# Zero-Coupon Curves and Forward Curves

Since the 2007 crisis, the equivalence between quoted interest rates is no longer observed in financial markets due to liquidity and counterparty risks. This phenomenon results from the non-respect of the no-arbitrage condition and the fact that interbank rates were considered until then as no risk rates have become riskier as their maturity increases (for example, EUR 12M is riskier than EUR 3M). Computing all IBOR forward curves from one sole curve is no longer in accordance with the market and the new rick notion. In this context, the approach used on trading floors –now used by Finance Active–consists of building a zero-coupon for each tenor (1M, 3M, 6M, 12M). We only use instruments which have the same tenor as underlying. Each zero-coupon curve is well defined by using simultaneously the techniques of bootstrapping and interpolation.

In the first part, we will provide all the steps for the construction of zero-coupon and forward curves and in the second part, we will give details for each step.

Building Zero-Coupon Curves and Forward Curves

For each currency, we build 4 zero-coupon curves for the 4 tenors: 1M, 3M, 6M, and 1Y. There is a reference tenor for each currency (for example, 3M for USD, 6M for EUR). We always begin with the construction of the reference tenor zero-coupon curve. The steps are as follow:

1. Select a panel of market instruments with the same underlying tenor as the reference tenor between Depo, FRA, Futures, and Interest Rate Swaps provided by our brokers.
2. Using the bootstrapping method, calculate the zero-coupon values at discrete times using the selected instruments.
3. Using the Tikhonov Regularization method, get a smooth zero-coupon curve, defined at each time, and with the good mathematical properties required for pricing financial products.
4. Calculate the forward curve of the reference tenor from the zero-coupon curve with the simple relation between forward and zero coupon.

Once we have built the zero-coupon curve of the reference tenor, we then build the curves for the other tenors. In this article, x will now represent the tenor of the curve that we are building. The steps are as follow:

1. Select a panel of market instruments with x as the underlying tenor between Depo, FRA, and Futures, but Interest Rate Swaps, with the reference tenor as the underlying tenor, and Basis Swaps between x and the reference tenor.
2. From the Interest Rate Swaps and the Basis Swaps, build synthetic Interest Rate Swaps with x for the underlying tenor.
3. Since we now have only instruments with x as the underlying tenor, we build the zero-coupon curve with the bootstrapping and Tikhonov regularization methods as the reference of the tenor curve.
4. Calculate the forward curve from the zero-coupon curve.

Market Instruments

The first thing is to select a set of maturities, called pillars, and a market instrument for each of these maturities that we will use to calculate the zero-coupon value at this maturity. The rule is to choose a market instrument which is liquid in the currency and for the maturity considered. For example, for the short maturities (between 0Y and 2Y), Forward Rate Agreements (FRA) are very liquids in the EUR market. For USD, Futures are also very liquids between 0Y and 2Y. For longer maturities, Interest Rate Swaps (IRS) are the instruments the most liquid.

We present briefly the most used market instruments: Deposit, FRA, Future, IRS, and Basis Swap. is the value of the zero coupon with tenor x at date t and with maturity T is the discounting zero coupon, in our case, it is the reference tenor zero coupon.

Deposit

Depo are standard money market contracts where, at start date (today or spot), counterparty A, called the Lender, pays a nominal amount N to counterparty B, called the Borrower, and at maturity date , the Borrower pays back the Lender the nominal amount N plus the interest accrued over the period [ ; ] at the Deposit Rate fixed at time .

For example, the EUR market quotes at time ="today" a standard strip of Deposits based on Euribor rates, with fixing dates , start date ="spot date"= +"2 business days", and maturity dates ,…, from 1 day up to 1 year.

We get the discount curve of maturity using the following relation: Where τ( , ) is the year fraction between and .

Forward Rate Agreement

Forward rate agreement (FRA) contracts are forward starting deposits. For instance, a 3x9 FRA is a six-month deposit starting three months forward. FRA contracts are quoted on the interbank OTC market for various currencies. For example, the EUR market quotes at =today three standard strips of Euribor FRA, starting at spot date = +2 business days with different forward start and end dates, and , and tenors x=1M,3M,6M,12M.

Let’s consider a x FRA. The underlying Euribor FRA rate fixes at time , two business days before the forward start date , and shares the same tenor of the FRA. We have the following relation: Where is the year fraction between and .

Futures

Futures are exchange-traded contracts similar to forward rate agreements (FRA). Any profit and loss are regulated through daily marking to market (margining process). Future contracts are rolled on the IMM dates:

• The 3rd Wednesday of March
• The 3rd Wednesday of June
• The 3rd Wednesday of September
• The 3rd Wednesday of December

The relation between future rates and the discount curve is the same as the one for the FRA.

Interest Rate Swaps

Interest rate swaps (IRS) are contracts in which two counterparties agree to exchange two streams of cash flows in the same currency, typically tied to a floating Libor rate versus a fixed rate K. These payments streams are called fixed and floating leg of the swap, respectively.

We consider a vanilla swap with a maturity (= ), starting in and with payment dates for the floating leg and for the fixed leg. We have: with the forward IBOR beginning in and maturing in and = - . We have the relation: The swap rate is the rate that makes the two legs of the swap equal: Interest rate swaps are quoted on the market with their swap rates.

Basis Swaps

Interest rate basis swaps are contracts in which two counterparties agree to exchange two streams of cash flows in the same currency, tied to two floating IBOR rates with different tenors x and y. Basis swaps are quoted on the market with their basis spread ( ) which makes the two legs of the swap equal: Examples of Maturities and Market Instruments

IRS xMyM is an interest rate swap with floating leg frequency x months and fixed leg frequency y months.

#### EUR 6M

• FRA: 0x6, 1x7, 2x8, 3x9, 4x10, 5x11, 6x12, 7x13, 8x14, 9x15, 10x16, 11x17, 12x18
• IRS 6M12M: 2Y, 3Y, 4Y, 5Y, 7Y, 10Y, 12Y, 15Y, 20Y, 25Y, 30Y, 40Y, 50Y

#### USD 3M

• Deposit: 3M
• Futures: 1, 2, 3, 4
• IRS 3M12M: 2Y, 3Y, 4Y, 5Y, 7Y, 10Y, 12Y, 15Y, 20Y, 25Y, 30Y, 40Y, 50Y

#### SEK 1M

• Deposit: 1M
• IRS 3M12M: 2Y, 3Y, 4Y, 5Y, 7Y, 10Y, 12Y, 15Y, 20Y, 25Y, 30Y
• Basis Swap 1M3M: 2Y, 3Y, 4Y, 5Y, 7Y, 10Y, 12Y, 15Y, 20Y, 25Y, 30Y

Bootstrapping Method

The bootstrapping method consists of calculating the value of the zero coupon at each maturity pillar using the associated market instrument in a recursive way starting from t (today) to the last maturity pillar. We always have: Bootstrapping from Deposits, Forward Rate Agreements, and Futures

There is no difficulty to deduce the zero coupon value from the deposit, forward rate agreement (FRA), and future rates. For the deposit, the zero coupon value depends only on the deposit rate. For FRA and futures, the zero coupon value of maturity depends on the FRA (or the future) rate and the value of the zero coupon of maturity .

Bootstrapping from Swap Rates

We recall the formula of the swap rate: There is two zero coupon in the formula, the zero coupon of tenor x that we are building, and the discounting zero-coupon curve. As said previously, we always build the reference tenor curve first. In this case = and: There is a geometric sum in the swap rate which is now simply: By inversing this formula and knowing that quoted swaps are starting at t ( =t) we have: This formula shows that the zero coupon of maturity depends only on the swap rate (t, ) and the zero coupon values of maturities ,i=1,..,m-1 that we know thanks to the bootstrapping method.

Once the reference tenor curve, used as the discounting curve, is built, the method is the same for the other tenors.

We do not give details on how we calculate swap rates for other tenors from reference tenor swap rates and basis swaps, and instead invite you to check out the reference article from Ametrano and Bianchetti.

Before the bootstrapping, we build synthetic swap rates between the pillars maturities by simple linear interpolation.

The zero-coupon values needed in the bootstrapping formula are in fact not always known, due to the difference between the fixed leg frequency and the floating leg frequency, for example. In this case, we deduce it by polynomial spline between the zero-coupon values previously calculated.

Tikhonov Regularization Method

The Tikhonov method is a method of regularization of ill-posed problems. In our case, we only have, from the bootstrapping method, zero-coupon values at discrete pillar maturities. The goal of the method is to find the values of the zero coupon at every time between today and the longest maturity but with good mathematical properties like smoothness and twice differentiability.

This is a regression problem of the type solving Ax=b with A being a matrix and b, a vector. The classical method is to minimize (least square method). With Tikhonov, we will minimize + where Γ is the "Tikhonov matrix."

Forward Curve from Zero-Coupon Curve

We have the following relation between a forward IBOR and a zero coupon: This shows that knowing the function T you can directly deduce the function T (T,T+δ).

Reference

Ametrano F., Bianchetti M. Everything You Always Wanted to Know about Multiple Interest Rate Curve Bootstrapping but Were Afraid to Ask (2013)