Download Hedging Prepayment Risk on Retail Mortgages

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
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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.
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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
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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.
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