# What is Bootstrapping using Discount Factors?

**Short answer**

Bootstrapping means building up a forward interest-rate curve that is consistent with the market prices of common fixed-income instruments such as bonds and swaps. The resulting curve can then be used to value other instruments, such as bonds that are not traded.

**Example**

You know the market prices of bonds with one, two three, five years to maturity. You are asked to value a four-year bond. How can you use the traded prices so that your four-year bond price is consistent?

**Long answer**

Imagine that you live in a world where interest rates change in a completely deterministic way, no randomness at all. Interest rates may be low now, but rising in the future, for example. The spot interest rate is the interest you receive from one instant to the next. In this deterministic interest-rate world this spot rate can be written as a function of time, *r(t).* If you knew what this function was you would be able to value fixed-coupon bonds of all maturities by using the discount factor

to present value a payment at time *T* to today, *t.*

Unfortunately you are not told what this *r* function is. Instead you only know, by looking at market prices of various fixed-income instruments, some constraints on this *r* function.

As a simple example, suppose you know that a zero-coupon bond, principal $100, maturing in one year, is worth $95 today. This tells us that

Suppose a similar two-year zero-coupon bond is worth $92, then we also know that

This is hardly enough information to calculate the entire *r(t)* function, but it is similar to what we have to deal with in practice. In reality, we have many bonds of different maturity, some without any coupons but most with, and also very liquid swaps of various maturities. Each such instrument is a constraint on the *r(t)* function.

Bootstrapping is backing out a deterministic spot rate function, r(t), also called the (instantaneous) forward rate curve that is consistent with all of these liquid instruments.

Note that usually only the simple 'linear instruments are used for bootstrapping. Essentially this means bonds, but also includes swaps since they can be decomposed into a portfolio of bonds. Other contracts such as caps and floors contain an element of optionality and therefore require a stochastic model for interest rates. It would not make financial sense to assume a deterministic world for these instruments, just as you wouldn't assume a deterministic stock price path for an equity option.

Because the forward rate curve is not uniquely determined by the finite set of constraints that we encounter in practice, we have to impose some conditions on the function *r(t).*

• Forward rates should be positive, or there will be arbitrage opportunities.

• Forward rates should be continuous (although this is commonsense rather than because of any financial argument).

• Perhaps the curve should also be smooth.

Even with these desirable characteristics the forward curve is not uniquely defined.

Finding the forward curve with these properties amounts to deciding on a way of interpolating 'between the points,' the 'points' meaning the constraints on the integrals of the *r* function. There have been many proposed interpolation techniques such as

• linear in discount factors

• linear in spot rates

• linear in the logarithm of rates

• piecewise linear continuous forwards

• cubic splines

• Bessel cubic spline

• monotone-preserving cubic spline

• quartic splines

and others.

Finally, the method should result in a forward rate function that is not too sensitive to the input data, the bond prices and swap rates, it must be fast to compute and must not be too local in the sense that if one input is changed it should only impact on the function nearby. And, of course, it should be emphasized that there is no 'correct' way to join the dots.

Because of the relative liquidity of the instruments it is common to use deposit rates in the very short term, bonds and FRAs for the medium term and swaps for the longer end of the forward curve.

Because the bootstrapped forward curve is assumed to come from deterministic rates it is dangerous to use it to price instruments with convexity since such instruments require a model for randomness, as explained by Jensen's Inequality.

Two other interpolation techniques are worth mentioning: first, that proposed by Jesse Jones and, second, the Epstein-Wilmott yield envelope.

The method proposed by Jesse Jones involves choosing the forward curve that satisfies all the constraints imposed by

Figure 2.13: **The Yield Envelope showing ranges of possible yields. The points at which the range is zero is where there are traded contracts.**

traded instruments but is, cricially, also not too far from the forward curve as found previously, the day before, say. The idea being simply that this will minimize changes in valuation for fixed-income instruments.

The Epstein-Wilmott model is nonlinear, posing constraints on the dynamics of the short rate. One of the outputs of this model is the Yield Envelope (Figure 2.13) which gives no-arbitrage bounds on the forward curve.

**References and Further Reading**

Epstein, D & Wilmott, P 1997 Yield envelopes. Net Exposure 2 August netexposure.co.uk

Epstein, D & Wilmott, P 1998 A new model for interest rates. *International Journal of Theoretical and Applied Finance 1* 195-226

Epstein, D & Wilmott, P 1999 A nonlinear non-probabilistic spot interest rate model. *Philosophical Transactions A* 357 2109-2117

Hagan, P & West, G 2008 Methods for constructing a yield curve. *Wilmott* magazine May 70-81

Jones, J 1995 Private communication

Ron, U 2000 A practical guide to swap curve construction. Technical Report 17, Bank of Canada

Walsh, O 2003 The art and science of curve building. *Wilmott* magazine November 8-10