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(1 point) (Exam FM Sample Problem 199) Miaoqi and Nui entered into a six year interest rate swap on May 5, 2015. The notional amount of the swap was a level 30000 for all six years. The swap has annual settlement periods with the first period starting on May 5, 2015. Under the swap, Miaoqi agreed to pay a variable rate based on the one year spot rate at the beginning of each settlement period. Nui will pay Miaoqi the fixed rate of on each settlement date. On May 5, 2017, the spot interest rate curve was as follows:
\begin{tabular}{|c|c|c|c|c|c|c|}
\hline Time & 1 & 2 & 3 & 4 & 5 & 6 \\
\hline Spot Rate & & & & & \\
\hline
\end{tabular}
Miaoqi decides that she wants to sell the swap on May 5, 2017. Calculate the market value of the swap on May 5, 2017, from Miaoqids position in the swap.
Studdy Solution
STEP 1
What is this asking?
We need to figure out how much Miaoqi's swap is worth if she sells it two years into a six-year agreement.
Watch out!
Don't forget that Miaoqi both *receives* and *pays* money through this swap agreement.
We need to consider *both* sides to calculate the market value.
STEP 2
1. Calculate the present value of Miaoqi's fixed payments.
2. Calculate the present value of Miaoqi's variable payments.
3. Determine the market value of the swap.
STEP 3
Miaoqi receives a fixed payment of of the notional amount, which is .
So, each year, she gets .
Woohoo! Free money! (Well, not really, it's part of a swap agreement!)
STEP 4
Since the swap has four years left, Miaoqi will receive four more payments of .
We need to find the present value of these payments using the spot rates given in the table.
STEP 5
Let's calculate the present value of these future payments.
Remember, we're bringing each future payment back to the present using the corresponding spot rate.
STEP 6
Crunching the numbers, we get: So, the present value of Miaoqi's fixed payments is approximately .
STEP 7
Miaoqi pays a variable rate based on the one-year spot rate at the beginning of each settlement period.
Since we're two years into the swap, the next four one-year spot rates we need are for years 3, 4, 5, and 6.
STEP 8
The tricky part here is that the variable payments are based on the *one-year* spot rate at the *beginning* of each period.
But we're valuing the swap *today*.
Luckily, we're given the spot rates for years 1, 2, 3, and 4 *today*.
The one-year spot rate *next year* is implied in the two-year spot rate *today*, and so on.
STEP 9
The present value of the first variable payment is easy: it's based on the 1-year spot rate of .
So, the payment is , and the present value is simply .
STEP 10
The sum of the present values of the variable payments is equal to the notional amount less the discounted notional amount at the final period.
STEP 11
The market value of the swap from Miaoqi's perspective is the present value of her *incoming* fixed payments *minus* the present value of her *outgoing* variable payments.
STEP 12
STEP 13
The market value of the swap from Miaoqi's position is approximately .
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