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Hedging Prepayment Risk on Retail Mortgages Hedging the interest rate risk of a retail mortgage portfolio is a difficult task for banks. Besides normal interest rate risk, there is risk caused by embedded options i.e. choices incorporated in the mortgage contract. In regular Dutch retail mortgages, the customer receives several embedded options. One of these options is the option to prepay the mortgage without incurring additional costs in case the customer moves to a new home. In this article we explain why there is risk when the customer receives this option. If the customer does not use the option, he will continue the current contract. We will also discuss how the resulting interest rate risk can be hedged and how to determine the hedging price. However let us first discuss the mortgage’s financing. Dirk Veldhuizen graduated in the master of Econometrics and OR and the master of Finance (honours track Quantitative Finance) at the Vrije Universiteit in September 2008. He now works at SNS Reaal as Risk Management Trainee. This article is based on his thesis for Quantitative Finance, written at SNS Reaal. Dr. Svetlana Borovkova (VU) and dr. Frans Boshuizen (SNS Reaal) supervised the thesis. How to finance a mortgage and why is there prepayment risk? When a customer acquires a mortgage loan, he or she usually borrows money for a long period of time, say 10 years. Moreover he will have to pay a fixed interest rate for the entire period. In order to fund this mortgage the bank could issue covered bonds or attract retail savings, which is cash on ordinary savings accounts. Additionally in order to prevent interest rate risks, the bank needs to make sure that the mortgage and funding have the same maturity time i.e. they have the same duration. This will ensure that the market value of the assets and liabilities remains the same. Furthermore in certain circumstances the bank may be unable to find funding with the same maturity rate as the mortgage. In this case, the bank could hedge the resulting interest rate risk by entering into a regular interest rate swap. This enables the bank to interchange the interest rate on the acquired funding with the interest rate on funding with the same maturity as the mortgage. We show how this works for an index amortizing swap in Figure 1. Now let us discuss how the embedded option can yield a loss to the bank. First of all, suppose that the bank matched its mortgages and funding perfectly. Now, suppose that the yield 32 AENORM 52 Juni 2006 curve otherwise known as the term structure of interest rates, decreases for all interest rates. A consumer who moves to a new home can now prepay his mortgage for free and purchase a new mortgage loan for a lower interest rate. However the bank’s funding cannot be prepaid without compensating the counterparties for the interest rate decrease. Moreover the bonds’ value increase as the interest rates decrease; this is due to discounting. Thus the bank makes a loss when it repays the funding. Another option for the bank would be keeping the relatively expensive funding, which eventually results in a greater loss over time compared to using new funding. Prepayment risk is defined as the interest rate risk due to early repayments or prepayments of mortgages. We will only focus on prepayments which are driven by the current interest rates on mortgages. We assume that non-interest rate driven prepayments are independent of interest rate movements. Hedging strategies A bank can set up a strategy to hedge prepayment risk. It needs to buy financial derivatives such that the cash of the prepaid mortgages can be lent to other parties while still yielding a sufficient interest rate. The bank needs a portfolio of receiver swaptions1 for this hedging strategy. It can also hedge its prepayment risk dynamically by checking every day, week or month what its current portfolio of mortgages and funding is. The mismatch can be hedged using regular swaps. Even though the costs of this last strategy are unknown beforehand, it does not require any option premium. The challenge with the first hedging strategy is to determine the required amount of receiver Interest payments and prepayments using IAS Customer Fixed interest rate payments Floating interest rate payments Bank Market Prepayments Prepayments as input Fixed i. r. p. Floating i. r. p. Funding for new mortgages: Notional - Prepayments IAPS Figure 1. Schematics of a hedge using an index amortizing payer swap (IAPS). Solid lines resemble the interest rate payments (i.r.p.). The dashed lines resemble the prepayments. The bank receives fixed interest rate payments on the mortgages, but borrows funding for a floating interest rate. This way the bank can re-use the cash of the prepaid mortgages for new loans for current market prices and prevent a loss on prepayments. The index amortizing swap with the prepayments, which equal the outcomes of the amortization function, exchanges exactly the right amount of fixed interest rate payments into floating interest rate payments to eliminate the interest rate risk of borrowing for a floating rate and lending for a fixed rate. swaptions. For example, when interest rates have been relatively high as in the fall of 2008, interest rate driven prepayments will be lower than expected. The receiver swaptions will expire worthless, because the fixed rate received on new swaps is higher, while the mortgage portfolio stays large. When interest rates fall for example in a few years, more customers than expected will be able to prepay their mortgage. Now the swaption portfolio is not large enough to cover all prepayments, which will result in a loss for the bank. Advanced hedging amortizing swaps strategy using index The use of index amortizing payer swaps, which is an over-the-counter contract, can be the answer to this challenge. The key feature of an index-amortizing swap is that the notional of the swap decreases or amortizes based on a function instead of a predetermined scheme. This function is usually based on a reference rate. Moreover we use the interest rate for a regular bond, which matures at the same time as our swap contract. The bank chooses a function which yields a smaller amortization of the notional when the reference rate is high and a larger amortization when interest rates are low. This corresponds to observations of the prepayments on Dutch retail mortgages. For an index amortizing payer swap to be effective, the bank needs to pay a floating interest rate on the funding preferably one which resets every month. Using an index amortizing payer swap2 on the entire mortgage portfolio, the bank changes the floating interest rate payments it has to pay into the required fixed rate payments. Disregarding credit risk and assuming that the amortization function perfectly matches the customer behaviour, Figure 1 shows how the index amortizing payer swap works. The risks associated with an index amortizing swap are described in Galaif (1993). There are the usual risks like counterparty risk and liquidity risk, an index-amortizing swap being an over-the-counter product. However there is also model risk, which we will address later on. The product also has an interesting interest rate risk, which is asymmetric for the fixed ratepayer and fixed rate receiver. Additionally the fixed ratepayer has less interest rate risk than the fixed rate receiver. When the yield curve decreases and hence the reference rate the fixed ratepayer incurs a loss. This loss is relatively small because the amortization is high and the size of the contract decreases rapidly. Also, when the yield curve increases, the amortization is low and the fixed ratepayer makes a relatively large profit. As mentioned in the above, there is also model A receiver swaption is an option to enter into an interest rate swap where you receive a fixed interest rate and pay a floating interest rate. The floating interest rate can be received on the lent cash. When the interest rates fall and you exercise the option, you can transform a relative low floating rate into a relatively high fixed rate. The net payoff of this swaption (when exercised) is the present value of the difference between the fixed interest rate received on the underlying swap of the swaption and the fixed rate which would be received when you enter a new at-the-money receiver swap. 2 When you enter a payer swap, you pay the fixed interest rate and receive the floating rate. When you enter a receiver swap, you pay the floating rate and receive the fixed rate. 7 AENORM 52 Juni 2006 33 risk associated with index amortizing swaps. Because the notional depends on a reference rate, the price of this product is path dependent. We need the entire history of the reference rate from the start of the swap in order to determine the current notional. One method to price this swap is simulation. For this we need an interest rate model, which may not capture the reality correctly. According to Fernard (1993) there is a relatively simple example of how to price an index-amortizing swap. However we will develop a general scheme for a more complex function. Another challenge when using an index- amortizing swap as a hedge is to determine the correct amortization function, i.e., the function which matches the prepayments of a portfolio of mortgages. This can prove to be rather tricky in practice. Nevertheless having an inaccurate amortization function will hurt the effectiveness of the hedge. This is firm specific basis risk when using this product for a hedging strategy. Furthermore we have seen before that the use of receiver swaptions yields the same type of risk. notional to the counterparty and borrowing the same amount from the same counterparty only increases counterparty risk while it does not change the net cash flows of the swap. However for the pricing, it is convenient to pretend we exchange the notional. Additionally when exchanging the notional, the value of a floating rate bond must equal the notional at the start of the contract. Setting the notional equal to 1, we have to find the fixed interest rate for which the expected present value of the interest payments, the prepayments and the remaining notional amount at the end of the contract, equals 1. It is important to note that we assume a risk-neutral world. Hence this translates into the following formula: E (rfixed ¦ t mn ¦ t mn 1 1 1 Lt dft m (Lt 1 Lt ) df Lmn df ) 1, Where rfixed is the fixed interest rate which needs to be paid, m is the number of payments per "Another challenge in using an index amortizing swap as a hedge is to determine the correct amortization function." Pricing an index amortizing swap In order to price an index amortizing swap, we determine the par fixed interest rate on the swap (the fixed interest rate such that the contract has no value to either parties). We thus need to equate the value of a bond, which pays a fixed interest rate and another, which pays a floating interest rate. Interest rate swaps exclude the exchange of the notional. Lending the year, n is the maturity of the contract in years, Lt is the notional amount at time t and dft is the discount factor for the tth interest payment. The notional amount Lt equals Lt= Lt-1— PRt-1, where PRt-1 is the prepayment in period t (we assume PR0=0 ), which is determined by our prepayment function. In order to determine the fixed rate of the index amortizing swap, we use the simulation scheme 1: Step 0: Choose and calibrate a model for the yield curve. Step 1: Simulate the relevant part of the yield curve for all future dates in a risk-neutral world. Step 2: Construct the reference rate. Step 3: Calculate the prepayments per period. Step 4: Determine the present value of the interest payments, prepayments and repayment of the remaining notional. Step 5: Repeat step 1-4, N times, where N is the number of simulations we want to run. Step 6: Calculate the par swap rate by solving the following equation: i N nm L N nm ¦ i 1 (rfixed ¦ t 1 mt dfti ) ¦ i 1 (¦ t 1 (Lit 1 Lit ) dfti Linm dfnmi ) N with i the index for the simulations and t the index for the time in a simulation. This yields: 1 − ¦ L = ¦ W = /LW − − /LW ⋅ GIWL + /LQP ⋅ GIWL 1 UIL[HG = Scheme 1 34 AENORM 52 Juni 2006 QP ¦ L = ¦ W = 1 QP /LW − ⋅ GIWL P Because the index-amortizing swap is path dependent, we have to use this somewhat large formula instead of using the average fixed rate of all separate simulation runs. The resulting outcomes depend on the realizations of the yield curve. Greeks of an index amortizing swap Besides calculating the par fixed rate of an index amortizing payer swap, calculating the Greeks or sensitivity with respect to several parameters is a challenge as well. Glasserman (2004) describes three different solutions to calculating the Greeks in simulation. Two of those are semi analytical solutions. These solutions are however infeasible, because the path dependence of the swap creates very complex derivatives with respect to parameters (high dimensional with many terms). In the third solution, the Finite-Difference Approximation, we calculate the Greeks using brute force. First, we calculate the value of many different values for the parameters and then calculate the derivative by computing: *UHHN = Glasserman, P. (2004). Monte Carlo Methods in Financial Engineering, New York : Springer Science and Business Media. Hull, J.C. (2006). Options, futures, and other derivatives 6th ed, Upper Saddle River: Pearson/Prentice Hall. Hull, J.C. and White, A. (1990). Pricing Interest Rate Derivate Securities, Review of Financial Studies, 3(4), 573-592. 9 ; + Δ − 9 ; − Δ Δ where v(x) is the value of the index amortizing swap at parameters X and Δ is a small change in one of the parameters. Summary In this paper we first discussed the principles of an advanced hedging strategy for prepayment risk using index amortizing swaps and then we computed the price and Greeks of an indexamortizing swap. Moreover we argued that when using an index-amortizing swap we could avoid answering the question of how large the position in hedging instruments should be. This can make index amortizing swaps more accurate than a portfolio of regular receiver swaptions. However, we still need to estimate a prepayment function, which can be chalenging in practice. References Boshuizen, F., Van der Vaart, A.W., Van Zanten, H., Banachewicz, K. and Zareba, P. (2006). Lecture notes course Stochastic Processes for Finance, Vrije Universiteit Amsterdam. Galaif, L. N. (1993). Index amortizing rate swaps, Quarterly Review Issue Winter, 63-70 Fernard, J. D. (1993). The pricing and hedging of index amortizing rate swaps, Quarterly Review Issue Winter, 71-74. AENORM 52 Juni 2006 35