Interest Rate Swap Sensitivity to Projection and Discount Curves
Summary
The document explains how an interest rate swap’s market value can respond to its projection curve and discount curve, particularly when the swap’s value is near zero. It separates the fixed and floating legs: fixed coupons are known and their present value depends on discounting, while unset floating coupons are projected from the forward curve and then discounted. When the two legs have similar values, their parallel discount-curve sensitivities can largely offset, leaving greater exposure to projection rates.
For the floating leg, an equal move in projection and discount curves can leave value close to par, because the projected coupons and their discounted present value move in offsetting ways. The remaining exposure is tied to changes in the spread between the curves. The answer cautions that this near-zero valuation case does not apply universally: a swap with a large value can have substantial discount-curve risk. It also emphasizes examining tenor buckets, where sensitivities may concentrate around principal repayment dates, especially maturity for a bullet swap.
Key ideas
- Fixed-leg coupons are known and their value depends on the discount curve.
- Unset floating coupons are projected from the estimation curve and discounted separately.
- Near zero value, fixed and floating legs can have offsetting parallel discount-curve sensitivities.
- The floating leg is sensitive to changes in the spread between projection and discount curves.
- A swap with a substantial market value can have large discount-curve sensitivity.
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Full text
# IRS - sensitivity to estimation (projection, coupon) curve and discounting curve # IRS - sensitivity to estimation (projection, coupon) curve and discounting curve The mark to market of an interest swap that is close to zero (e.g., at the swap's inception) has more sensitivity to which curve - the estimation (projection, coupon) curve or the discount curve? And why? ## Answer by Dimitri Vulis (score 1, accepted) https://quant.stackexchange.com/a/67875 Let us consider separately the interest rate sensitivities of the fixed leg and the floating leg. Let us assume notional exchange, so each leg looks like a bond. If the mark to market of the swap is zero, then the mtm's of the legs are the same, with opposite sign; but their cash flows and risks are not the same. All the coupons of the fixed leg are known in advance and are equal. The leg's mtm depends only on the discount curve, not the projection curve. If we look in more sensitivity by tenor bucket, most of it is to the rate used to discount the notional repayment - at maturity if the swap is bullet, or at times of amortizations. In contrast, the coupons of the floating leg that are not yet set are unknown. If the coupons are set in advance and the next coupon is already set; it then behaves like a fixed coupon- i.e. has sensitivity (small) to the discount curve at the time of the cash flow and no sensitivity to the projection curve. To calculate the mtm, we project the unset coupons using the projection curve. If the projection curve is upward sloping, then the floating coupons far in the future wlll be projected to be larger than the coupons in the near future. Nevertheless, if the mtm of the fixed coupons is close to the mtm of the floating coupons, except for the sign, then their sensitivities to a parallel shift in discount curve is also similar, except for sign, and the net sensitivity of the swap to the discount curve is small. The sensitivity to other shifts of the discount curve (slope, curvature) may be greater. Observe please that if the mtm of the swap is not zero, e.g., if the swap was done when the interest rates were 10-15% and the fixed coupon is this big, but now the floating coupons are projected to be close to 0, even negative, then the swap can have a huge mtm and a correspondingly huge sensitivity to the discount curve. Anyway, let us look at the risk of the floating leg. It's a mathematical fact (a common interview question is to explain why) that an identical change in the discount and projection curve has no effect on the leg's mtm, which stays close to par until the spread changes between the discount and projection curves. The change in the present value of the floating coupons and the notional (from the discount curve movement) is offset by the change in floating coupon amounts (from the projection curve movement). The sensitivity to one curve moving and the other staying put is the same as the sensitivity to the other curve moving the other way and the first curve staying put, the spread sensitivity. So in the special case when the discount curve sensitivities of the fixed and floating legs nearly offset each other, the swap is left with a larger sensitivity to the projection curve. If you further break down the risk by tenor buckets (which one always should), the sensitivities are mostly to the projection rates at the time of notional payments, i.e. maturity if the swap is bullet. ## Answer by twzlk (score 3) https://quant.stackexchange.com/a/67869 the estimated forward curve actually predicts the amount of future cash flows of floating leg while the discount curve is used to compute the PV of those cash flows based on a supposedly risk free rate or OIS rate which is deemed to be lower thus, the IRS is more sensitive to estimation curve
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