Financial Risk Manager Part 1

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Explanation:

To compute the 1.5-year spot rate, we will first derive discount factors from the given swap rates and then calculate the spot rate from the derived discount factors.

Step 1: Deriving discount factors from swap rates:
Consider an interest rate swap. If we assume that the notional amount is exchanged, the fixed leg of the swap would resemble a fixed coupon-paying bond, with fixed leg payments acting like semiannual, fixed coupons, and the notional amount acting like the principal payment. We can, therefore, write an equation for each “bond” that equates the present value of its cash flows to its price of par.

(100 + \frac{1}{2}) \times d(0.5) = 100 \Rightarrow d(0.5) = 0.9950

\frac{1.6}{2} \times d(0.5) + (100 + \frac{1.6}{2}) \times d(1.0) = 100 \Rightarrow d(1.0) = 0.9842

\frac{1.88}{2} \times d(0.5) + \frac{1.88}{2} \times d(1.0) + (100 + \frac{1.88}{2}) \times d(1.5) = 100 \Rightarrow d(1.5) = 0.9723

Step 2: Deriving the spot rate from discount factors:
Spot rates, $z(t)$ and discount factors, $d(t)$ are related as shown in the following formula, assuming semi-annual coupons:

z(t) = 2 \left[ \left( \frac{1}{d(t)} \right)^{\frac{1}{2t}} - 1 \right]

\Rightarrow z(1.5) = 2 \left[ \left( \frac{1}{0.9723} \right)^{\frac{1}{2 \times 1.5}} - 1 \right] = 1.88\%

Explanation:

To compute the 1.5-year spot rate, we will first derive discount factors from the given swap rates and then calculate the spot rate from the derived discount factors.

(100 + \frac{1}{2}) \times d(0.5) = 100 \Rightarrow d(0.5) = 0.9950

\frac{1.6}{2} \times d(0.5) + (100 + \frac{1.6}{2}) \times d(1.0) = 100 \Rightarrow d(1.0) = 0.9842

\frac{1.88}{2} \times d(0.5) + \frac{1.88}{2} \times d(1.0) + (100 + \frac{1.88}{2}) \times d(1.5) = 100 \Rightarrow d(1.5) = 0.9723

Step 2: Deriving the spot rate from discount factors:
Spot rates, $z(t)$ and discount factors, $d(t)$ are related as shown in the following formula, assuming semi-annual coupons:

z(t) = 2 \left[ \left( \frac{1}{d(t)} \right)^{\frac{1}{2t}} - 1 \right]

\Rightarrow z(1.5) = 2 \left[ \left( \frac{1}{0.9723} \right)^{\frac{1}{2 \times 1.5}} - 1 \right] = 1.88\%

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Q.48 A trader at a derivatives trading desk is explaining the relationship between spot and swap rates to his intern. As an example, he wants to extract the 1.5-year spot rate from the swap rates structure presented below:

Term in Years Swap Rate
0.5 1.00%
1.0 1.60%
1.5 1.88%

What is the implied 1.5-year spot rate?

Term in Years	Swap Rate
0.5	1.00%
1.0	1.60%
1.5	1.88%

Other

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CChannel

Last updated: June 22, 2026 at 14:25

1.70%.

100.0%

0.80%.

0.0%

1.88%.

1.93%.