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INTEREST RATE EXPOSURE
MANAGEMENT OF INTEREST RATE EXPOSURE
The important thing to note is that there is no exchange of principal amount. If the settlement rate on the settlement date 5 is above the contract rate, the seller compensates the buyer for the difference in interest on the agreed upon principal amount for the duration of the period in the contract. Conversely, if the settlement rate is below the contract rate, the buyer compensates the seller. The compensation is paid up-front on the settlement day and therefore has to be suitably discounted since interest payment on short-term loans is at maturity of the loan. One of the following two formulas is used for calculating settlement payment from the seller to the buyer: P = (L - R) x DF x A / [(B x 100) + (DF x L)] P = (R - L) x DF x A / [(B x 100) + (DF x L)]
This means that the FRA is only a hedge; the actual underlying deposit or loan is a separate transaction which may not be-and most often is not-with the same bank that traded the FRA. The settlement rate is the rate with which the contract rate is to be compared to compute the settlement payment. In each market there is a clearly specified procedure to determine the settlement rate. The fixing date is the day on which the settlement rate is determined. For US dollar FRAs, fixing date is the settlement date itself i.e. t=S. while for other currencies it is two business days before the settlement date. In the Indian rupee market it is one day before the settlement date. See calculation of settlement payment discussed below. Here the notation is L: The settlement rate (%) R: The contract rate (%) DF: The number of days in the contract period A: The notional principal B: Day count basis (360 or 365) The first formula is used when L > R and the payment P is from the FRA seller to the FRA buyer; the second formula is used when L < R and the payment is from the buyer to the
seller. In effect, if the settlement rate is higher, the FRA seller compensates the buyer for the extra interest; if the settlement rate is lower, the buyer surrenders the interest saving to the seller. 6 Let
In a forward foreign currency contract, the parties fix the rate of exchange between two currencies for future delivery. In a FRA, the rate of interest on a future borrowing or lending is locked in. Just as the forward exchange rate reflects the market's expectations regarding the future spot rate, the rate fixed in an FRA reflects the market's expectations of future interest rates. The expectations theory of the term structure says that forward interest rates implicit in a given term structure equal the expected future spot interest rates. Thus, the 3 month rate expected to
rule 6 months from today is implied by the 6 and 9 months actual rates today: where, as usual, the superscript " e " denotes expected. In general, given the spot interest rates for a short and a long maturity, the rate expected to rule for the period between the end of short maturity and the end of long maturity is given by DS, DL and DF are as explained above. B is the day count basis (360 or 365 days). Interest rates i o,s^ i^ O,L^ stated as fractions, (not per cent) are the spot interest rates at time t = 0 for maturities Sand L respectively. When L> R the FRA buyer incurs extra interest cost equal to (L - R)/ 100(DF/B). This is discounted by a discount factor equal to [1 + (L/100)(DF/B)] , This gives the formula above. Note that the rate so calculated will only serve as a benchmark for a FRA quotation. The actual quote will be influenced by demand-supply conditions in he market and the market's expectations. We will now illustrate applications of FRAs for borrowers and investors the former to lock in the cost of short term borrowing and the latter to lock in the return on short-term investment. FRA for a Borrower A firm plans to borrow £5 million for 3 months, 6 months from now. The current 3 month
Euro-sterling rates are 10.50-10.75%. The firm has to pay a spread of 25 b.p. (0.25%) over LIB OR. The treasurer is apprehensive about the possibility of rates rising over the coming six months. He wishes to lock in the cost of loan. Sterling 6/9 FRA is being offered at 10.8750%. The treasurer decides to buy it. We will work out the firm's cost of borrowing under alternative scenarios of 3month rates 6 months from today. The anticipated borrowing is for 91 days. Scenario 1: Six months later, sterling settlement LIBOR is 11.50. The bank, which sold the FRA compensates the firm by immediately paying an amount A calculated as A = (0.1150 - 0.10875) x 5,000,000 x (91/365) / [1 + 0. (91/365)]
= £7,573.
on the settlement date. The firm borrows £5 million at 11.75% including a spread of 25 b.p. The compensation received can be invested at 11.25% (This is the LIBID). The cost of the loan is Interest on 5 million at 11.75% for 91 days = (0.1175) x 5,000,000 x (91/365) = 146,472.
From this we must subtract the compounded value of the compensation received from the
FRA selling bank. This is given by
(7573.94) x [1 + 0.1125(91/365)] = 7,786.
So the net cost is £1, 38,686.23 which works out to an annual rate of 11.1254%. This is the rate locked in by the firm (10.8750 + 0.25 = 11.1250). Scenario 2: 6 months later the settlement rate LIBOR is 10.25% The firm pays the bank an amount A given by A = (0.10875 - 0.1025) x 5,000,000 x (91/365) = 7,596. [1+0.1025(91/365)] The firm has to borrow this at 10.50% in addition to the loan of £5 million. Its total cost now consists of interest on 5 million plus the repayment of the loan taken to pay the compensation. This works out to £1 38,686.23 which is again an annual cost of 11.1254%. FRA for an Investor
A fund manager is expecting to have $5 million 3 months from now to invest in a 3 month (92 days) Eurodollar deposit. The current 3 month rates are 8.25-8.375%. The $3/6 FRA bid rate is 8.1250. The manager sells a FRA for $5 million.
- 3 months later, the settlement rate is 7.50% The bank pays the manager an amount A given by
A = (0.08125 - 0.0750)(5,000,000)(92/360) = $7 835 92
[1 + 0.075(92/360)]
The manager invests this along with $5 million at 7.50%. His total return is $103,819. which is 8.125% annual return contracted in the FRA.
on a notional principal wherein the FRA buyer agrees to pay interest at a fixed rate (the contract
rate) while the seller pays interest at the settlement rate. Settlement is done by payment of the net difference by one party to the other. Here is an example: Bank A and Bank B enter into a 6 x 9 FRA. Bank A pays fixed rate at 6.50%. Bank B pays a rate based on 91 day T -bill yield fixed the day before the settlement date. Other details:
- Notional principal = Rs. 10 crore
- FRA start and settlement date 10/12/04, Maturity date 10/3/
- T bill yield on fixing date (say 9/12/04) = 5.50%
- Determine cash flow at settlement (assume discount rate as 7%) The calculations are as follows: (a) Interest payable by bank A = (10 crore) (0.065) (91/365) = Rs 16,20,547.9 (b) Interest payable by bank B = (10 crore) (0.055) (91/365) = Rs 1,371,232.8 (c) Net payable by bank A on maturity date {(a) - (b)} = Rs 24,9315. (d) Discounting (c) to settlement date
= (c)/(1+ discount rate*discount period).
= Rs 2,49,315.1/[1 + 0.07(91/365)] = Rs 2,45,038.
Amount payable on settlement date = Rs 2,45,038.67 payable by Bank A.
RBI guidelines state that corporate are permitted to do FRAs only to hedge underlying exposures while market maker banks can take on uncovered positions within limits specified by their boards and vetted by RBI. Capital adequacy norms are applicable and the minimum required capital ratio would depend upon the underlying notional principal, the tenor of the agreement and the type of counterparty. INTEREST RATE OPTIONS
premium stated as a fraction of the face value of the contract or the underlying notional principal. An interest rate cap consists of a series of call options on interest rate or a portfolio of calls. A cap protects the borrower from increase in interest rates at each reset date in a medium- to- Iong-term floating rate liability. Similarly, an interest rate floor is a series or portfolio of put options on interest rate which protects a lender against fall in interest rate on rate rest dates of a floating rate asset. An interest rate collar is a combination of a cap and a floor. In the following subsection we will analyze simple interest rate options.
A Call Option on Interest Rate Consider first a European call option on 6-month LIBOR. The contract specifications are as follows: Time to expiry: 3 months (say 92 days) Underlying Interest Rate: 6-month LIBOR Strike Rate: 9% Face Value: $5 million Premium or Option Value: 50 b.p. (0.5% of face value) = $25, The current three and six month LIBORS are 8.60 and 8.75% respectively. Let us work out the pay-off to a long position in this option. Assume that the option has been purchased by a firm, which needs to borrow $5 million for six months in three months time. The pay-off to the holder depends upon the value of the 6 month LIBOR 3 months later: The option is not exercised. The firm borrows in the market. The pay-off is a loss of compounded value of the premium paid three months ago. The present value of the loss (at the time of option expiry) is the premium compounded for three months at the 3-month rate, which prevailed at option initiation. In the above example it is $25,000[1 + 0.0860(92/360)] = $25,549.
If the loss is to be reckoned at the maturity of the loan, this amount must be further compounded for 6 months at the 6-month LIBOR at the time the option expires. The option is exercised. The option writer has to pay the option buyer an amount, which
3-month Euro-deposit. The amount involved is $10 million. The current 3-month rate is
10.50%, which the investor considers to be satisfactory. A put option on LIB OR is available with the following features: Maturity : 3 months (91 days) Strike Rate : 10.50%
Face Value : $10 million
Underlying : 3-month LIBOR.
Premium : 25 b.p. (0.25% of face value) = $25,
To hedge the risk, the investor goes long in the put. Three months later, if the 3-month LIBOR is less than 10.50% he will exercise the option or else let it lapse. Suppose the 3- month LIBOR at option expiry is 9.5%. The option writer must pay the option buyer a sum equal to (0.105 - 0.095)(10,000,000)(91/360) = $25,277.
This is paid 3 months after option exercise or its discounted value at option
exercise.
The break-even rate is the value of i satisfying the following equality:
A[l + i(M/360)] = A[l + R(M/360)] - P[l + i t,T (T/360)][1 + i(M/360)]
Where P is the put premium and other notation is same as in the case of a call option. In the example, A = 10 million, R = 0.105, T = 91, M = 91, it,T= 0.105 and P = 25,000. The break- even rate works out to 9.46%. If the 3-month LIBOR 3 months later are less than this, the put buyer makes a net gain. Interest rate options are thus similar to currency options in their pay-off profiles and hedging applications. Valuation of these options also has many similarities with valuation of currency options.
A Put-call Parity Relation
It is easy to see that a long position in a call option with strike rate R and a short position in a put with the same strike and same maturity, both on the same underlying index (such as 6-
- A put option on M-day LIB OR, at strike rate R, maturing T-days from today, face value A, premium P.
- An FRA, on M-day LIB OR, maturing T-days from today, face value A, contract rate R. Thus irrespective of the outcome you gain. The discounted value of the gain at time t (today) is (P - C) as it should be since the FRA is costless. Now suppose C > P. You can verify that by selling a call, buying a put and buying an FRA profit can be made irrespective of what the interest rate is at option expiry. Thus, if the strike rates in the put and the call both equal the current rate in a corresponding FRA, the call and put must have identical premia. To put it in another manner, a long position in a call and a short position in a put both with same maturity, the same strike rate and the same underlying interest rate is equivalent to buying an FRA 'on the same interest rate at a contract rate equal to the strike rate in the put and call. INTEREST RATE CAPS, FLOORS AND COLLARS Interest rate caps and floors are portfolios respectively, simple calls and puts on interest rate. We will begin by looking at examples of applications of caps and floors. Interest Rate Caps A corporation borrowing medium-term floating rate funds wishes to protect itself against the risk of rising interest rates. It can do so by buying an interest rate cap for the duration of the loan. The following example illustrates the working of this instrument. , A corporation has borrowed $50 million on floating rate basis for 3 years. The interest rate reset dates are March 1 and September 1. The spread over LIBOR is 25 b.p. (0.25%). It is a bullet loan I (i.e. repayment of the entire principal is at maturity). It buys a 3 year cap on 6-month LIBOR with the following features: lf i > R, your gain from exercising the call exactly offsets what you have to pay the buyer of the
FRA leaving you with the difference between the compounded values of the put premium you received and the call premium you paid. The put you sold will lapse. If i < R, the put will be exercised against you but the loss will be offset by your gain from the
