Yoshinao Kiyama, Tsukasa Yamashita,
Toshinao Yoshiba, and Toshihiro Yoshida
Yoshinao Kiyama: Derivatives and Fixed Income Division, The Sakura Bank, Limited
Tsukasa Yamashita: Risk Management Advisory, Bankers Trust Company,
(E-mail: tsukasa.yamashita@bankerstrust.com)
Toshinao Yoshiba: Research Division 1, Institute for Monetary and Economic Studies, Bank of Japan (E-mail: toshinao.yoshiba@boj.or.jp)
Toshihiro Yoshida: Quantitative Analysis Department, Global Fixed Income and Derivatives Division Japan, UBS Securities Limited, Tokyo Branch (E-mail: toshihiro.yoshida@ubs.com)
This study was started by Yoshida, who contracted with the Institute for Monetary and Economic Studies, Bank of Japan, with Yamashita. Later, Yoshiba and Kiyama joined the
project. The original Japanese version of this paper was submitted to the workshop on financial risk management that was held by the Bank of Japan in June 1996, and was published in Kin’yu Kenkyu (Monetary and Economic Studies), 15 (4), Bank of Japan, 1996.
In order to measure the interest rate risk of banking accounts such as deposits and loans, this paper extends the value at risk (hereafter, VaR) analysis framework, which is useful for the riskevaluation of trading accounts.In order to apply the VaR concept derived from trading
accounts to banking accounts, we should take into account thefollowing issues: (1) the longer risk evaluation period because of the inflexibility of adjustability of banking account positions;(2) the evaluation of risk included in the administered rates (thelong-term prime rate and the short-term prime rate); and (3) theprepayment risk (associated with housing loans, etc.). Therefore,in this paper we first construct a VaR model including a termstructure
model to express the stochastic process of market rate, theadministered rate model, and the prepayment function model.Then, we perform a simulation using an imaginary portfolio to
analyze the factors determining interest rate risk.In conclusion, it has been proved that the factor of administeredrates increases interest rate risk both in single products and
in a portfolio. Taking into account the behavior of customers whowant better interest rate conditions, the factor of prepayment decreases the present value, which is itself the basis of calculating risk. Finally, we perform a sensitivity analysis of model parametersto show the magnitude of model risk.
Key words: Value at risk; Monte Carlo simulation; Holding period; Term-structure model; Heath-Jarrow-Morton model; Administered rate; Prepayment
I. Introduction
It is essential to measure the risk of financial transactions as accurately as possible in order to price individual transactions, to hedge for risk control, to calculate the required total capital prepared for the risk of the whole portfolio, and to make decisions on the best resource allocation, paying attention to the trade-off between profitability and risk. In this context, we can say that the quantitative method of risk measurement is the most important technique for the management
of financial institutions. Financial institutions confront various kinds of risks. Of those risks, market risk and credit risk seem most important. In this paper, we focus on market risk,
especially the market risk of whole banking account (or non-trading account) portfolios, which account for the most typical portfolios of Japanese banks. As is widely recognized, value at risk (hereafter, abbreviated as VaR), a kind of stochastic risk evaluation method conducted on a present value (hereafter, PV) basis, has by now taken root for the interest rate risk evaluation of trading accounts. On the other hand, as to the interest rate risk evaluation of banking accounts including loans and deposits, there is no consensus on the definition of risk. In addition, in the process of risk measurement, technical skills are not yet mature enough. First, there is no
consensus on whether we should measure risk on an earnings basis (so-called earning at risk [EaR]) or on a PV basis (VaR). Second, some features of banking account transactions (difficulties in closing positions, the possibility of prepayment, and the linking of products to administered rates) make risk calculation more difficult than in the case of trading accounts.
It is preferable to use the same rule to measure the aggregate risk of portfolios of
financial institutions. From this point of view, we propose a theoretical framework to
measure the interest rate risk of banking accounts within the VaR framework, which
has become a popular method of risk measurement for trading accounts. Using this
theoretical framework and an imaginary portfolio, we perform a simulation to analyze what kinds of factors determine the interest rate risk of banking accounts. Of course, it is difficult at this stage to judge which of EaR and VaR is superior.
However, it seems very meaningful to study the possibilities of the extension of the risk measurement method with VaR, for the following reasons. (1) If credit risk can also be measured under the VaR framework, as shown by Oda and Muranaga (1997), we can expect to be able to manage market risk and credit risk in aggregate on a PV basis. (2) If credit liquidation and the disposal of deposit business by selling branches become more popular, as in the
This paper consists of the following parts. First, in Section II, we give an overview of the differences between trading accounts and banking accounts. We list the points that we should consider when we apply the VaR framework, which has been developed as the risk management method for trading accounts, to banking accounts. In Section III, we show how to approach model-building when we extend the VaR model, and explain the outline of the model that we use in this paper. In Section IV, we perform a simulation under the extended VaR framework with the imaginary portfolio and analyze the various factors that change the interest risk of banking
accounts. Finally, in Section V, we derive some implications for risk management.
II. The Approach to Extending the VaR Model:
Taking into Account Features of Banking Accounts
A. Features of Banking Accounts: Comparison with Trading Accounts
When we calculate the interest rate risk of banking accounts within the VaR framework, we have to list sequentially the types of financial transactions that are actually included in banking accounts, and state what kinds of features those transactions have by comparison with trading accounts. There are two definitions of banking accounts: (1) the total of non-trading accounts; and (2) non-trading accounts excluding investment securities accounts. We use the second, narrower definition for banking accounts in this paper. Because we exclude investment securities such as bonds and stocks, we focus mainly on loans on the assets side and deposits and financial
debentures on the liabilities side. When we calculate risk, we may deal with offbalance- sheet positions such as swaps, futures, and options held in banking accounts mainly for hedging purposes, and the exchange rate risk of foreign currency assets and liabilities. However, because we can deal with these transactions basically in the same way as the off-balance-sheet transactions of trading accounts and exchange rate risk, which have already been discussed actively in the literature, we do not deal with these in this paper. Next, Appendix Table 1 shows the product lineup of standard city banks. We can derive the features of deposits and loans that are representative products of banks as follows.
• Most products do not possess liquidity.
• Most loans and deposits are on a fixed rate basis (the applied interest rate does not change until the contractual maturity).
• Some long-term loans and long-term deposits are on a floating rate basis; therefore, the applied rate is settled every renewal period under predetermined interest rate conditions. The interest rates are market rates in some cases and administered rates, such as the short-term prime rate and the long-term prime rate, in other cases.
• Even in the case of fixed-rate loans and deposits, when maturity arrives and products are rolled over, the applied rate is renewed according to the market rate, shortterm prime rate, and long-term prime rate at that time. Especially in the case of non-maturity loans and non-maturity deposits, maturities arrive randomly and applied interest rates are renewed according to the change in the corresponding interest rate applicable at the time.
• In the case of corporate loans, there are few prepayments during the transaction term. On the other hand, in the case of individual loans (especially housing loans), there are frequent prepayments. Recently, it has become usual that the applied interest rate is fixed for the first few years or based on a floating rate such as the short-term prime rate. However, there remain some housing loans that started in the past and whose applied interest rate is floating based on the long-term prime rate or on a fixed rate until maturity.1
• There are frequent prepayments in the case of individual term deposits.
As compared to trading accounts, the three main differences are as follows.
[1] It is difficult to close a position immediately.
[2] There are products such as housing loans and time deposits that allow customers to cancel before maturity (hereafter, we always call the cancellation before maturity “prepayment”).
[3] There are not only products linked to market rates but also products linked to administered rates (e.g., the prime rate).
B. Approach to Extension of VaR
It is widely recognized that the VaR framework which measures portfolio risk in terms of the probability distribution of changes in PV is effective for the risk calculation of trading accounts. However, considering the features of banking account portfolios outlined above, what kind of modifications are required in methods for calculating current value and risk? The first change is due to the problem of liquidity ([1] in Section II.A). The liquidity of trading account products is high, and the banks usually hold such products only for a short term. Therefore, it is usual that the holding period of the portfolio for risk calculation (the required period for liquidation)—or risk calculation period2—is usually one day, or at the most two weeks. On the other hand, for banking accounts, the banks intend to hold a product for a long time and the liquidity of the products is low. Therefore, the risk calculation period should be assumed to be longer than that of a trading account. In calculating risk, it is necessary to assume that the interest rate fluctuation process lasts longer, and to explicitly account for the fluctuation pattern of the position itself (dynamic simulation). In addition, because of [2] in Section II.A, it is essential to consider prepayment (cancellation of loans as interest rates decrease and cancellation of deposits as interest rates
rise) to be a kind of option and to recognize its value. Finally, because of [3] in Section II.A, it is necessary to make an explicit model of the relationship between market rates, which are risk factors for trading account VaR, and administered rates such as prime rates. In the following sections, we elaborate these points and discuss our approach to the extension of the VaR model to calculate the interest rate risk calculation of banking accounts.
1. Lack of liquidity
When we calculate the risk of trading accounts, we can assume relatively short holding periods such as one day or two weeks because of the specific characteristics of such accounts. Then, using the VaR model’s variance-covariance method, we calculate the risk amount in a simple way, usually under the following assumptions.
[1] We assume that the value of the position changes linearly according to the
change in the interest rate (that is, we omit convexity risks).
[2] We assume that market rates of various maturities fluctuate according to a
1. Financial institutions except for city banks still sell housing loans linked to long-term prime or to fixed interest rates.
2. In this paper, we use both terms “holding period” and “risk evaluation period” to mean the same thing, without distinction.
multivariate lognormal process with specified drift rates,3 volatilities, and correlations.4
On the other hand, banking account products possess little liquidity, so their holding period is longer and it is necessary to address the following four problems.
[1] The longer the holding period, the larger the fluctuation of interest rates, so convexity risk may be non-negligible. We cannot capture this kind of nonlinear risk accurately under the ordinary variance-covariance method, so we have to introduce a simulation method.
[2] The VaR model assumes that market rates of various terms are random variables under a multivariate lognormal process. On the one hand, this assumption makes the calculation of risk easy. However, on the other hand, when considering yield curve fluctuations over a long time, such an assumption is likely to produce an unrealistic yield curve that contradicts the non-arbitrage conditions. Therefore, when making a simulation, it is necessary to use some term-structure model to solve this problem.
[3] It is necessary to take account explicitly of position fluctuation during the holding period.
[4] Because it is not always the case that loss is maximized at the end of the holding period, it is necessary to calculate the loss amounts at each time during the holding period.
2. Prepayment
In banking accounts, unlike trading accounts, there are products that allow customers to cancel before maturity, e.g., mortgage loans and time deposits. Though the interest rate to be applied and the maturity are decided when the contract is made, it is possible for borrowers and depositors to cancel at any time during the term of the loan or deposit. The right of prepayment belongs not to banks but to customers. Unless we take account of the value of prepayment, we cannot accurately
evaluate the value of housing loans and time deposits, and cannot calculate their risk. Therefore, we need some valuation model of prepayments in the simulation.
3. Administered rates
The price of interest rate-sensitive products in trading accounts usually depends on the market rate. Therefore, we consider only the market interest rate as a risk factor. However, products of banking accounts require not only market rates but also administered rates as index rates. For example, a floating-rate housing loan changes its applied rate when an index rate such as the short-term prime rate or the long-term
3. The drift rate is often ignored.
4. The following equations specify this process.
dri = miridt + siridWi (i = 1, 2, . . . , n),dWidWj = rijdt,where
ri: the market interest rates selected for the risk factor,
Wi: the standard Wiener process,
mi: drift rate,
si: volatility,
rij : correlation.
prime rate changes by more than a specific amount. In addition, the short-term prime rate is applied to short-term loans in many cases. When the loan is rolled over at maturity, the short-term prime rate at that time is applied to the renewed loan. In order to calculate the value and risk of these products, we have to make models of administered rates as well as of the market rate. Market rate-linked products do not owe interest rate risk to the cash flow after interest rate renewal in the future. On the other hand, because administered rate-linked products owe interest rate risk also after the interest rate novation, we have to take care of the cash flows after renewal.
4. Definition of interest rate risk
In this paper, as in the case of the VaR of trading accounts, we calculate the interest rate risk of banking accounts as the fluctuation of the discounted PV of the portfolio over a specific holding period. So we must use two time axes, one to represent time during the holding period (risk measurement period), and the other to describe the yield-curve term structure seen at each holding. Figure 1 shows conceptually how one simulation generates a series of yield curves during the holding period.5 First, we generate the shape of the yield surface (interest rate environment) under the termstructure model. Next, we calculate the portfolio PV at each point during the holding period. To obtain the PV, we take the expected value of all cash flows. In other words, based on the yield curve of every point generated along the holding period axis by the simulation, we have to make a simulation to generate the yield curve along the yield construction period.6 Using the future path of market rates generated in this way, we construct the future path of administered rates. The development of the yield curve determines future cash flows, which are discounted by the series of generated market rates to obtain the PV in one simulation. The simulation is repeated many times, and the average value is set to be the PV at the given time. We measure the risk by how much the PV changes as time passes.7
5. Technically speaking, when we adopt the Heath-Jarrow-Morton (HJM) model for the stochastic development of
the yield curve, we should be careful that the yield curve construction period becomes shorter as time passes.
6. We can also use a lattice method.
7. For exact simulation, we should use the original probability measure of the stochastic process of the yield curve.
Yield curve term
Holding period
Interest rate
Figure 1 Diagram of Yield Curves Simulated during the Evaluation of Interest Rate
Risk Using a Term-Structure Model
5. Position change
In considering a trading account, we assume that positions are closed out in the near future, and deal only with the positions which we actually hold. We usually measure risk statically, not taking account of any position changes in the future. In considering a banking account, because of the longer holding time involved, it seems more realistic to assume that some matured or prepaid products are not removed from the portfolio but keep rolling over with interest rate renewal (the rest become products linked to the market rates). We perform a dynamic simulation to calculate risk taking account of changes in portfolio composition.
What kinds of new assets and liabilities are accepted as present positions mature depends on the management strategy of a financial institution. Therefore, it may be appropriate for an institution to assume that its portfolio will develop following its own strategy and to calculate risk accordingly. When we make simulations, we assume several scenarios for position transition so that we can understand how such assumptions affect the size of the calculated interest rate risk. When we deal with position dynamics, the holding period should be considered to be time actually passing. Therefore, as with the rollover rule along the yield construction axis, some of the matured products rollover and the rest switch to the market rate-based products. Of course, the interest rate is renewed. However, from this point of view, because the portfolio whose risk we are going to calculate itself changes, we measure both “the PV change resulting from change of interest rate environment” and “the PV change resulting from position change” at the same time.8
III. Details of the Model Used in the Simulation
In this section, we build the basic framework for calculation of interest rate risk of banking accounts, as analyzed in the last section. We precisely define the measurement of interest rate risk discussed in the last section. Then, we explain how to model each risk factor: the market rate, administered rates, and prepayments.
A. Definition of Interest Rate Risk
As described in the first section, in calculating the interest rate risk of banking accounts, we think of an account as a portfolio and analyze how much its current value is likely to decrease during the holding period due to interest rate fluctuations. This point of view means that we capture the risk of banking accounts by applying VaR, a risk methodology for trading accounts. Concretely speaking,
• We discount all cash flows that result from assets and liabilities during the specific period according to the interest rate environment in order to calculate the PV of banking accounts.
8. In this paper, we compare the PVs of the following seven years’ cash flows from each time grid in the holding
period. This means that we neglect the net cash flow of interest payments and receipts during the holding period.
Although this problem relates to the appropriate definition of risk amount, we should analyze the transition of the
portfolio value including the accumulated cash flows arising in the holding period in order to evaluate the future
value of the existing portfolio.
• We calculate the minimum of the PVs determined by the change of interest rate term structure during a specific holding period.
• We calculate the a percentile point of the distribution of the difference between the above minimum and the initial PV, which we call the a percentile point “risk amount.”9
B. The Process of Market Rate Fluctuation
There is much research, both theoretical and empirical, on time-series models of market interest rates. There are well-known models that include one risk factor and describe the dynamics of instantaneous spot rates, such as the Vasicek (1977) model or the Cox-Ingersoll-Ross (1985) (CIR) model.
These one-factor models have the advantage of simple mathematical form. On the other hand, they depend on the assumption that all interest rates in the term structure are perfectly correlated with each other. They are fine for the purpose of valuing interest rate derivative products with a short maturity, but it is possible that this kind of model does not satisfactorily describe the various fluctuations of yield curves over a long time period, as we attempt to do in this paper.
Therefore, it is necessary for us to devise an interest rate model that is consistent with the fluctuations of the term structure seen in the real world. Taking account of this requirement, we adopt the HJM model as our stochastic model of market rates.10
1. Outline of the HJM model
Heath, Jarrow, and Morton (1992) focused on the instantaneous forward rate [f (t, T ), 0 ² t ² T ] at future time T seen from time t , and developed a multi-factor model as follows:
N
df (t ,T ) = mf (t ,T )dt +åsf i (t ,T )dWi (t ), 0 ² t ²T, (1)
i =1
where mf (t ,T ) is a drift term, sf i (t ,T ) are volatility terms, and each [Wi (t ), t ³ 0] is an independent standard Wiener process. The price of a discount bond is given under the equivalent Martingale measure as
P (t ,T ) = exp [–ºtT f (t , s )ds ]. (2)
By assuming various forms of volatility function, this flexible model can express various kinds of yield curve fluctuations. Many famous one-factor models fit into this framework. In addition, the HJM scheme has the advantage that it does not require a market price of risk, which is usually essential for pricing derivative securities using no arbitrage arguments. The framework requires only that volatility structures be specified. However, especially in
9. Please refer to Appendix 1 for the mathematical expression of the interest rate risk defined in this way.
10. Please refer to Appendix 2 for other typical multi-factor models such as the Brennan-Schwartz model or the
Cox-Ingersoll-Ross model.
2. The HJM model in this paper
We adopt the two-factor HJM model in order to describe the fluctuation of market rates because (1) the HJM model is superior to other multi-factor models in terms of completeness; (2) there is some degree of freedom about the identification of a volatility function, so that we can set up the model flexibly for specific purposes; and (3) empirical analyses such as the “principal component analysis” of interest rates show that two or three factors can explain more than 90 percent of all fluctuations.11
There exist various views about what kind of volatility functions we should take into account. In this paper, after considering the result of empirical analysis, we comply with Heath, Jarrow, and Morton (1992) and Ritchken and Sankarasubramanian (1995). We assume the following function for the volatility term sf i (t ,T ) of equation (1).
sf 1(t ,T ) =s1e –k(T–t ), sf 2(t ,T ) =s2. (s1,s2, k: constant parameters) (3)
The former term implies that the volatility of forward rates decreases exponentially with time, consistent with a change in yield curve slope. The latter term influences the volatility of every forward rate equally, in both the near and the distant future, describing parallel shifts.
In our simulation, we do not directly model the forward rates but model P (t , T ),
the price of a zero-coupon bond at time t with maturityT, as follows.12
é ¶logP (t ,T )ù 2 dP (t ,T ) = – ————— P(t ,T )dt +åspi(t ,T )P (t ,T )dWi (t ). (4) ë ¶T ûT=t i=1
The volatility terms for the zero-coupon bond spi (t ,T ) are derived as s1 sp1(t ,T ) = – —[1 – e–k(T–t )], sp2(t ,T ) = –s2(T – t ). (5) k
C. Stochastic Process Describing Administered Rates
In order to calculate the interest rate risk of banking accounts, it is necessary to discuss how to model not only market rates but also administered rates such as the short-term prime rate and the long-term prime rate. It is true that we have said less about modeling administered rates than about modeling the market rate. Our reasons are that (1) administered rates are used only in limited cases; and (2) the administered rates are to some extent linked to market rates, so we have not considered them as independent risk factors. However, as we discuss below, it is possible
that risks due to changes in administered rates are bigger than market risk. If so, we cannot neglect this source of risk.
11. For example, please refer to Litterman and Scheinkman (1991).
12. Please refer to Heath, Jarrow, and Morton (1992), which shows how to transform the forward rate stochastic
process described by equation (1) to the zero-coupon bond stochastic process described by equation (4).
1. Short-term prime rate
The short-term prime rate is an average of the funding rates of liquid deposits or from the interbank market or the open market, and the expense rate. However, it does not always respond immediately to changes in market rates. In this paper, we construct a model of the procedure for revising the short-term prime rate, as follows.13
• We generate the three-month interest rate in accordance with the market rate model (the HJM model).
• When the three-month interest rate changes by more than 0.25 percent since the last time the short-term prime rate was revised, we decide to revise the short-term prime rate.
• However, the revision does not take place immediately. It is revised after some time lag.
• The change in the short-term prime rate is equal to the difference between the three-month interest rate at the revision time (i.e., after the time lag) and the three-month interest rate at the previous revision time, and is always a multiple of 0.125 percent.
If we make such a model, we should consider what risk premium should be paid for the new risk factor (uncertainty due to the randomness of the time lag) introduced into the administered rate.
2. Long-term prime rate
The coupon rate of financial debentures is revised when the difference between the current coupon rate and the yield of financial debentures in the secondary market becomes more than 0.2 percent. The minimum unit of revision is 0.1 percent. The coupon rate is checked every month in this way. The long-term prime rate is determined as the coupon rate plus 0.9 percent. In this paper, we model this as follows.
• We assume that the spread between an index market rate (five-year swap rate) and the secondary market yield of a financial debenture obeys a normal distribution.We estimate its average and variance from historical data.
• We generate the five-year market rate in accordance with the stochastic model of market interest rates (i.e., the HJM model).
• We calculate the secondary market yield of financial debentures by adding a spread (which obeys a normal distribution, generated as above) to the five-year market rate.
• When the difference between the calculated secondary market yield and the current coupon rate of the financial debenture (the current long-term prime rate minus 0.9 percent) becomes more than 0.2 percent, we revise the coupon rate by a minimum unit of 0.1 percent.
• When the coupon rate is revised, we revise the long-term prime rate to the new coupon rate plus 0.9 percent.
13. We can devise another model of the short-term prime rate as follows. First, we divide the level taken by a market
interest rate into several ranges. Each range has a corresponding short-term prime rate. The time lag between the
time at which the market rate moves from one range to another and the time at which the prime rate actually
changes to the corresponding level follows a probability distribution—for example, an exponential distribution.
(Please refer to Appendix 3.)
D. Evaluation of Prepayment Value
In the field of mortgage securities, there has been discussion of how to make a model of prepayment that includes option value. On the other hand, because there are few publicly available data about prepayment behavior in time deposits, the discussion has been less active than in the case of mortgage securities. In this paper, we review the research on mortgage securities, and develop a prepayment model for banking accounts. The candidates for consideration are rational models, prepayment function models, discrete models, etc.14
In the case of banking accounts, we should consider prepayments of loans, especially housing loans, and time deposits. We can directly adapt a model of mortgage securities to housing loans. In addition, except for the difference between assets and liabilities and the problem of parameters, we can apply the analysis of mortgage securities to time deposits. For example, the rational prepayment model assumes that investors compare current applied interest rates with interest rates on other products, take account of future changes of interest rates before maturity, and
exchange their assets or liabilities for other, more favorable products. In this paper, for compatibility with the
1. Overview of prepayment function model
In the prepayment function model, we assume a function that expresses the prepayment and statistically estimate parameters. In this kind of model, we assume that when mortgagors think about prepayment, they take account of interest rates and the economic situation, but only that of the present and the past, not of the future, as is assumed in the rational prepayment model. Therefore, we can use this model for estimating value in a
For example, Schwartz and Torous (1989) suggest the following function model.
Let v be a vector of explanatory variables that affect prepayment, and b be a vector of coefficients, then the prepayment function p (t ) can be described as
p (t ) = p0(t ) exp (b'v), (6)
where p0(t ) is the baseline function that expresses the prepayment rate, independent of the change in interest rate environment, and the prime symbol means the transpose of a vector. We assume the following log-logistic function for the baseline function.
gp(gt )p–1
p0(t ) = ————. (7)
1 + (gt )p
As one of the explanatory variables, for example, we use the difference between the current applied interest rate and the interest rate of other candidate products. In addition, we can use a dummy variable in order to express the seasonal fluctuation.
14. Please refer to Appendix 4 for the details of each model.
Finally, because the attributes of investors may significantly affect the function’s form and its parameters, we can make a separate model for each attribute group, just as we can do in the rational prepayment model.
2. Prepayment function model in this paper
Under the prepayment function model in this paper, following Schwartz and Torous (1989), we use the following as explanatory variables.
• The difference between interest rates
(the interest rate that has already been applied to the product) minus (the interest rate that will be applied to it if we make a new contract for it).
• The third power of the above difference
This represents the fact that the larger the difference becomes, the faster prepayment activity accelerates.
• The ratio of the remaining amount (the remaining amount when we take account of prepayment) divided by (the remaining amount when we assume no prepayment).
However, because we have no data to help us evaluate these parameters, we set the value of the parameters to several levels and estimate the magnitude of the effect of prepayment.
When examining products that include the option to prepay, as we discussed in Section II.B.4, we should strictly calculate their value (including the time value of the option) by a Monte Carlo or lattice method.15 However, if we attempt to value the option at the same time as we simulate the fluctuation of portfolio value with a Monte Carlo method, the calculation load becomes too heavy. In this paper, we do not carry out a so-called “simulation on the simulation.” We first generate yield curves for each time.
Then, the future prepayment schedule is determined based on the forward rates implied by the yield curve and the assumed prepayment function model. Finally, we discount the fixed future cash flows to obtain the PV of the product.
IV. Detail of the Method and Result of the Simulation
In this section, taking account of the above theoretical considerations, we carry out a simulation using an imaginary portfolio, and explore how the features of banking accounts actually affect the PV and risk due to interest rates.
A. Composition of an Imaginary Portfolio
In this subsection, we explain the content of the basic form of the imaginary portfolio that we used in our simulation (see Appendix Table 2). The numbers in the table show the share of each asset and liability in total assets and liabilities. When we made the imaginary portfolio, we referred to the balance sheets of several city banks as of March 1995. We assumed that both total assets and total liabilities were ¥30 trillion. In the actual simulation, we varied the composition of the portfolio so that we could explore how PV and risk were affected.
15. If we adopt the rational model, we must use the lattice method, just as we do when we calculate prices of
American-type options.
First, the portfolio has products divided into three categories determined by the method for setting the interest rate (linking the products to the market rate, the short-term prime rate, or the long-term prime rate). The products linked to the market rate are further divided into two categories: one has a constant spread to the market rate (perfect linkage), the other has a weaker correlation with the market rate (imperfect linkage). There are four categories in total. Next, in order to specify which interest rate is applied to the initial term and to each term after rollovers, we assigned the composition ratio of interest rate renewal time-span. In such cases, we should classify not on the basis of the product’s remaining term but of the term of the applied rate.
In addition, maturity information is also important. For example, we compare a three-year floating rate loan with interest rate reset every six months to a six-month fixed rate loan. After six months have passed, the interest rate on the floating rate note should be reset. However, after the same six months, at maturity of the fixed rate loan, there is a possibility that the product is replaced by other products. In Appendix Table 2, we classify the term of a product according to the term of the applied interest rate. However, when the term of the applied interest rate and final
maturity differ from each other (i.e., in the case that interest payments are made before maturity), we show the final maturity in the remarks column. In addition, to be strictly accurate, the PVs of the differences between the three kinds of index interest rates described above and the actual applied interest rates fluctuate as the yield curves change. However, for simplicity, we assume no spread irrespective of the market rate, the short-term prime rate, and the long-term prime rate. We also neglect products in which the customer can choose which index rate to use.
B. Simulation Method
1. Setting the yield curve span and the length of holding period
In this paper, we generate yield curves during the holding period by means of the HJM model, with the initial yield curve derived from the
2. Assumption about position change
First, among financial products that have no explicit maturity, there are products with no interest rates, such as current deposits and cash. If we assume that the balances of these products do not change in the future, which means that there is
16. It is a provisional assumption that the holding period is not longer than three years, so that the assumption is not
based on the specific theory.
17. Because we recognize the PV of a portfolio as the value of the discounted cash flows that arise during the
“limited” yield curve span (seven years in this paper), the PV is not exact. Moreover, in this simulation, we
assume the products are rolled over only in the case that the renewed maturity after rollover is not beyond seven
years from the current grid point. Regarding the products without specific maturity—for example, a liquid
deposit—we always evaluate just the cash flows occurring within seven years.
no fluctuation of PV (that is, risk) associated with interest rate fluctuations, we can
dispense with these products in this simulation. Second, as for financial products
with an interest rate, such as liquid deposits (partially linked to the market rate) and
overdrafts (linked to the short-term prime rate), we assume for convenience that the
interest rate renewal term is one month and that the balance does not change.
A rollover ratio of 100 percent means that the amount that comes to maturity is
fully rolled over to the same product. We vary the rollover ratio up to 100 percent
and observe the effect on the PV and risk.
As for prepayment, if it occurs along the axis of the yield curve term, we have to
invest or fund with the simulated market rate applying immediately afterward. For
convenience, prepayment does not occur along the axis of the holding period.
3. Configuration of parameters
a) Parameters of the HJM model
As for the standard parameters of the HJM model, we assumed that the parameters
s1 and k, which determine the volatility term relating to the difference between the
long-term and short-term interest rates, were 4.56 ´ 10–5 and 6.32 ´ 10–2, and that
the volatility term for parallel shifts, s2, which affects the whole term until maturity,
was 4.04 ´ 10–5.18
b) Parameters that prescribe the stochastic model of the prime rate
1) Short-term prime rate
We assume that the time lag between the fluctuation of the short-term prime rate
and the fluctuation of the market rate obeys an exponential distribution.19 When we
estimated the parameter l of the exponential distribution, we used the data of the
end day of every month from January 1989 (when the “new short-term prime rate”
was introduced) to February 1996. For each of the 23 times of short-term prime rate
renovation, we specified how many months of time lag the short-term prime rate
was revised afterward (see the table below). We can calculate the average time lag of
1.065 with this histogram and obtain the parameter l of 0.939, which is equal to the
reciprocal of the expected value of the exponential distribution.
Time lag Frequency
Zero months 14
One month 6
Two months 2
Three months 1
Total 23
18. We used the parameters estimated from weekly data for government bonds from April 1988 to July 1995. We are
grateful to Katsuya Komoribayashi of the Dai-Ichi Mutual Life Insurance Company, who gave us much
assistance with this estimation.
19. The assumption that the random variable X which represents the time lag follows an exponential distribution
means that the probability density function of X [f (x)] can be described as follows with parameter l.
f (x) = l exp (–lx), for x ³ 0.
2) Long-term prime rate
The secondary market rate of financial debentures consists of the five-year market rate
(five-year swap rate) and the spread, which obeys a normal distribution with mean m and
standard deviation s. In order to determine m and s, we used the data of the spread
between the secondary market rate of financial debentures and the five-year swap rate
from October 1987 to February 1996, so that we acquired m of –0.360 and s of 0.161.
c) Parameters of prepayment function
We adopted in equation (6) as explanatory variables the difference between interest
rates, the third power of the difference, and the ratio of the remaining amount.
Because we have no data to estimate the parameters b1, b2, and b3, we quote the
result of
b1 = 0.39678, b2 = 0.00356, b3 = 3.74351. (8)
Parameters b1 and b2 show how much the prepayment rate fluctuates relative to the
baseline function as the interest rate difference fluctuates. In the above case, if there is
1 percent of interest rate difference, the prepayment rate increases 1.5 times as before.
Parameter b3 shows that as the ratio of remaining amount decreases, prepayment
becomes unlikely to occur. For example, if 10 percent of the balance has already been
prepaid, the prepayment ratio decreases as much as 30 percent compared to the initial
level. We set up the parameters of baseline functions according to the maturity.20
C. Simulation Result
1. Trial calculation of yield curve and risk amount
Here, we confirm the yield curve form of the market rate and the administered rate
used in this simulation. For example, the yield curve of the forward rate that we use
in order to evaluate the PV of the initial time (PV0) is shown in Figure 2. Although
20. We set up the value of parameters p and g as the following table. With these parameters, the simulation shows the
unrealistic result that almost all the product should be prepaid before maturity. Therefore, we introduce another
parameter, a, so that we use p ~ (t ) = ap(t ) instead of p as our prepayment function.
0
1
2
3
4
5
6
Percent
0 6 12 18 24 30 36 42 48 54 60
Long-term prime rate
66 72 78
Short-term prime rate
Five-year market rate
Three-month market rate
Months
Maturity One year Three years Five years Seven years
p 3 3 3 3
g 0.30 0.10 0.05 0.03
a 0.05 0.10 0.15 0.20
Figure 2 Implied Market Interest Rates on the Initial Time (End of 1995) and Future
Paths of Administered Rates
in Section II.B.4 we described that it is necessary to carry out a “simulation on the
simulation” in order to calculate the precise PV, we do not carry it out because it
requires a huge calculation load. Therefore, we regard the forward interest rate
implied by the yield curves at each time that is generated along the direction of the
holding period as the future path (along the construction term of the yield curve)
of the market rate. Similarly, we generate the future path of administered rate by
including factors such as the time lag.
Next, we confirm what kind of yield curves are formed at the end of the holding
period (three years later) if we generate many paths along the direction of the holding
period. Figure 3 shows the result when we generated 10 paths.21
Figure 4 Distribution of minPV (Basic Portfolio, Perfect Rollover, No Prepayment)
0
1
2
3
4
5
8
7
6
Percent
1.7 1.8 1.9
99 percentile
¥ trillions
Risk amount
2.0 2.1
PV0
Figure 3 Yield Curve Movement Three Years Later (10 Paths)
3 9 15 21 27 33 39 45 51 57 63 69 75 81
Months
2.0
3.0
4.0
5.0
1.5
2.5
3.5
4.5
Percent
21. Figure 3 shows a yield curve derived from the HJM model with the initial parameters. In this figure, we cannot
clearly recognize the transition of the yield curve slope. However, changing the parameters of the HJM model, we
can generate various shapes of yield curves as shown in Appendix 5.
In our simulation, we distribute the minimum PVs (minPVs) of 36 timings on
each path, and define the risk amount as the difference between the 99 percentile
point and the PV0. Figure 4 shows how the minPVs are distributed. We carried out
5,000 simulations, using a basic portfolio and assuming perfect rollover, and drew
a histogram of the result. The distribution has an almost smooth shape with the
limitation on the right-hand side, PV0.
2. The effect on the PV of individual products and risk according to the features
of banking accounts
First, we carry out a simulation with the individual products in order to capture the
effect of administered rates and prepayment, which we consider to be the representative
risk factors of banking accounts. In the simulations below, we always carry out
500 simulations (500 paths). Because one simulation generates 36 PVs, in fact, we
calculate the risk amount based on 18,000 PVs.
a) The effect of the administered rate on the risk amount
We assume the three kinds of interest rates (the market rate, the short-term prime
rate, and the long-term prime rate) to be the loan rate, and verify the effect that the
administered rate has on the risk amount (Table 1). The object product is ¥5 trillion
of a six-month loan. We assume that the balance will be constant and will be rolled
over for seven years.22
The risk of short-term prime rate-linked and long-term prime rate-linked
products is larger than that of market rate-linked products. Because the transition of
the short-term prime rate and the long-term prime rate is a kind of stepwise
function, it does not link perfectly to the transition of the market rate so that it leads
to an increase in risk. In addition, as a factor specific to the short-term prime rate, it
has the effect on the risk that the short-term prime rate is revised with some time lag
as the market rate fluctuates. On the other hand, as for the long-term prime ratelinked
products, because the basis between the secondary market rate of financial
debentures and the market rate (five-year swap rate) fluctuates, the risk is larger than
that of market rate-linked products. In addition, in our simulation we discount the
interest cash flow of the long-term prime rate, which is a kind of five-year interest
rate, with the six-month interest rate. In this case, the initial yield curve becomes
upward-sloping. Therefore, the spread effect diminishes as time passes so that the PV
decreases. For this reason, it seems that the risk of a three-year holding period is
much larger than that of a one-year holding period.
Table 1 Effect of Administered Rates (Risk of Individual Products)
¥ billions
Market rate-linked Short-term prime rate-linked Long-term prime rate-linked
One-year holding 1.3 88.5 160.5 period
Three-year holding 1.4 97.0 371.3 period
22. In the following section, we show the risk amount of just a few kinds of holding periods because of the limited
space of this paper. However, actually, we calculated the risk amounts of holding periods up to three years with
one-month intervals, i.e., 36 holding periods (please refer to Appendix Table 3).
b) The effect of prepayment on the PV
With a five-year housing loan (¥1.5 trillion, market rate-linked [2.8 percent fixed]),
we verify the effect that prepayment has on the PV0 and on risk (Table 2). As Table 2
shows, when we compare the prepayment by baseline function (independent of
the interest rate difference)23 with the one including the customers’ action based on
the interest rate difference, we can verify that the PV decreases (¥8.5 billion Þ
¥4.7 billion) because of the option (etc.) held by customers.
3. The effect that the features of banking accounts have on the PV of the
portfolio and the risk amount
In this subsection, we proceed further, calculating the PV0 and risk of a portfolio
that consists of various individual products.
a) The effect of the administered rate on the risk amount
In order to capture clearly the effect of the administered rate, just for this simulation
we increase the linkage ratio to the market rate of liquid deposits in the basic portfolio
from 20 percent to 100 percent. In this case, the risk of the basic portfolio is
¥666.6 billion under the conditions of no prepayment and a three-year holding
period (Table 3). In addition, by exchanging all the administered rate-linked
products into market rate-linked products (see Appendix Table 4), the risk decreases
greatly, so that the risk of a six-month holding period is ¥24.7 billion and that of a
three-year holding period is ¥133.6 billion. Therefore, we can say that administered
rate-linked products have much influence on the risk not only of individual products
but also of portfolios.
Table 3 Effect of the Products Linked to Administered Rate
¥ billions
Portfolio Basic portfolio Market rate-linked products only
Six-month holding period 338.0 24.7
Three-year holding period 666.6 133.6
b) The effect of prepayment
When we take account of prepayment in the basic portfolio only by baseline function,
the PV0 is ¥2,215.7 billion (Table 4). Moreover, if we include the prepayment based
on the interest rate difference, the PV0 decreases by as much as ¥5.8 billion to
¥2,209.9 billion, and the risk of a three-year holding period is ¥326.1 billion. It is true
23. In this case, the prepayment by the baseline function is a less-favored form of prepayment for customers because of
the normal slope yield curve (i.e., the longer-term interest rate is larger than the shorter-term interest rate). Therefore,
the PV with prepayment of the baseline function is larger than that without prepayment (0 Þ ¥8.5 billion).
Table 2 Effect of Prepayment (Five-Year Housing Loan)
¥ billions
Prepayment
Baseline only Plus interest All factors None rate factor
PV0 8.5 4.7 3.4 0
One-year holding period risk 43.3 46.1 46.0 47.5
Three-year holding period risk 64.1 65.7 64.7 64.5
that the level of risk itself decreases compared with that without the factor of interest rate
difference (¥359.1 billion). The more a position is reduced by prepayment, the less the
probability of PV fluctuation becomes. In this sense, prepayment has the effect of
decreasing the risk. On the other hand, the possibility of prepayment immediately changes
the PV. Consequently, prepayment has the effect of increasing the risk. It seems that the
former effect surpasses the latter effect, so that the risk decreases in this case.
24. In order to analyze the difference of
the yield curve, we should compare the case with the multivariate lognormal distribution process of yield-curve
fluctuation and the case with the HJM model. However, in this paper, we compare the HJM model with the
variance-covariance method, which is generally the most popular.
5. The effect of model parameters on risk amount
a) Parameters of the HJM model
Here, we study the sensitivity of risk to the parameters of the HJM model in this
paper. Using the basic portfolio, we increase the amount of s1, k, and s2 as much as
five times so that we can compare the resulting risk with that before the change is
made. In order to facilitate comparison, we assume no prepayment and assume
perfect rollover. While s1 affects mainly the short-term interest rate, s2 affects
interest rates of all terms (Table 6). In other words, because s1 and s2 express the
fluctuation of the yield curve slope and the fluctuation of parallel shift for each, as
the amount of these increases, the risk increases. On the other hand, k expresses the
extent to which the influence of s1 on the far-future forward rate is smaller than that
Table 5 Comparison with the Variance-Covariance Method
¥ billions
Method Variance-covariance HJM HJM
Rollover — None Perfect
Prepayment None None None
One-month holding period 29.1 3.7 3.7
Six-month holding period 72.6 24.7 24.7
Three-year holding period 178.0 120.9 133.6
4. Comparison with the usual VaR
In the most popular VaR calculation method of trading account so far, we set up
several term interest rates as risk factors, assume the multivariate lognormal process
among them, and then calculate VaR using a variance-covariance matrix. Under this
method, as the holding period t become longer, the risk simply increases Ö
t times.
On the other hand, under the HJM method adopted in this simulation, we actually
generate the fluctuation of the yield curve for three years. Comparing the risk from
both methods,24 using the portfolio that consists of market rate-linked products
(see Appendix Table 4), we obtained the result shown in Table 5.
Table 4 Effect of Prepayment (Three-Year Holding Period)
¥ billions
PV0 (baseline) PV0 (plus interest rate factor) Risk amount
2,215.7 2,209.9 326.1
V. Conclusion: Implication for Risk Management
In this paper, we have tried to suggest the basic framework with which we calculate
the interest rate risk of banking accounts. By using this framework to calculate the
risk of an imaginary portfolio, we could capture an approximation of the interest rate
risk of banking accounts, and the relation between the administered rate and the risk
amount or prepayment and the risk amount, etc., which are features of banking
accounts. It is true that the simulation result in this paper is no more than a calculation
example that is derived under various assumptions. However, the result proves
that the interest rate risk of a banking account is affected by various factors.
It is true that the models of market rates, prepayment, and administered rates in
this paper are no more than examples. There can be various types of models in the
case of risk management of actual financial institutions. In addition, in this paper,
we have made many assumptions in order to simplify the simulation as much as
possible. Nevertheless, we faced the problem that the calculation workload becomes
too heavy after the process of including the various features of banking accounts.
In actual banking operations, we should take account of the trade-off between the
accuracy of risk measurement and the calculation workload in order to choose the
most suitable model. When doing this, we should first carefully consider each building
block such as the administered rate factor or the prepayment factor. Then, we
should continue to simplify each building block.
Table 7 Effect of Parameter Change of Prepayment on PV
¥ billions
b1 = 0 b1 = 0.39678 b1 = 0.39678 ´ 2
b2 = 0 b2 = 0.00356 b2 = 0.00356 ´ 2
PV0 2,215.8 2,209.9 2,200.6
b) Parameters of prepayment
When we take account of interest rate differences as described above, prepayment
decreases the PV0, which is itself the basis of risk. Then, using a basic portfolio
assuming perfect rollover, we increase the parameters b1 and b2 twice so that we get
the result that the PV0 decreases by a further ¥9.3 billion, as Table 7 shows. (In this
comparison, we take account not of the ratio of the remaining amount but of the
baseline function.)
Table 6 Effect of Parameter Change of the HJM Model
¥ billions
Standard s1 k s2
Six-month holding period 256.0 332.4 255.9 351.7
One-year holding period 326.9 352.3 303.7 459.7
Three-year holding period 359.1 409.3 342.7 650.5
on the near-future forward rate. Therefore, as the k increases, the far-future forward
rate tends not to fluctuate, so that the risk decreases.
We neglect the following points from the standpoint of simplification in this
model: (1) measuring the risk of non-maturity products; (2) including the interest
rate spread and interest rate cash flow when we calculate (the discounted) cash flow;
and (3) integrated management of the interest rate risk of banking accounts together
with that of trading accounts or credit risk. These are important problems that
we should study from now on. In addition, in order to estimate the prepayment
function realistically, it is essential to include data on prepayments or cancellations.
It is very important to arrange the transaction data from the viewpoint of actual
risk management business. We hope that more theoretical and quantitative studies
are cultivated in this field, in which little work has been conducted in
APPENDIX 2: MODELING OF MARKET RATES
A. The Brennan-Schwartz Model
The model of Brennan and Schwartz (1979) is a representative two-factor model.
The model assumes stochastic processes for two interest rates: the instantaneous spot
rate and the long-term bond rate (consol rate). Because these two factors are directly
connected with short-term and long-term interest rates, it is easy to understand
the model intuitively. However, when it comes to pricing interest rate derivative
securities, the model can only derive a partial differential equation that should be
satisfied given a market price of risk. Therefore, we must solve this partial differential
equation numerically in order to calculate a price.
Let [r (t ), t ³ 0] be the instantaneous spot rate, [l (t ), t ³ 0] be the interest rate
of a long-term bond (consol bond), and [W1(t ), t ³ 0], [W2(t ), t ³ 0] be two standard
APPENDICES
In these appendices, we show (1) measure of risk; (2) major multi-factor termstructure
models; (3) modeling of administered rates; and (4) mathematical
expression of prepayment models.
APPENDIX 1: MEASURE OF RISK
Let [ f (s , u), 0 ² s ² u] be the instantaneous forward rate at future time u evaluated at
time s . And let CFi [s, u, f (s , u)] be the future cash flow at time u foreseen at time s
of each asset or liability i. We set the risk evaluation period as [t, t]. We consider
sample paths (w) of change of the term structure of interest rates {f (s , u), s Î[t , t],
u Î[s, s +T ]} and let R (t, t, w) be the set of term structures on the paths {f (s , u),
s Î[t , t]}. Then, given the term structure of interest rate f (s , u)ÎR (t, t, w) at each
time s,25 PVf (s , T ), the PV of the banking account evaluated in the period
u Î[s, s +T ], can be obtained as follows with the information available at time s F
s ,
PVf (s ,T ) = åE˜{ºs
s+T
exp [–ºs
u
r (u)du]CFi [s, u, f (s, u)]du| F
s }, (A.1)
t ² s ² u ² s +T, s Î[t , t],
r (t ) = f (t, t ),
where E˜ [•] means the expected value under the risk-neutral probability measure (the
equivalent Martingale measure). Then, we define VaRa (t ) as
Pr{ min [PVf (s ,T ) – PV0(t ,T )]² – VaRa (t )} = a, (A.2)
f ÎR(t, t, w)
the potential loss, with probability a, due to changes in the whole term structure of
interest rates at the time t .
25. As for products with fixed cash flow, we can calculate the discounted value using this term structure of interest
rates. On the other hand, as for products with option value, we calculate the expected value by generating some
paths on the basis of this initial term structure.
Wiener processes defined on some probability space (½, F, P ). The Brennan and
Schwartz (1979) model assumes that the interest rates follow the processes below.
dr (t ) = b1(r, l, t )dt + h1(r, l, t )dW1(t ),
dl (t ) = b2(r, l, t )dt + h2(r, l, t )dW2(t ), (A.3)
dW1(t )dW2(t ) = rdt,
where bi(• ), hi(• ), i = 1, 2 represent functions that depend on the short-term interest
rate, the long-term interest rate, and time. r is a constant number that represents the
correlation between the two Wiener processes. Under the assumption that there is an
active market for consol bonds, a discussion about non-arbitrage conditions determines
that the market price of risk for long-term interest rate [l2(r, l, t ), t ³ 0] must
satisfy the following.
l2(r, l, t ) = – h2(r, l, t )/l + [b2(r, l, t ) – l 2+ rl ]/h2(r, l, t ). (A.4)
In addition, if we assume
h1(r, l, t ) = rs1,
h2(r, l, t ) = ls2, (A.5)
l 1 b1(r, l, t ) = r [aln(—) + ––s1
2], pr 2
given the market price of risk for short-term interest rate [l1(r, l, t ), t ³ 0], the price
of a discount bond B (r, l, t ) satisfies the following partial differential equation.
1 1 1 —Brrr 2s1
2 +Brlrlrs1s2+ —Bll 2s2
2 + Brr [aln(l /pr)+ —s2
2 – l1s1]
2 2 2
+Bll (s2
2 + l – r ) – Bt – Br = 0, (A.6)
B(r, l , 0) = 1,
where Bx represents the first partial derivative of B (r, l, t ) with respect to x and Bxx
represents the second partial derivative.
B. The Multi-Factor CIR Model
The multi-factor model within the framework of Cox, Ingersoll, and Ross (1985) is
also well known. In this model, we assume that we can describe the instantaneous
spot rate [r (t ), t ³ 0], as the sum of k state variables [y j (t ), t ³ 0, i = 1, . . . , K ] and
that each state variable fluctuates according to a square root process with mean
reversion property. In other words, we can describe [r (t ), t ³ 0] as follows,
K
r (t ) = åy j (t ), (A.7)
j=1
and we can assume that the stochastic process for each state variable follows the
square root process below.
dyj (t ) = kj[qj – y j (t )]dt + sjÃy j (t )dWj (t ), j = 1, . . . , K, (A.8)
where each [Wj (t ), t ³ 0] is an independent standard Wiener process. It is known that
in this model, we can calculate the price of a discount bond, P (t , t), at time t with
maturity t as a natural extension of the pricing formula of the one-factor CIR model.
K K
P (t, t) = ÕAj(t, t) exp [–åBj(t, t)y j (t )], (A.9)
j =1 j =1
é2gj exp [1/2(kj + lj + gj)(t – t )]ù
Aj(t, t) = ç—————————————ç , (A.10)
ë 2gj + (kj + lj + gj)(e g j(t – t) – 1) û
2(e gj (t – t ) – 1) Bj(t, t) = ————————————, (A.11)
2gj + (kj + lj + gj)(e g j(t – t) – 1)
gj = Ã(kj + lj)2 + 2sj
2 , (A.12)
where lj y j is the risk premium for each state variable, and lj is constant.
Longstaff and Schwartz (1992) paid attention to the point that, within the twofactor
CIR framework, we can describe both the instantaneous spot rate [r (t ), t ³ 0]
and its volatility [V(t ), t ³ 0] by linear equations in the same state variables. They
suggested estimation of the parameters with a generalized autoregressive conditional
heteroskedasticity (GARCH) model. Formally, when state variables are assumed to
satisfy the following partial differential equation,
dyj (t ) = kj[qj – y j (t )]dt + sjÃy j (t )dWj (t ), j = 1, 2, (A.13)
we can describe the instantaneous spot rate [r (t ), t ³ 0] and its volatility [V(t ), t ³ 0]
as follows:
r (t )= ay1(t ) + by2(t ),
(A.14)
V 2(t ) = a2y1(t ) + b2y2(t ).
Alternatively, Chen and Scott (1995) suggest, assuming K is equal to two or three,
the estimation of parameters with a Kalman filter, and discuss its features. They
confirm that there are two principal variables which are estimated from actual
interest rate changes: (1) a factor that affects the whole term structure of interest rates
equally; and (2) a factor that mainly affects the short-term interest rate and to a lesser
extent affects the long-term interest rate.
2kjqj ——— sj
2
The disadvantage of this model is that the state variables are not directly observable
and are hard to interpret, so an intuitively clear picture of the changes in interest
rates is not available. Moreover, in practice, it is difficult to estimate the parameters.
C. The HJM Model
We assume the following process describing the instantaneous forward rate [ f (t ,T ),
0 ² t ²T ] at future time T viewed from time t ,
N
df (t , T ) = a(t , T )dt + åsn(t , T )dWn(t ), 0 ² t ²T, (A.15)
n=1
where each [Wn(t ), t ³ 0] is an independent standard Wiener process, [a(t , T ),
0 ² t ²T ] and [sn(t ,T ), 0 ² t ²T ], n = 1, . . . , N are a drift term and a volatility
term that satisfy some regularity conditions. Now, letting bn(t ), n = 1, . . . , N be the
market prices of risk, then under the equivalent Martingale measure,
N
df (t , T ) = a(t , T )dt + åsn(t , T )dW
~
n (t )
(A.16) N
n=1
+ åbn(t )sn(t , T )dt , t Î[0, T ].
n=1
We can derive the following relation based on a consideration of the absence of
arbitrage:
N
a(t , T ) = – åsn(t , T )[b(t ) – ºt
Tsn(t , v)dv], 0 ² t ²T. (A.17)
i=1
Using this equation, we can obtain a formula for bond prices that does not
include the market price of risk but contains volatility functions under the equivalent
Martingale measure. We calculate the discount bond price P (t ,T ) on the equivalent
Martingale measure as follows:
P (t , T ) = exp[–ºt
Tf (t , s )ds ]. (A.18)
Therefore, we can describe the process by which the discount bond price fluctuates
as follows:
N
dP(t , T ) = r (t )P (t , T )dt –å[ ºt
Tsn(t , s )ds ]P (t , T )dW
~
n (t ), (A.19)
n=1
r (t) = f (t , t ).
APPENDIX 3: MODELING ADMINISTERED INTEREST RATES
A. The Model That Fixes the Level of Administered Rate Corresponding to the
Let [rs (t ), t ³ 0] be the short-term prime rate, and [r (t, t), t > t ³ 0] be the market
rate (actually, the three-month CD rate) at time t with fixed maturity of t,
26. In our simulation, we do not take account of this time lag in the movement of the long-term prime rate.
r 1s
, 0 ² r (t, t ) ² r 1,
rs(t + Æt) = {r i
s, r i–1 ² r (t, t ) ² r i, (A.20)
r s
m, r m–1 ² r (t, t ),
where Æt is the time lag since the time when r (t ) crossed a rate boundary most
recently and obeys the exponential distribution EXP(l).
B. The Model That Describes the Revision of the Short-Term Prime Rate
Assume
r3CD(t ): three-month CD rate at time t ,
t¯ : the time at which the short-term prime rate was last revised,
t¯ : the time, > t¯ , at which the condition abs [r3CD(t ) – r3CD(t¯ )] ³ 0.25 first becomes
true,
abs(x ): the absolute value of x.
Then,
ér3CD(t¯¯ + Æt ) – r3CD(t¯ ) ù rs(t¯ + Æt ) = rs(t¯ ) + int ———————— + 0.5 ´ 0.125, (A.21) ë 0.125 û
Æt : a random variable that represents a time lag with exponential distribution
EXP(l),
int[X ]: the largest integer that does not exceed X.
C. Modeling of the Long-Term Prime Rate
Assume
rl(t ): the long-term prime rate at time t ,
rc(t ): the coupon rate of a five-year financial debenture at time t ,
r¯c (t ): the secondary market rate of a five-year financial debenture at time t ,
t¯ : the time at which the condition abs [r¯c (t ) – rc(t )] ³ 0.20 first becomes true.
Then,
ér¯c (t¯ + Æt ) – rc(t¯ + Æt) ù rc(t¯ + Æt ) = rc(t¯ ) + int ———————— + 0.5 ´ 0.1, (A.22) ë 0.1 û
Æt : a random variable that obeys EXP(l)26,
int[X ]: the largest integer that does not exceed X.
Here, we assume that the secondary market rate of a debenture can be obtained
by adding some spread to the market interest rate of the same maturity (i.e., a
five-year swap rate),
r¯c (t ) = r (t) +rspread(t ), (A.23)
rspread(t )~N(m, s2), the normal distribution with mean m and variance s2.
... ...
APPENDIX 4: PREPAYMENT MODELS
A. Rational Models
We can trace back the research that tries to explain the prepayment behavior of a
mortgagor in terms of rational decision-making at least to Dunn and McConnell
(1981). This class of models assumes that the mortgagor takes account of the value of
the remaining liabilities along the path of future interest rates and prepays only when
this value exceeds the sum of the remaining principal amount and the refinancing
cost. Johnston and Van Drunen (1988),
Singh (1994) represent this class of model.
“Rational” does not always mean that everyone acts according to a rational standard
without exception. Some models assume a time lag (i.e., see Dunn and Spatt
[1986]). Some models divide the mortgagors into several groups according to the
level of refinancing cost at prepayment in order to reflect borrowers’ attitudes to the
fluctuation of interest rates. Other models also take into account prepayments that
are generated independently of economic conditions (
In this kind of model, we have to compare the values mentioned above during the
construction of the yield curve, working backward from maturity, just as we do when
we evaluate an American-type option. It is difficult for the
in this paper to operate this kind of calculation, which is better suited to a lattice
method.
McConnell and Singh (1994) first calculate with a lattice method the interest rate
boundary at which prepayment happens. Then, they suggest a model that accelerates
the prepayment at the boundary rate during
be the value of the mortgagor’s liabilities, and assume the refinancing cost to be equal
to F (t ), the remaining principal at prepayment times the constant ratio RF. In this
situation, the mortgagor always compares V [r (t ), t ] and (1 + RF )F (t ). He prepays
just after the value of his liabilities exceeds the sum of the remaining principal and
the refinancing cost. Under a model with continuous evaluation, the interest rate
boundary for prepayment rc(t ) satisfies
V [rc(t ), t ] = (1+RF )F (t ). (A.24)
In addition, it is not a realistic assumption that all the borrowers decide to prepay
simultaneously. So, using a time lag a, we assume that the prepayment ratio p(t )
changes according to
p0(t ), r (t ) > rc(t ),
p(t ) = { (A.25)
p0(t ) + a, r (t ) < rc(t ).
Here, p0(t ) is the prepayment ratio that is generated independently of the interest
rate (the so-called background prepayment). In addition, if we classify the
mortgagors into several groups on the basis of their levels of “rationality,” we allocate
a different refinancing cost to each group so that we can vary the probability of
rational prepayment.
B. The Prepayment Function Model
Schwartz and Torous (1989) suggest the following model. Let v be an explanatory
vector variable that affects the prepayment, and b be the coefficient vector. Then, the
prepayment function is described as follows.
p(t ) = p0(t ) exp (b'v), (A.26)
where p0(t ) is the prepayment ratio that is independent of environmental change.
We assume that p0(t ) is the log-logistic function as follows.
gp(gt )p–1
p0(t ) = ————. (A.27)
1+ (gt )p
As explanatory variables, we take account of the difference between the mortgage
contract rate c and the refinancing rate l (t ) (replacing this with the long-term
interest rate).
v1(t ) = c – l (t – s ), s ³ 0, (A.28)
where s is a parameter that represents the time lag between the decision to prepay
and actual prepayment.
Furthermore, we introduce a variable that represents the acceleration of
prepayment due to the widening of the interest rate differential, a variable that
represents the ratio of the balance assuming prepayment to the balance assuming no
prepayment, and a variable that represents the seasonal fluctuation of prepayment,
and so on.
C. The Discrete Model
The simplest example is the Zipkin (1993) model. When Zipkin formulated the
prepayment model of mortgage securities, he assumed the discrete interest rate model
based on a Markov chain and proposed a model with a prepayment ratio that is a
deterministic function of the interest rate. When we assume a continuous model for
the interest rate, we divide interest rates into several ranges, and assume a different
prepayment function for each range.
p1, r (t ) ² r1,
p(t ) = {pi, ri–1 ² r (t ) ² ri, (A.29)
pn, rn–1 ² r (t ).
In the case of time deposits, r (t ) should be the difference between the interest
rate of existing time deposits and the current rate of a new time deposit.
... ...
APPENDIX 5: GENERATING YIELD CURVES USING THE HJM
MODEL
By changing the parameter of the HJM model, we can generate various kinds of yield
curves. For example, if we set s1 = 0.000365, k = 0.0632, and s2 = 0.000202, we get
the following curves after three years.
0
1
2
3
4
5
6
7
Percent
3 9 15 21 27 33 39 45 51 57 63 69 75 81
Months
Appendix Table 1 The Features of the Main Products in Banking Accounts
Assets
Loans
Overdrafts Short-term loans Long-term loans
Holding period — Until maturity Until maturity
(several months) (mid- and long-term)
Liquidity No No No
Interest rate Linked to base Fixed until maturity
type interest rate Fixed until maturity or linked to base
interest rate
Principally no, but
Cancellation of Withdrawal at any time Principally, no frequent cancellation as
contract for consumer loans (in
particular, housing loans)
Characteristics Principal: withdrawal One-time repayment One-time repayment
of cash flow at any time; at maturity at maturity or by
interest: periodical installments
Deposits
Liabilities Liquid deposits Time deposits
Current Ordinary Saving Notice Time
Holding period Until withdrawal Until withdrawal Until withdrawal Until withdrawal Until maturity
(several years)
Liquidity No No No No No
Interest rate No interest Fixed until Fixed until Fixed until Fixed, partly
type withdrawal withdrawal withdrawal variable
Cancellation Withdrawal at Withdrawal at Withdrawal at Withdrawal with Yes of contract any time any time any time prior notice
Characteristics
of cash flow
Note: 1. The holding period alluded to in this table indicates the banks’ actual holding period, which is
a different concept from the “holding period” as a measurement period in the main text.
Principal:
withdrawal at
any time;
interest:
periodical
Principal:
withdrawal at
any time
Principal:
withdrawal at
any time;
interest:
periodical
Repayment of
principal and
interest at
maturity
Repayment of
principal and
interest at
maturity and
mid-term
interest
payment
Appendix Table 2 Imaginary Portfolio of Banking Accounts (Basic Form)
Liabilities Variable/fixed Interest rate Linkage Non- 1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y Total Remarks maturity
Current deposits No interest — 5 5
Other demand
deposits (ordinary, Non-maturity Market Imperfect — 20 20
saving, notice)
Time deposits Fixed rate Market Perfect — — 25 35 — 10 — — — — 70
Time deposits Variable rate Market Perfect — 5 — 5 Maturity is
3 years
100
Assets Variable/fixed Interest rate Linkage Non- 1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y Total Remarks maturity
Cash and deposits No interest — 5 5
Short-term loans Short-term (bills discounted, Fixed rate prime — — — 20 — 20
loans on bills)
Long-term loans Fixed rate Market Perfect — — 5 — 5 (loans on deeds)
Long-term loans Variable rate Short-term — — — 25 — 25 Maturity is
(loans on deeds) prime 3 years
Long-term loans Variable rate Long-term — —— 5 — 5 Maturity is
(loans on deeds) prime 3 years
Long-term loans Fixed rate Market Perfect 5 — — 5 — — 10 (loans on deeds)
Housing loans Variable rate Short-term — —— 5 — 5 Maturity is
prime 7 years
Housing loans Fixed rate Market Perfect — — — — 5 — — 5
Overdrafts (corporate Non-maturity Short-term — — 20 20 loans, card loans) prime
100
Note: 1. The table shows the share of each product assuming that the total sums of liabilities and assets are 100, respectively.
Appendix Table 3 Simulation Results
Target products: individual loans
Perfect rollover
Holding period Market rate-linked Short-term prime Long-term prime
(months) (risk, ¥ billions) rate-linked rate-linked
(risk, ¥ billions) (risk, ¥ billions)
1 1.0 50.1 27.2
2 1.0 51.1 29.1
3 1.0 51.1 33.8
4 1.0 51.1 33.8
5 1.0 51.1 33.8
6 1.0 71.1 94.3
7 1.3 71.5 94.9
8 1.3 71.5 94.9
9 1.3 71.5 94.9
10 1.3 71.5 94.9
11 1.3 71.5 94.9
12 1.3 88.5 160.5
13 1.4 88.5 161.4
14 1.4 88.5 164.3
15 1.4 88.5 164.3
16 1.4 88.5 164.3
17 1.4 88.5 164.3
18 1.4 89.8 224.3
19 1.4 89.8 225.9
20 1.4 89.8 225.9
21 1.4 89.8 225.9
22 1.4 89.8 225.9
23 1.4 89.8 225.9
33 1.4 91.4 331.5
34 1.4 91.4 331.5
35 1.4 91.4 331.5
36 1.4 97.0 371.3
PV0 0 269.2 467.5
Appendix Table 4 Imaginary Portfolio of Banking Accounts
(No Administered Rate-Linked Products)
Liabilities Variable/fixed Interest rate Linkage Non- 1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y Total Remarks maturity
Current deposits No interest — 5 5
Other demand
deposits (ordinary, Non-maturity Market Imperfect — 0 0
saving, notice)
Time deposits Fixed rate Market Perfect — — 45 35 — 10 — — — — 90
Time deposits Variable rate Market Perfect — 5 — 5 Maturity is
3 years
100
Assets Variable/fixed Interest rate Linkage Non- 1M 3M 6M 12M 2Y 3Y 4Y 5Y 6Y 7Y Total Remarks maturity
Cash and deposits No interest — 5 5
Short-term loans Short-term (bills discounted, Fixed rate prime — —— 0 — 0
loans on bills)
Long-term loans Fixed rate Market Perfect — — 80 — 80 (loans on deeds)
Long-term loans Variable rate Short-term — —— 0 — 0 Maturity is
(loans on deeds) prime 3 years
Long-term loans Variable rate Long-term — —— 0 — 0 Maturity is
(loans on deeds) prime 3 years
Long-term loans Fixed rate Market Perfect 5 — — 5 — — 10 (loans on deeds)
Housing loans Variable rate Short-term — —— 0 — 0 Maturity is
prime 7 years
Housing loans Fixed rate Market Perfect — — — — 5 — — 5
Overdrafts (corporate Non-maturity Short-term — — 0 0 loans, card loans) prime
100
Brennan, M. J., and E. S. Schwartz, “A Continuous Time Approach to the Pricing of Bonds,” The
Journal of Banking and Finance, 3, 1979, pp. 133–155.
Chen, R. R., and L. O. Scott, “Multi-Factor Cox-Ingersoll-Ross Model of the Term Structure:
Estimates and Tests from a Kalman Filter Model,” working paper,
Cox, J. C., J. E. Ingersoll, and S. A. Ross, “An Intertemporal Equilibrium Model of Asset Prices,”
Econometrica, 53, 1985, pp. 363–384.
Dunn, K. B., and J. J. McConnell, “Valuation of GNMA Mortgage-Backed Securities,” The Journal of
Finance, 36, 1981, pp. 599–617.
———, and C. S. Spatt, “The Effect of Refinancing Costs and Market Imperfections on the Optimal
Call Strategy and the Pricing of Debt Contracts,” working paper,
1986.
Heath, D., R. A. Jarrow, and A. Morton, “Contingent Claim Valuation with a Random Evolution of
Interest Rates,” The Review of Futures Markets, 9, 1990, pp. 54–76.
———, ———, and ———, “Bond Pricing and the Term Structure of Interest Rates: A New
Methodology for Contingent Claim Valuation,” Econometrica, 60, 1992, pp. 77–105.
Johnston, E. T., and L. D. Van Drunen, “Pricing Mortgage Pools with Heterogeneous Mortgagors:
Empirical Evidence,” working paper,
Litterman, R., and J. Scheinkman, “Common Factors Affecting Bond Returns,” Journal of Fixed Income,
1, 1991, pp. 54–61.
Longstaff, F. A., and E. S. Schwartz, “Interest Rate Volatility and the Term Structure: A Two-Factor
General Equilibrium Model,” The Journal of Finance, 47, 1992, pp. 1259–1282.
McConnell, J. J., and M. Singh, “Rational Prepayments and the Valuation of Collateralized Mortgage
Obligations,” The Journal of Finance, 59, 1994, pp. 891–921.
Mori, A., M. Ohsawa, and T. Shimizu, “Calculation of Value at Risk and Risk/Return Simulation,”
IMES Discussion Paper Series 96-E-8, Bank of
Oda, N., and J. Muranaga, “A New Framework for Measuring the Credit Risk of a Portfolio: The
‘ExVaR’ Model,” Monetary and Economic Studies, Institute for Monetary and Economic Studies,
Bank of
Ritchken, P., and L. Sankarasubramanian, “A Multifactor Model of the Quality Option in Treasury
Futures Contracts,” The Journal of Financial Research, 18, 1995, pp. 261–279.
Schwartz, E. S., and W. N. Torous, “Prepayment and the Valuation of Mortgage-Backed Securities,”
The Journal of Finance, 55, 1989, pp. 375–392.
Stanton, R., “Rational Prepayment and the Valuation of Mortgage-Backed Securities,” working paper,
———, “Pool Heterogeneity and Rational Learning: The Valuation of Mortgage-Backed Securities,”
working paper,
Vasicek, O. A., “An Equilibrium Characterization of the Term Structure,” The Journal of Financial
Economics, 5, 1977, pp. 177–188.
Wright, D., and J. Houpt, “An Analysis of Commercial Bank Exposure to Interest Rate Risk,” Federal
Reserve Bulletin, 82, 1996, pp. 115–128.
Zipkin, P., “Mortgages and Markov Chains: A Simplified Evaluation Model,” Management Science, 39,
1993, pp. 683–691.
References
