# WHAT DO WE MEAN BY CURVE CONSTRUCTION?

Assessing the present value (or discount) of a future cash flow is the single most important problem in fixed income; to generate correct index-linked future cash flows comes at a close second. By index-linked future cash flow we mean a cash flow linked to a rate *Ls,t* setting at some future time *S* and paying at a time *T > S.* This cash flow, if applied to a unit principal, can be written as

(2.1)

where *dt*s,t is the day count fraction to assign the portion of L (which is quoted annually) to the correct accruing period^{[1]} between *S* and T. Two questions arise here. First, what is the value of *L*s,t that I am expecting *today* with the information in my possession *today*? We stress the role played by the present: in all fair value and risk-neutral calculations of financial derivatives, we only expect to produce a price correct in the instant. Second, once I have established the value of the cash flow in the future, what is the value of that cash flow now? Because of the time value of money, calculating the present value of a future cash flow is a crucial way of comparing different cash flows in different context and/or currencies. We could see it as a way of normalizing all cash flows by their natural appreciation.

The two problems above are closely related but, perhaps counterintuitively, the second one is more challenging. While certainly elegant, to see discounting as some sort of normalization is not immediately helpful in practice. If we look closely, however, it offers an important clue to the process. We are trying to normalize *all* cash flows, meaning that whatever calculation we intend to carry out needs to apply to our particular cash flow and to all cash flows of traded instruments in a selected financial landscape. The term *selected financial landscape* is willfully vague. Let us clarify it with a more specific example.

Let us call V the present value of equation 2.1 and let us write it as

where *D*T is the discount factor. Crucially, *D*T is unknown to us. Let us now imagine that we are operating in a very primitive market where the only type of instruments actively traded are bonds that promise the investor a unit of principal at a future maturity. By coincidence there is one with maturity T, trading at a price *P.* If one unit of principal paid at time T is worth *P* now, then any other cash flow paid at time *T* will be worth the cash flow amount times *P.* This means that we have found the value of our discount factor, namely *D*T = *P.* While this helps us in determining the value of the discount factor, it does not help us yet in determining the value of the floating cash flow. We have said that in our market there are only simple bonds: by another happy coincidence there is another one actively traded expiring at time *S,* from which we know immediately the value of the discount factor is *D*S. By using a relationship

defining a forward rate,^{[2]} we can easily find the value of our future cash flow and hence the value of V.

In the preceding, we show the principles of *curve construction* (or *curve calibration),* the process by which we establish, using information provided by traded instruments, the values of future rate resets *L*i,j and at the same time a discount function *D*t enabling us to assess their present value. These principles can be summed up in the following:

┠We use *simultaneously* all the information available in the market, which we believe is relevant for our curve construction.

┠Once we have constructed our curve, we should expect to be able to reprice *only* the instruments we have used.

At the moment, judging from our simple example, the terms stressed previously might seem puzzling: we shall see later how important they become when we embark on a real-world curve construction.

Since we have established that curve construction consists of building a set of functions that need to be calibrated to a set of market instruments, we now introduce what these market instruments are.

- [1] An important problem when dealing with interest rates is one of accrual since an interest rate is always a number that is valid within two moments in time
*and it is quoted annually.*How do we actually count the time elapsed between the two points and therefore establish what portion of the interest rate has accrued? Let us imagine that we have an interest rate*R – 5%*and this value is agreed between a borrower and a lender on March 10 on a sum of $100,000. Over the entire period of one year, 5% is paid, that is, $5,000. However, how do we calculate the fraction of that interest rate that is paid between any two points in time, for example, between August 4 and October 10? What is the fraction of one year that needs to be applied to the rate? The answer is far from unique and it is based on literally counting the days between the two points in time, which is why it is often called*day count fraction.*The way we count the days is based on day count conventions and the borrower and the lender will agree on one at the beginning of the transaction. Two common conventions are Act/360 and Act/365, which dictate that to obtain the fraction we count the actual number of days between the two points in time and divide by, respectively, 360 and 365 (which we have assumed to be the number of days in a year). In our example, the interest accrued between August 4 and October 10 amounts to $930.56 using an Act/360 day count convention and $917.81 using an Act/365 day count convention. Another type of convention, 30/360, dictates that every month brings to the calculation only 30 days even if the month in question has 31 (or fewer in the case of February); once we sum them, we divide them by 360. In our example, the accrued interest with a 30/360 convention is $916.67. There are many conventions (South American currencies have some of the strangest ones) that could apply to a time interval and, although very important in practice, they are mostly irrelevant to our subsequent discussion. Whenever we indicate a time interval (as*dt*s,t*•*or*dt*i or*T*i –*T*i-1, etc.) we shall always assume that the proper day count convention has been applied. - [2] The attentive reader must have recognized in the above equation the definition of LIBOR. Its absence, in name, from our primitive market is purely for illustration purposes.