In the following we use the continuously compounded interest convention. So that one dollar put in the bank at a constant rate of interest r would grow exponentially, ert .This is the convention used outside the fixed-income world. In the fixed-income world where interest is paid discretely, the convention is that money grows according to
where n is the number of interest payments, t is the time interval between payments (here assumed constant) and r' is the annualized interest rate.
To convert from discrete to continuous use
The yield to maturity (YTM) or internal rate of return (IRR)
Suppose that we have a zero-coupon bond maturing at time T when it pays one dollar. At time t it has a value Z(t; T). Applying a constant rate of return of y between t and T, then one dollar received at time T has a present value of Z(t; T)at time t, where
It follows that
Suppose that we have a coupon-bearing bond. Discount all coupons and the principal to the present by using some interest rate y. The present value of the bond, at time t,is then
where P is the principal, N the number of coupons, Ci the coupon paid on date ti. If the bond is a traded security then we know the price at which the bond can be bought. If this is the case then we can calculate the yield to maturity or internal rate of return as the value y that we must put into the above to make V equal to the traded price of the bond. This calculation must be performed by some trial and error/iterative procedure.
The plot of yield to maturity against time to maturity is called the yield curve.
Since we are often interested in the sensitivity of instruments to the movement of certain underlying factors it is natural to ask how does the price of a bond vary with the yield, or vice versa. To a first approximation this variation can be quantified by a measure called the duration.
By differentiating the value function with respect to y we find that
This is the slope of the price/yield curve. The quantity
is called the Macaulay duration.(The modified duration is similar but uses the discretely compounded rate.) The Macaulay duration is a measure of the average life of the bond.
For small movements in the yield, the duration gives a good measure of the change in value with a change in the yield. For larger movements we need to look at higher order terms in the Taylor series expansion of V(y).
The Taylor series expansion of V gives
where Sy is a change in yield. For very small movements in the yield, the change in the price of a bond can be measured by the duration. For larger movements we must take account of the curvature in the price/yield relationship.
The dollar convexity is defined as
and the convexity is
Yields are associated with individual bonds. Ideally we would like a consistent interest rate theory that can be used for all financial instruments simultaneously. The simplest of these assumes a deterministic evolution of a spot rate.
The spot rate and forward rates
The interest rate we consider will be what is known as a short-term interest rate or spot interest rate r(t). This means that the rate r(t) is to apply at time t. Interest is compounded at this rate at each moment in time but this rate may change, generally we assume it to be time dependent.
Forward rates are interest rates that are assumed to apply over given periods in the future for all instruments. This contrasts with yields which are assumed to apply from the present up to maturity, with a different yield for each bond.
Let us suppose that we are in a perfect world in which we have a continuous distribution of zero-coupon bonds with all maturities T. Call the prices of these at time t, Z(t; T). Note the use of Z for zero-coupon.
The implied forward rate is the curve of a time-dependent spot interest rate that is consistent with the market price of instruments. If this rate is r(r)at time t then it satisfies
On rearranging and differentiating this gives
This is the forward rate for time T as it stands today, time t. Tomorrow the whole curve (the dependence of r on the future) may change. For that reason we usually denote the forward rate at time t applying at time T in the future as F(t; T)where
Writing this in terms of yields y(t; T)we have
This is the relationship between yields and forward rates when everything is differentiable with respect to maturity.
In the less-than-perfect real world we must do with only a discrete set of data points. We continue to assume that we have zero-coupon bonds but now we will only have a discrete set of them. We can still find an implied forward rate curve as follows. (In this I have made the simplifying assumption that rates are piecewise constant. In practice one uses other functional forms to achieve smoothness.)
Rank the bonds according to maturity, with the shortest maturity first. The market prices of the bonds will be denoted by ZiM where i is the position of the bond in the ranking.
Using only the first bond, ask the question 'What interest rate is implied by the market price of the bond?' The answer is given by yi, the solution of
This rate will be the rate that we use for discounting between the present and the maturity date Ti of the first bond. And it will be applied to all instruments whenever we want to discount over this period.
Now move on to the second bond having maturity date T2. We know the rate to apply between now and time Ti, but at what interest rate must we discount between dates Ti and T2 to match the theoretical and market prices of the second bond? The answer is r2, which solves the equation
By this method of bootstrapping we can build up the forward rate curve. Note how the forward rates are applied between two dates, for which period I have assumed they are constant.
This method can easily be extended to accommodate coupon-bearing bonds. Again rank the bonds by their maturities, but now we have the added complexity that we may only have one market value to represent the sum of several cash flows. Thus one often has to make some assumptions to get the right number of equations for the number of unknowns.
To price non-linear instruments, options, we need a model that captures the randomness in rates.