor his obligation for selling the t-period bond short at time zero. The holder will receive B(0, t)/B(0, j) at time j.
The effective interest rate over the periods t to j of this investment is
[B(0, t)/B(0, j)] 1/(j-t) - 1 := f(t, j).
2.Consider, for example, t = 2 and j = 5. Here, B(0, 2) = 0.867908 and B(0, 5) = 0.666957. The ratio B(0, 2)/B(0, 5) = 1.301295, and so f(2, 5) = (1.301295) 1/(5-2) - 1 = (1.301295) 1/3 - 1 = 0.091756 or 9.1756%.
3.From (a), we have
[1 + f(t, j)] (j-t) B(0, j) = B(0, t).
Recall that B(0, t)*(1 + s(t))t = 1---see IV(a) above. Thus, B(0, t) = (1 + s(t))-t . Substituting this identity for both B(0, j) and B(0, t) in the above equation yields
(1 + s(t))t (1 + f(t, j)) (j-t) = (1 + s(j)) j.
In the example of (b), we have (1 + 0.073404)2 (1 + 0.091756)3 = (1 + 0.084377)5.
4.The identity above presents two alternative ways to invest 1 at time 0 for j periods:
Time:012345
A:1 ------>1.0843775
B:1------> (1.073404)2 ------>1.0843775
In (A), you purchase 1.0843775 5-period zero coupon bonds whereas in (B) you first purchase (1.073404)2 2-period coupon bonds, and then reinvest this sum at the forward rate of 9.1756% over the next 3 periods.
VI.COUPON-BEARING BOND PRICES:
Each coupon bond has a maturity, T, that represents the “life” of the bond. It also has a face value, F, and a coupon rate c, 0 < c < 1. A coupon bond pays a coupon, cF, at the end of each period t = 1, 2, … , T, as well as a payment of F at time T. Typically, F = 1000. Here, we will normalize F to 1.
ZERO COUPON BOND PRICES B(0, t)1 / 2 / 3 / 4 / 5 / 6
0.934579 / 0.867908 / 0.800610 / 0.733378 / 0.666957 / 0.602103
1.Consider a 6% coupon bond with a maturity of 4 periods. The associated cash flow vector is (0, 0.06, 0.06, 0.06, 1.06). What is its no-arbitrage value at time 0?
We can use the zero coupon bond prices to calculate the value of this coupon bond, as follows.
The bond holder receives 0.06 at time 1 regardless of the state. The value at time 0 of receiving 1 at time 1 is B(0, 1) = 0.934579. Hence:
The value at time 0 of receiving 0.06 at time 1 is (0.06)(0.934579) = 0.056075.
The bond holder receives 0.06 at time 2 regardless of the state. The value at time 0 of receiving 1 at time 2 is B(0, 2) = 0.867908. Hence:
The value at time 0 of receiving 0.06 at time 2 is (0.06)(0.867908) = 0.052075.
It should be clear now that the value of this bond can be calculated as:
0.06[ B(0,1) + B(0,2) + B(0.3) + B(0,4) ] + B[0,4] = 0.06(3.336475) + 0.733378 = 0.933567.
2.The value of a coupon bond or any cash flow stream can also be calculated using the spot rates in a conventional present value analysis.
SPOT RATES s(t)1 / 2 / 3 / 4 / 5 / 6
0.070000 / 0.073404 / 0.076944 / 0.080608 / 0.084377 / 0.088232
For example, the value of the 6%, 4-period bond is
0.933567 = 0.06/(1 + 0.07) + 0.06/(1 + 0.073404)2 + 0.06/(1 + 0.076944)3 + 1.06/(1 + 0.080608)4.
3.This bond is selling at less than 1, and so is said to be “selling at a discount.” The coupon rate of 6% is less than the prevailing spot rates, and so to compensate the cost of the bond is less than 1.
4.The value of a 9%, 4-period bond is
0.09[ B(0,1) + B(0,2) + B(0.3) + B(0,4) ] + B[0,4] = 0.09(3.336475) + 0.733378 = 1.033661.
This bond is selling at greater than 1, and so is said to be “selling at a premium.” The coupon rate of 9% is greater than the prevailing spot rates, and so to compensate the cost of the bond is more than 1.
5.A coupon bond whose price equals the face value (here set to 1) is said to be “selling at par.” What is the coupon rate c that would yield a 3-period par bond? It must be the value c that satisfies
c(3.336475) + 0.733378 = 1, which implies that c = 0.079911 or 7.9911%.
VII.FIXED RATE VALUATION
Consider a contract in which the seller will receive at each time t a fixed rate r multiplied by some notational amount N. For convenience, we take N = 1. What is the no-arbitrage time 0 value of this contract? Let FixedRateVal[t, j] denote the value of this contract at state (t, j). We shall illustrate the calculations with our example. Suppose r = 0.0864 in our example.
SHORT RATE PROCESS r[t, j] / u = 1.3 / d = 0.90 / 1 / 2 / 3 / 4 / 5
0.07 / 0.091 / 0.1183 / 0.15379 / 0.199927 / 0.259905
0.063 / 0.0819 / 0.10647 / 0.138411 / 0.179934
0.0567 / 0.07371 / 0.095823 / 0.124570
0.05103 / 0.066339 / 0.086241
0.045927 / 0.059705
0.041334
FIXED RATE VALUATION (6 Periods) / Rate = 0.0864
0 / 1 / 2 / 3 / 4 / 5
0.397918 / 0.325597 / 0.257523 / 0.193143 / 0.131092 / 0.068577
0.353147 / 0.280130 / 0.210034 / 0.141800 / 0.073224
0.297861 / 0.223312 / 0.150193 / 0.076829
0.233388 / 0.156551 / 0.079540
0.161246 / 0.081532
0.082970
1.Consider FixedRateVal[5, 0]. At this state the seller will receive 0.0864 at time t = 6. Since the observed short rate is 0.259905, the value to the seller at this state is 0.0864/(1 + 0.259905) = 0.068577.
2.Consider FixedRateVal[5, 1]. At this state the seller will receive 0.0864 at time t = 6. Since the observed short rate is 0.179934, the value to the seller at this state is 0.0864/(1 + 0.179934) = 0.073224. The rest of the values in column 5 are computed in the same way.
3.Consider FixedRateVal[4, 0]. At this state the seller will receive 0.0864 at time t = 5. The value today for this cash flow is 0.0864/(1 + 0.199927) = 0.072004. If the short rate moves up to 0.259905 at t = 6, the value of this contract to the seller is 0.068577; if, on the other hand, the short rate moves down to 0.179934, the value of this contract to the seller is 0.073224. The value today for this future value is
[0.5*(0.068577) + 0.5(0.073224)]/(1 + 0.199927) = 0.059087.
The total value to the seller of the contract at this state is the sum(0.072004 + 0.059087) = 0.131092. Continuing backwards through the binomial lattice in this fashion yields the value of 0.397918.
4.The value of this contract is more easily calculated as
0.0864[B(0,1) + B(0,2) + B(0, 3) + B(0, 4) + B(0, 5) + B(0, 6)] = 0.0864(4.605535) = 0.397918.
VIII.FLOATING RATE VALUATION
Consider a contract in which the seller will receive at each time t the short rate r(t-1) observed in the previous period multiplied by some notational amount N. For convenience, we take N = 1. What is the no-arbitrage time 0 value of this contract? Let FloatingRateVal[t, j] denote the value of this contract at state (t, j). We shall illustrate the calculations with our example.
SHORT RATE PROCESS r[t, j] / u = 1.3 / D = 0.90 / 1 / 2 / 3 / 4 / 5
0.07 / 0.091 / 0.1183 / 0.15379 / 0.199927 / 0.259905
0.063 / 0.0819 / 0.10647 / 0.138411 / 0.179934
0.0567 / 0.07371 / 0.095823 / 0.124570
0.05103 / 0.066339 / 0.086241
0.045927 / 0.059705
0.041334
FLOATING RATE VALUATION (6 Periods)
0 / 1 / 2 / 3 / 4 / 5
0.397897 / 0.404737 / 0.398740 / 0.373079 / 0.316119 / 0.206289
0.306763 / 0.302397 / 0.282142 / 0.237211 / 0.152495
0.223781 / 0.208385 / 0.174212 / 0.110771
0.151155 / 0.125857 / 0.079394
0.089819 / 0.056341
0.039694
1.Consider FloatingRateVal[5, 0]. At this state the seller will receive 0.259905 at time t = 6. Since the observed short rate is 0.259905, the value to the seller at this state is 0.259905(1 + 0.259905) = 0.206289.
2.Consider FloatingRateVal[5, 1]. At this state the seller will receive 0.179934 at time t = 6. Since the observed short rate is 0.179934, the value to the seller at this state is 0.179934/(1 + 0.179934) = 0.152495. The rest of the values in column 5 are computed in the same way.
3.Consider FloatingRateVal[4, 0]. At this state the seller will receive 0.199927 at time t = 5. The value today for this cash flow is 0.199927/(1 + 0.199927) = 0.166616. If the short rate moves up to 0.259905 at t = 6, the value of this contract to the seller is 0.206289; if, on the other hand, the short rate moves down to 0.179934, the value of this contract to the seller is 0.152495. The value today for this future value is
[0.5*(0.206289) + 0.5(0.152495)]/(1 + 0.199927) = 0.149502.
The total value to the seller of the contract at this state is the sum (0.166616 + 0.149502) = 0.316119.
Continuing backwards through the binomial lattice in this fashion yields the value of 0.397918.
4.The value of the floating rate contract at time zerois …. 1 – B(0, 6)!!
How can it be this easy? A floating rate contact can be thought of as 6 individual contracts, one for each period. Therefore, its value is the sum of these 6 contracts.
Consider the contract in period t ε {1, 2, …, 6}, the one that pays off r(t-1) at time t. Suppose at time zero you purchase a zero coupon bond with maturity t-1 and sell a zero coupon bond with maturity t. The cost of this portfolio is B(0, t-1) – B(0, t). You will receive 1 at time t-1. Now, reinvest this at the observed short rate of r(t-1). You will receive 1 + r(t-1) at time t. However, you must pay back 1 to the holder of the zero coupon bond with maturity t, thereby leaving a cash flow of r(t-1) at time t. This trading strategy exactly matches the payoff associated with the period t portion of the floating rate contact! The law of one price says then that the value of this portion is B(0, t-1) – B(0, t).
How can we be sure of this? We know the values in each state of the t=6 portion of the floating rate contract; they are provided in column 5 of the above table. We also know the Elementary Prices associated with each of these states. If everything is consistent, then it must be the case that the t=6 portion has value
Value = FRB(5, 0)*EP(5, 0) + FRV(5, 1)*EP(5, 1) + … + FRV(5, 6)*EP(5, 6)
= (0.206289)(0.17290) + (0.152495)(0.094323) + … + (0.039694)(0.023652)
= 0.064854,
which just happens to equal B(0, 5) – B(0, 6) = 0.666957 – 0.602103!
The total value of the floating rate contract is therefore
[B(0,0) – B(0, 1)] + [B(0,1) – B(0, 2)] + … + [B(0, 4) – B(0, 5)] + [B(0,5) – B(0, 6)] = 1 – B[0, 6].
5.Keep in mind that the table determines the value of the floating rate contract at all states over time.
6.The values of the fixed rate and floating rate contracts are equal. This is not a coincidence. The fixed rate of 0.0864 was chosen to make this true. The value of the fixed rate contract is c(4.605535)---see VI(d) above. The value of the floating rate contract is 1 – B(0, 6). Equating the two values yields the value of c.
7.The value of the floating rate contract can be obtained using discounted expectation as
IX.SWAP VALUATION
What is the value of a contract that allows one to swap a floating rate contact with a fixed rate contract? It is merely the difference in the values of these two contracts. By design, the values are equal, so at a fixed rate of 0.0864, the swap has value zero.
1.Suppose the fixed rate 0.08 instead? What then would be the value of the swap?
X.EUROPEAN SWAPTION VALUATION
Consider a contact that allows the holder an option to swap a floating rate contact with a fixed rate contract at time t = 2. This type of option is called a swaption. What is no-arbitrage value of this contract? Let Swaption(t, j) equal the value of this contract at state (t, j).
Fixed rate valuation (first 2 periods) / Rate = 0.08640 / 1 / 2
0.397918 / 0.325597 / 0.257523
0.353147 / 0.280130
0.297861
Floating rate valuation (first 2 periods)
0 / 1 / 2
0.397897 / 0.404737 / 0.39874
0.306763 / 0.302397
0.223781
SWAPTION VALUATION (2 Period Maturity)
0 / 1 / 2
0.016283 / 0 / 0
0.034845 / 0
0.074080
1.At time t = 2, the holder would choose the floating rate contract if its value were greater than the value of the fixed rate contract. Therefore, Swaption(2, 2) = 0.297861 – 0.223781 = 0.074080. The values of the Swaption in the other states at time t = 2 are zero.
2.Swaption(1, 1) = 0.5(0.074080)/(1 + 0.063) = 0.34845.
3.Swaption(0, 0) = 0.5(0.034845)/(1 + 0.07) = 0.016283.
4. Suppose you had a fixed rate contract for six period with a notional amount of 1 million. You thought interest rates might go down. It will cost you 16,283 to buy a 2-period option to swap to a floating rate contract.
XI.CAP ANALYSIS
The holder of a cap with strike price r receives max{r(t-1) – r, 0} at each time t. (Again, the notional amount N is set to one.) If you purchased a floating rate loan, you could acquire a cap to limit the interest you would pay in any period to the strike rate r. The floating rate loan requires you to pay r(t-1) at time t. If r(t-1) > r, then the cap would pay you the difference r(t-1) – r, and thus the net cash flow for you would be simply r. Let CAP(t, j) denote the value of the cap at state (t, j).
SHORT RATE PROCESS r[t, j] / u = 1.3 / d = 0.90 / 1 / 2 / 3 / 4 / 5
0.07 / 0.091 / 0.1183 / 0.15379 / 0.199927 / 0.259905
0.063 / 0.0819 / 0.10647 / 0.138411 / 0.179934
0.0567 / 0.07371 / 0.095823 / 0.124570
0.05103 / 0.066339 / 0.086241
0.045927 / 0.059705
0.041334
CAP ANALYSIS (6 Periods) / Strike = 0.086395
0 / 1 / 2 / 3 / 4 / 5
0.050107 / 0.086610 / 0.141244 / 0.179948 / 0.185035 / 0.137717
0.020620 / 0.038529 / 0.072149 / 0.095420 / 0.079275
0.005309 / 0.011219 / 0.024092 / 0.033946
0.000000 / 0.000000 / 0.000000
0.000000 / 0.000000
0.000000
1.Consider CAP(5, 0). The short rate is r(5, 0) = 0.259905, which exceeds the strike price of 0.086395 by 0.173510. This will be paid at time 6, and therefore has a value of (0.173510)/(1.259905) = 0.137717 at state (5, 0).
2.Consider CAP(5, 1). The short rate is r(5, 1) = 0.179934, which exceeds the strike price of 0.086395 by 0.093539. This will be paid at time 6, and therefore has a value of (0.093539)/(1.179934) = 0.079275 at state (5, 1). The rest of the column associated with t=5 is calculated in the same manner.
3.Consider CAP(4,0). The short rate is r(4, 0) = 0.199927, which exceeds the strike price of 0.086395 by 0.113532. This will be paid at time 5, and therefore has a value of (0.113532)/(1.199927) = 0.094616 at state (4, 0). The value at state (4, 0) for the future value of the cap is
[0.5(0.137717) + 0.5(0.079275)]/(1.199927) = 0.090419. The total value at state (4, 0) is the sum (0.094616) + (0.090419) = 0.185035. The rest of the table is obtained in a similar fashion.
XI.FLOOR ANALYSIS
The holder of a floor with strike price r receives max{r – r(t-1), 0} at each time t. (Again, the notional amount N is set to one.) If you sold a floating rate loan, you could acquire a floor to ensure that the interest you would receive in any period will be no less than the strike rate r. For the floating rate loan you receive r(t-1) at time t. If r(t-1) < r, then the floor would pay you the difference r – r(t-1), and thus the net cash flow for you would be simply r. To value the floor, proceed in the usual manner via discounted expectation in an exactly analogous manner as was used above to value the cap. Details omitted.
SHORT RATE PROCESS r[t, j] / u = 1.3 / d = 0.90 / 1 / 2 / 3 / 4 / 5
0.07 / 0.091 / 0.1183 / 0.15379 / 0.199927 / 0.259905
0.063 / 0.0819 / 0.10647 / 0.138411 / 0.179934
0.0567 / 0.07371 / 0.095823 / 0.124570
0.05103 / 0.066339 / 0.086241
0.045927 / 0.059705
0.041334
FLOOR ANALYSIS (6 Periods) / Strike = 0.086395
0 / 1 / 2 / 3 / 4 / 5
0.050105 / 0.007451 / 0.000013 / 0.000000 / 0.000000 / 0.000000
0.066984 / 0.016246 / 0.000029 / 0.000000 / 0.000000
0.079371 / 0.026133 / 0.000065 / 0.000000
0.082220 / 0.030685 / 0.000142
0.071417 / 0.025186
0.043272
1.Note how the floor value equals the cap value. Coincidence? Consider the portfolio consisting of 1 floating rate contract, 1 floor contract, and -1 cap contact, i.e., Floating + FLOOR – CAP. (The floor and cap contracts have the same strike price of r.) The payoff at each time t is ….. r! So, this portfolio replicates a fixed rate contract with rate r. SO:
Value of FIXED = Value of (Floating + FLOOR – CAP)
= Value of Floating + Value of FLOOR - Value of CAP
The rate r above was chosen so that the values of the floating rate contract and the fixed rate contract would be equal. And, this gives the result.
XII.ADJUSTABLE RATE LOAN ANALYSIS
Consider a bank who sold an adjustable rate loan for a period of 5 periods. The initial loan balance is N, which we set to one. In each time t, the bank takes the outstanding loan balance and computes the mortgage payment associated with a fixed rate loan at an interest rate equal to r(t) plus an increment (in this case 0.02) with duration equal to the remaining life of the loan. The buyer of the loan will pay this amount at time t+1. The new loan balance is computed by first incrementing the current loan balance by the interest due determined by r(t) and then deducting the payment received.
The valuation of this loan (to the bank) proceeds using the idea of leveling. At each state, we imagine that the outstanding loan balance is 1, and then compute the value of the loan. This is using the property that the value of this loan scales with the outstanding loan balance. First, we have to compute the “annuity” and “loan balance” factors at each state. In what follows, we let r’(t, j) = r(t, j) + 0.02 denote the rate the bank charges above the observed short rate.
1.Let AF(t, j) denote the annuity factor at state (t, j). It represents the fixed-rate mortgage amount that would be paid on a loan with loan balance 1, interest rate of r’(t), and a duration of 5-t periods. We have:
AF(t, j) = r’(t, j)/[1 – (1 + r’(t, j))–(T-t) ].
ANNUITY FACTORS0 / 1 / 2 / 3 / 4
0.257092 / 0.323015 / 0.429503 / 0.633816 / 1.219927
0.303939 / 0.403461 / 0.596733 / 1.158411
0.385725 / 0.571331 / 1.115823
0.553882 / 1.086339
1.065927
Examples:AF(0, 0) = 0.09/[1 – (1.09)-5 ] = 0.257092. AF(1, 0) = 0.111/[1 – (1.111)-4 ] = 0.323015.
2.Let LBF(t, j) denote the loan balance factor at state (t, j). Assuming a current loan balance of 1, it represents the new loan balance at time t+1. We have: