U K Revoery
MGT 547 Fixed Income Security Analysis
Professor Hongjun Yan
YALE SCHOOL OF MANAGEMENT
Solutions to Problem Set 7: Caps, Floors, Collars
1) Use the interest rate tree below. The first number at each node is the 0.5-year rate at that node. The next number is the price of $1 par of a 0.5-year zero at that node. The next number, at the time 0 node, is the price of $1 par of a 1-year zero.
Time 0
Interest Rate Model
Time 0.5
5.526%
0.973113
4.9%
0.976086
0.952587
4.345%
0.978737
a) What is the value of a 5.25% interest rate cap on $100 notional amount of a 1-year semi-annual floating rate note?
The cap does not affect the time 0.5 coupon on the floater, which is based on the time 0 rate of 4.9%. The cap can only get in the money if the time 0.5 rate goes up to 5.526%, in which case it has a positive payoff at time 1. The tree below gives the values of this possible payoff.
Time 0 cap (0.1342+0)×0.976086
Time 0.5
0.1342=0.973113×(5.526-5.25)/2
0.0655=0.5×
0
b) What is the value of $100 par of a 1-year semi-annual floating rate note that is capped at 5.25%?
100 - 0.0655 = 99.9345
c) Consider a 1-year semi-annual interest rate collar with a 5.25% cap. What floor rate makes the collar worth zero?
The cap is currently out of the money, so it is safe to guess that the same-cost floor must be too, that is, the floor rate is below 4.9%. Then the floor payoff depends only on the realization of the time 0.5 rate. Let k be the floor rate.
Time 0 floor Time 0.5
0
0.0655
0.978737×100×(k-0.04345)/2
k must solve 0.0655 = 0.976086×0.5×0.978737×100×(k-0.04345)/2 ⇒ k = 4.619%.
1
2) Use the interest rate tree below. The first number at each node is the 0.5-year rate at that node. The second number at each node is the price of $1 par of a 0.5-year zero at that node. The third number (if it appears) is the price of $1 par of a 1-year zero at that node. The fourth number (if it appears) is the price of $1 par of a
Professor Hongjun Yan
YALE SCHOOL OF MANAGEMENT
Solutions to Problem Set 7: Caps, Floors, Collars
1) Use the interest rate tree below. The first number at each node is the 0.5-year rate at that node. The next number is the price of $1 par of a 0.5-year zero at that node. The next number, at the time 0 node, is the price of $1 par of a 1-year zero.
Time 0
Interest Rate Model
Time 0.5
5.526%
0.973113
4.9%
0.976086
0.952587
4.345%
0.978737
a) What is the value of a 5.25% interest rate cap on $100 notional amount of a 1-year semi-annual floating rate note?
The cap does not affect the time 0.5 coupon on the floater, which is based on the time 0 rate of 4.9%. The cap can only get in the money if the time 0.5 rate goes up to 5.526%, in which case it has a positive payoff at time 1. The tree below gives the values of this possible payoff.
Time 0 cap (0.1342+0)×0.976086
Time 0.5
0.1342=0.973113×(5.526-5.25)/2
0.0655=0.5×
0
b) What is the value of $100 par of a 1-year semi-annual floating rate note that is capped at 5.25%?
100 - 0.0655 = 99.9345
c) Consider a 1-year semi-annual interest rate collar with a 5.25% cap. What floor rate makes the collar worth zero?
The cap is currently out of the money, so it is safe to guess that the same-cost floor must be too, that is, the floor rate is below 4.9%. Then the floor payoff depends only on the realization of the time 0.5 rate. Let k be the floor rate.
Time 0 floor Time 0.5
0
0.0655
0.978737×100×(k-0.04345)/2
k must solve 0.0655 = 0.976086×0.5×0.978737×100×(k-0.04345)/2 ⇒ k = 4.619%.
1
2) Use the interest rate tree below. The first number at each node is the 0.5-year rate at that node. The second number at each node is the price of $1 par of a 0.5-year zero at that node. The third number (if it appears) is the price of $1 par of a 1-year zero at that node. The fourth number (if it appears) is the price of $1 par of a