Zero-Coupon Swaps

Zero-coupon swaps come up when hedging a loan whose interest is added to principal and paid at maturity, and when valuing a trade from market rates. Payments fall on different dates than in an ordinary interest rate swap, so the payment formula and the value formula change.

How the lump-sum payment works

A zero-coupon swap is an interest rate swap in which the fixed-rate interest is not paid along the way but compounded and paid once at maturity¹. The interest accrues on the notional amount and is compounded as often as the trade specifies². The other leg follows the same payment dates as an ordinary interest rate swap. If that leg is floating, it can also be compounded and paid once at maturity¹. The ordinary exchange of fixed and floating interest is covered on the Interest Rate Swaps page.

Consider a hypothetical example in which Company P and Bank Q enter a 5-year zero-coupon swap with a notional of 10 billion yen. Company P pays the fixed rate of 1.10% per year, compounded annually. The floating leg pays annually. Company P makes no fixed payments for the first 4 years and pays the following amount at year 5.

$$N\left((1+K)^{n}-1\right)$$

Here $N$ is the notional, $K$ is the fixed rate, and $n$ is the number of years. The payment is 10 billion yen × (1.011 multiplied five times − 1) = 562,233,834 yen, or about 562.2 million yen. Paying the same 1.10% interest every year would cost 110 million yen a year, or 550 million yen over 5 years. The zero-coupon payment is 12.23 million yen larger because the accumulated interest itself earns interest.

A 5-year example

The floating leg is forecast from the following market spot rate curve (annual compounding): 0.60% for 1 year, 0.75% for 2 years, 0.90% for 3 years, 1.00% for 4 years, and 1.10% for 5 years. Deriving forward rates from this curve gives the forecast floating rate for each year. The table shows the payments seen by Company P, with receipts positive and payments negative³.

The forecast floating receipts total 550,248,009 yen over the 5 years, which is 11,985,825 yen less than the fixed payment of 562,233,834 yen. The market value at inception is still 0 because the fixed payment arrives in one sum at year 5, and its present value is small. Figure 1 shows the payments.

Figure description

The cash flows, before discounting, of a zero-coupon swap with a notional of 10 billion yen, a fixed rate of 1.10% per year compounded annually, and a term of 5 years, seen from Company P, the fixed payer. Amounts are in million yen. The floating receipts occur every year, and their amounts are forecasts. There are no fixed payments through year 4, and a fixed payment falls only at year 5. The fixed lump sum is a longer arrow than any single floating receipt.

Market value depends on the spot rate

The spot rate is defined as the compound yield of a zero-coupon bond, which pays no interest along the way, and is also called the zero-coupon rate⁴. The discount factor $DF(n)$, which converts an amount paid in n years to present value, follows from the n-year spot rate $r_n$⁵.

$$DF(n) = \frac{1}{(1+r_n)^{n}}$$

The fixed leg of a zero-coupon swap pays once at maturity, so its present value is a single number: the payment times the maturity discount factor. If the same curve is used for forecasting and discounting, the present value of the floating leg is $N(1-DF(n))$, whether it pays on a schedule or in a lump sum at maturity⁶. The value $V$ to the fixed payer is the floating present value minus the fixed present value.

$$V = N\bigl(1-DF(n)\bigr) - N\left((1+K)^{n}-1\right)DF(n) = N\left(1-\left(\frac{1+K}{1+r_n}\right)^{n}\right)$$

The market value is 0 at inception. In the formula above, $V=0$ when $K=r_n$, so with annual compounding the fixed rate at inception equals the spot rate for the same maturity. The ICE system documentation likewise sets the initial value of the fixed rate (called the Zero Rate in the system) to the rate at which the swap's NPV is 0². On the example curve, the 5-year discount factor is 0.946769. The floating leg's present value is 10 billion yen × (1 − 0.946769) = 532,305,801 yen. The fixed leg's present value is 562,233,834 yen × 0.946769 = 532,305,801 yen. The two match.

If the 5-year spot rate moves right after inception, the market value changes as shown in the table. The fixed payer is Company P and the fixed receiver is Bank Q. Valuation is covered on the Valuing an Interest Rate Swap page.

The fixed payer, Company P, gains when the spot rate rises, because the present value of the floating receipts increases. It loses when the spot rate falls, because the fixed payment stays the same while the floating receipts shrink. The fixed receiver, Bank Q, has the opposite gains and losses.

Deep diveWhy paying the floating leg in a lump sum does not change its present value

Suppose the floating leg compounds each year's floating rate and pays once at maturity. Each period is one year, and with $F_i$ as the forward rate for year $i$, the payment at maturity is as follows.

$$N\left(\prod_{i=1}^{n}(1+F_i)-1\right)$$

A forward rate follows from a ratio of discount factors, $1+F_i = DF(i-1)/DF(i)$⁷. In the product, adjacent terms cancel, so $\prod(1+F_i) = DF(0)/DF(n) = 1/DF(n)$. The payment at maturity is $N(1/DF(n)-1)$, and multiplying by $DF(n)$ gives the present value $N(1-DF(n))$, the same as for the scheduled payments. On the example curve, the product is 1.0562234, and 10 billion yen × 0.0562234 is about 562.2 million yen. This equals the lump sum on the 1.10% fixed leg because the fixed rate equals the 5-year spot rate.

Uses

Corporations use the swap to hedge borrowing or bonds whose interest is added to principal and paid at maturity⁸. Banks use it when hedging for customers, either in bond issuance or in corporate hedging.

Consider a hypothetical example in which Company P borrows 10 billion yen at a floating rate for 5 years, equal to the notional, and pays the interest together with the principal at maturity. The loan and the swap's floating leg use the same index, and the floating leg is compounded and paid at maturity. Company P makes the same fixed lump-sum payment as in the example above and receives the floating lump sum. The table shows the result if the floating rate stays at the same level every year for 5 years, with swap receipts positive and payments negative.

At a floating rate of 2.00%, the swap receipt covers part of the loan interest, and Company P pays 562.23 million yen, which is 478.57 million yen less than without the swap. At 0.50%, the loan interest falls to 252.51 million yen, but Company P pays 309.72 million yen on the swap, so it pays 562.23 million yen, which is 309.72 million yen more than without the swap. In both cases the combined interest on the loan and the swap is fixed at the amount of the fixed lump sum.

Differences from an ordinary interest rate swap

For swaps of the same 5-year term, the fixed leg's payments and the way the fixed rate is set differ. The table also gives the values on the example curve.

Notes

1. The opening description of ICE, "SDX Interest Rates Help: Zero Coupon Swap" (external site). It explains that the fixed cash flows are not paid periodically but compounded and paid once at maturity, and that a floating leg can also be compounded and paid once at maturity. ↩ ↩²

2. ICE, "SDX Interest Rates Help: Zero Rate" (external site). It explains that the fixed interest of a zero-coupon swap is compounded according to the compounding frequency and paid once at maturity, and that the initial value is the rate at which the swap's NPV is 0. ↩ ↩²

3. Amounts in the table are calculated without rounding the discount factors and forward rates, and then rounded to the nearest yen. Floating rates are displayed rounded to four decimal places. Totals are the sums of the unrounded values, rounded afterward. ↩

4. Bank of Japan, Financial System and Bank Examination Department, Center for Advanced Financial Technology, "【補足資料】スポットレートとその応用" (external site) (Supplementary material: spot rates and their applications, August 2012, in Japanese), "２．スポットレート" (2. Spot rate), page 6. ↩

5. The same material (external site), "３．ディスカウント・ファクター" (3. Discount factor), page 11. ↩

6. Society of Actuaries, "Interest Rate Swaps" (external site) (May 2017), chapter 4, "Special Formula," formula (4.2) (page 12). The $P_n$ in the source corresponds to $DF(n)$ on this page. ↩

7. Same material (external site), Example 1 (page 5). It calculates the rate from year 1 to year 2 from current spot rates and calls it a forward rate. ↩

8. ICE, "SDX Interest Rates Help: Zero Coupon Swap" (external site), "Advantages of a Zero Coupon Swap." It lists uses for corporations and banks. ↩

9. The fixed rate of an ordinary 5-year interest rate swap with annual payments, calculated from the discount factors of the example curve as (1 − 5-year discount factor) ÷ sum of discount factors (4.860431). The formula is (4.2) of the same material (external site) (page 12). ↩