Use the interest rate tree below. The first number at each node is the 0.5-year

Question

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.

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?
b) What is the value of $100 par of a 1-year semi-annual floating rate note that is capped at 5.25%?

Please explain

Explanation / Answer

The interest rate cap will be triggered only in the case the interest rate goes up to 5.526%

The payout in that case will be 100 x (5.526% - 5.25%) x 0.973113 x 0.976086 = 0.262156

0.973113 discounts cashflow receivable at time 1 to time 0.5 and 0.976086 discounts cashflow from time 0.5 to time 0

The probability of interest rate going up is 50% and hence value of the cap will be 0.262156 / 2 = 0.131078

The value of a $100 par of a 1 year semi annual floating rate note that is capped at 5.25% can be thought of as a combination of two securities 1) A normal floating rate note & 2) An interest rate cap @ 5.25% which has been sold by the purchaser / bought by the issuer of the security. Since the purchaser of the bond has sold an additional interest rate cap its value should be lower by the value of the cap to compensate him for the same. Hence the value of the floating rate note = 100 – 0.131078 = 99.868922

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