Prepayment Option and the Case of Development Institutions

As we mentioned earlier, in the great majority of loans the borrower can choose to repay the principal before the maturity of the loan and doing so results in the presence of a financial option embedded in the loan. In order to model this option, we assume that the borrower acts according to sound financial reasons and we can identify two main drivers behind these reasons: credit and interest rates.

The basic question a borrower asks himself during the life of a loan is, if I were to repay my loan now, could I get something better elsewhere? If the loan is a fixed-rate loan, the rate was fixed at the moment of entering the contract and it matched the rate landscape at the time (it was basically that fixed rate that would make the loan be worth par). In case the landscape changes in the direction that results in overpaying on the part of the borrower, this might be tempting to prepay the existing loan and enter into a new one at a lower rate. If the rate of the loan is variable, the value of the interest rate option is very small since, with only a little time lag between rate resets, a resetting loan rate always reflects the current rate level, taking away the borrower's incentive to prepay. Let us always remember the fact (see Section 2.2.5.2) that a floating leg is always more or less worth par.

The other driver is the credit standing of the borrower. At the moment of entering the loan the lender would have set a spread s over LIBOR matching the creditworthiness of the borrower: the lower the latter the higher the former. If during the life of the loan the credit standing of the borrower improves, there is a chance that it might obtain a new loan at a spread lower than s.

If we want to describe the sensitivity of the option to credit and interest rates we could say that the option is a put on the prevailing interest rate (if the loan is a fixed-rate loan, the more interest rates decrease the more the option of switching loans is valuable) and a call on the credit risk of the borrower, the more this improves the more valuable the option. Of course if we describe the credit risk of the borrower, as we should after Sections 3.2.1 and 3.2.2, using CDS spreads and hazard rates, we could say that the option is a put on these, too: the more they decrease the more valuable the option.

How is this option priced in practice? Loans are fairly simple instruments and short rate models are usually sufficient to price the option embedded in them. Of course we will be dealing here with a model belonging always to the spread-based family of credit models: there are alternatives in terms of pricing the optionality with credit rating models (see for example Engelmann [36]), but they are outside the scope of a financial institution practicing investment banking in the traditional sense of the word. These institutions, which are the focus of our attention, tend to prefer, as we have highlighted in Section 3.1, risk-neutral models based on trading activity.^[1]

Since, as we said, loans are fairly simple instruments, a further goal would be an analytic implementation (i.e., one not relying on numerical simulations). We offer an example in Appendix C where we will follow closely Schonbucher's [74] description of a PDE-based implementation.

Before discussing the behavior of the option with respect to its variables, let us first describe the special case of development institutions. Development institutions can be considered as some sort of credit cooperatives, a cooperative that, in the case of the larger development institutions, can reach a very large membership. Members pool funds together that are then either lent internally or used to back a solid credit that can be used to borrow at favorable terms. This, combined with the fact that they are nonprofit organizations, has of course some advantages for its receiving members but also entails some restrictions on the decisions these can take with respect to a loan.

The first one has to do with fixed-rate loans. Let us first consider the normal situation found in the for-profit world. Let us imagine that we receive a loan at a fixed-rate C1 and let us also assume, for simplicity, that the principal will be repaid at the end, that the fixed coupon is exactly like the prevailing swap rate S1 at the time of entering the contract, and that we are discounting our cash flows simply using interest rate swaps.^[2] This means that, in the case of a unit principal, we must have

(3.25)

where we have stressed the dependence of the discount factor on the swap rate. Let us imagine that the prevailing swap rate decreases considerably and now, at time T2, is equal to S2 < S1. The loan now will be worth

(3.26)

(as we have seen in the previous chapter, a lower swap rate results in higher discount factors, so the above follows). The lender is now holding a more valuable asset since the present value of the loan is higher than the principal amount. We have assumed that the fixed rate on a loan follows closely the swap rate: this means that we could obtain a loan somewhere else at a rate C2 whose present value would be approximately par. By switching loans it is easy to see that we would gain the present value of the difference between the two rates. Let us define it as P1

(3.27)

The situation in development institutions is different. Being nonprofit, the amount charged on loans is simply used to cover operating costs. These costs would increase considerably should borrowers switch from loan to loan too often. As a consequence, if a borrower chooses to prepay a fixed- rate loan it will be charged the exact amount P1. This means that, irrespective of interest rate moves, a country borrowing from a development organization has no interest rate driven incentive to prepay a loan.

We have seen in the introduction, and we will analyze it in more detail later, that development institutions using the tools of investment banking toward development basically borrow using their good credit and pass on this good credit (plus a small spread to cover operating costs) to the borrowers. This results in the borrowers being able to obtain a loan at a level close to the one of a AAA-rated institution and certainly lower than the one they could obtain in the market.

Let us imagine that at time T1 a development institution issues floating rate loans at Li + s1, that is, at LIBOR (we have shown the suffix to stress its floating aspect) plus a spread s. Let us imagine that at time T2 the development bank manages to issue debt at an even lower level resulting in the ability to issue loans at Li + s2 with s2 < s1. A country might be tempted to prepay the loan with spread s and enter into a loan with spread s2: to prevent this the development institution would charge the borrower

(3.28)

This means that the borrower has no incentive driven by the credit standing of the development institution to prepay the loan. The combination of the two penalties P1 and P2 results in a prepayment option driven purely by the credit of the borrowing country and not, in any way, by the interest rate environment or other considerations. This means that from the discussion leading to Equation C.5 (in Appendix C) we need to ignore stochastic interest rates. If at first the presence of the above-mentioned penalties might seem a little harsh, let us remind the reader that development institutions provide loans to borrowers that are sometimes cheaper in the order of a few percentages with respect to what the same borrower could get, if at all, in the market. P1 and P2 are a small price to pay for this advantage.

The final point, originating from the special status of development organizations, is the fact that, being a credit cooperative, all loans, at a specific time, are issued at the same level irrespective of the credit standing of the borrower. All fixed-rate loans issued at a certain time will have the same fixed rate and all floating-rate loans issued at a certain time will have the same spread s over LIBOR. This has very important consequences, particularly when fair valuing loans. Although all loans are issued at the same level, we have seen in Equations 3.23 and 3.24 that the survival probability used to fair value a loan is driven by that borrower-specific credit standing (in practice, its CDS level).

The first consequence is that the lender is holding assets with very different values. A normal (for-profit) institution choosing to fair value its loans would issue loans at levels such that, roughly speaking, the combination of floating rate plus spread would balance the effect of discount factor plus survival probability. In practice, in the typical cash flow,

(3.29)

the discount factor Di would be driven by the interest rate Li (as we have seen in the previous chapter) and the spread cs charged to the borrower would match the survival probability Si: the lower this one, the higher cs (in practice cs follows crudely, as we shall see later, the CDS level of the borrower). The combination of the two would make sure that a strip of cash flows such as the one in Equation 3.29 would be roughly equal to the principal amount of the loan. In the case of a development institution we have said that cs remains constant for all borrowers, meaning that to the institution, as an asset, the loan to a borrower with a poor credit standing will be worth less than the loan to a borrower with a better credit standing.

The second consequence appears when dealing with the prepayment option. We have said that valuing the prepayment option is akin to answering quantitatively the question asked by the borrower, is it more advantageous to me to remain with this loan or to switch to a different one? Fair valuing loans is a considerable advantage when dealing with this question because it enables us to value the loan in a way considered objective by the lender, the borrower, and a potential third party to whom the borrower might turn for an alternative loan.

• [1] There is also, as an aside, the world of mortgage modeling that needs to rely heavily on empirical data (see for example Kau et al. [57]) and needs to value financial options that are not always exercised according to optimal reasoning (see for example Stanton [77] or Goncharov [43]).
• [2] This means that we are ignoring the curve construction techniques highlighted in Section 2.2.5.1. While this is not entirely correct as described in the previous chapter, for the situation, for example, of a USD loan for a USD-centered institution, it could be a fair approximation.