# 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?

Let us consider separately the interest rate sensitivities of the fixed leg and the floating leg. Let us assume notional exchange ge, so each leg looks like a bond.

If the mark ro market of the swap is zero, then the mrm'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 ro the rate, used ro 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 fnext coupon is already set, then it behaves like fixes coupon- 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 mrm of the xied coupons is close ro the mtm of the floating coupons, except for the sign, then their sensitivities ro 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 mrm of the swap is not zero, eg, 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 ro 0, even negative, then the swap can have a huge mrm 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 ro explain why) that an identical change in the discount and projection curve has no effect on the leg's mtm, which stays close ro 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, it, the spread sensitivity.

So in the special case when the discount curve sensitivituea 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.

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