A COMPARISON OF CALL STRATEGIES FOR CALLABLE ANNUITY MORTGAGES PETER LØCHTE JØRGENSEN, KRISTIAN R. MILTERSEN, AND CARSTEN SØRENSEN
Abstract. In this paper we consider and compare different strategies for calling callable anuity mortgages. It turns out that a key figure called percentage critical gain gives a remarkable good indication of when to call. We compare the proposed simple strategy with the optimal strategy where the valuation criterion is the loss of net present value of the mortgage by using the simple strategy. The analysis is carried out using the Cox-Ingersoll-Ross term structure of interest rate model with parameters estimated using Danish non-callabale government bonds. We consider the three different tax clientelles relevant for Danish investors and we take call fees etc. into consideration.
1. Introduction A callable mortgage can be viewed as a combination (portfolio) of an otherwise equivalent non-callable mortgage and a call option. The option to call the mortgage is similar to an American type of call option with the non-callable bond as the underlying asset. The call option provides the debtor with the right but not the obligation to repurchase the non-callable bond payments by prepaying the remaining debt (plus the first comming interest payment). The option is of the American type since the option to call the mortgage applies at all times during the remaining life of the mortgage. In this paper we analyze the pricing of the callable mortgages from the point of view of the debtor by applying methods from option pricing theory. The value of the callable mortgage depends on the strategy followed by the debtor. An optimal call strategy is a strategy that maximizes the present value of the callable mortgage (which from the debtors point of view is negative). Pricing the callable mortgage and determining the optimal call strategy is a complicated matter for several reasons. First, the call option is an interest rate related contingent claim. This puts strong demands on the applied model, since in pricing interest rate contingent claims one generally has to model the dynamics of the entire yield curve. Second, we wish to take into account various details of the contract terms, transactions costs in relation to the call decision, and to some extent the taxation of the debtor. As a model for the dynamics of the term structure of interest rates we choose to work with the model formulated by Cox, Ingersoll, and Ross (1985). Apart from being fairly well-known one advantage of this model is that it allows us to calculate callable as well as non-callable mortgages and bond prices, and as we will demonstrate later, also to determine the optimal call strategy in relation to the callable mortgage. We estimate the parameters of this model from historical data for the spot interest rate and from observed bond prices in Denmark at the end of 1995. For purposes of characterizing the optimal call strategy we record some key figures in relation to callable mortgages. In order to be able to evaluate the applied call strategy we compare optimal call strategies Date: September 1995. This version: April 28, 1999. Key words and phrases. Mortgages, call strategy. We would like to thank Bjarne G. Sørensen for comments and sugestions which have improved this paper both concerning the contents and readability. 1
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PETER LØCHTE JØRGENSEN, KRISTIAN R. MILTERSEN, AND CARSTEN SØRENSEN
b1
b2
-
2 months
Notification date
t0
t1
t2
bN
,A , A A , A,
tN
Figure 1. Timeline. in different term structure scenarios and for debtors with different tax rates with some so-called simple strategies. One criterion for evaluation of the various simple strategies is the value of the callable mortgage when the given call strategy is pursued. We therefore compare the value of the mortgage when the optimal call strategy is pursued, with the mortgage value when the simple strategy is pursued. 2. Calling Mortgages In relation to Danish callable annuity mortgages there is normally a two month notification period prior to the coupon payment dates. The timeline in Figure 1 serves to illustrate this problem in relation to the call decision. The dates t0 , t1 , . . . , tN denote succeeding coupon dates. If a debtor decides to call the mortgage at time t0 this must be noticed two months prior to t0 at the latest. In execution of the call the debtor pays the remaining debt plus interest at time t0 . In return he receives, in principle, the otherwise identical non-callable bond, the payments from which net out with the payments on the debtor’s short position. From option pricing theory it is well-known that American call options should be exercised only just prior to coupon/dividend payment dates or at the maturity date1 . Similarly, and in relation to the present problem, it can be shown that a call decision should always be postponed to a date just prior to a notification date. Suppose for example, that a debtor decides to call at some intermediate date. It can then easily be shown that the debtor can only be worse off than in the case where the call decision is postponed to the notification date, since at the notification date the debtor can decide to call and be precisely as well off as if the call decision was taken at an earlier date. Alternatively the interest rate might have increased to a level where the debtor can profit more from simply buying the bond in the market than from prepaying at par value.2 At a given notification date the debtor must decide whether to call or to wait. If the debtor decides to call his gain equals the value of the callable mortgage minus the remaining debt minus the call costs. On the other hand he looses the right to call at a later date on what might be more advantageous conditions. In relation to the determination of the optimal date and interest rate level for a call it is important to be able to determine the value of the lost option. In the following we have applied the one-factor model of 1 Cf.,
e.g., Cox and Rubinstein (1985, p. 139). argument is valid independent of the debtors option of choosing a fixed price aggrement. The situation is exactly the same as exercising an American call option on a non-dividend paying stock. If you think you have some information indicating that the stock is overvalued you should still not exercise your option. There are better strategies. E.g., you can short forward contracts on the stock which actually is the same as entering a fixed price agreement about taking a new mortgage at a fixed interest rate at the next notification date. Later on you can decide whether you prefer to call the mortgage or to buy the underlying bonds in the market.
2 This
A COMPARISON OF CALL STRATEGIES FOR CALLABLE ANNUITY MORTGAGES
3
Cox, Ingersoll, and Ross (1985) (CIR-model) to estimate the value of the lost option in relation to the call decision. 3. Model and Methodology Below we describe model assumptions and the method that we have used in obtaining values for the callable mortgage and in determining optimal call strategies. We assume that the debtor finances his needs for cash on normal market conditions. The evolution of the short term interest rate, rt , evolves according to the stochastic differential equation √ drt = κ(θ − rt )dt + σ rt dZt , where the parameter κ denotes the mean reversion speed towards a mean reversion level of θ. σ is the volatility of the short interest rate and dZ is a noise term. The relevant interest rate in the following analysis is the after-tax interest rate faced by the debtor. We compute present values on an after-tax basis and we take account of fees in relation to the call. Since we assume simultaneous tax payments at the rate s, the short after-tax interest rate is (1 − s)r.3 The value of the callable mortgage at the n’th notification date and at the interest rate r is denoted by C(r, n), n = 1, . . . , N . At the last coupon date the call option has no value since the remaining debt matures at par in any case, i.e., C(r, N ) = P V (−bN , r, 16 ), where P V (x, y, z) denotes the present value of the riskless payment x in z years of time if the short interest rate is y. This zero-coupon bond price is obtained as a closed formula in the CIR model. At earlier notification dates we have ( P V (−(RG(n) + O(n)), r, 16 ), if we call, C(r, n) = if we do not call, C+ (r, n), where C+ (r, n) denotes the value of the callable mortgage immediately after a decision not to call at the n’th notification date. C+ (r, n) of course contains the value of the option to call at a later date. RG(n) denotes the remaining debt, and O(n) is the call costs in relation to the call at the n’th coupon date. All payments are computed at the notification date where the decision to call or not must be taken. For later use we also introduce K(r, n) which denotes the present value of an otherwise identical non-callable mortgage. The value can be calculated using the closed formula for zero-coupon bonds in the CIR model. Debtor’s optimal call decision is determined by maximizing present value. Consequently, the debtor will call at the n’th notification date if and only if 1 P V (−(RG(n) + O(n)), r, ) ≥ C+ (r, n). 6 Note that both sides of this expression will be negative. The critical interest rate, r n , at the n’th notification date is defined as the interest rate that makes the debtor indifferent between calling and waiting. At interest rates below the critical interest rate the debtor will call, and for interest rates above r n the debtor will wait. We apply the following method in computing the value of the callable mortgage and in determining the critical interest rates. We start out from the value of the callable mortgage at the last notification date, i.e., C(r, N ) = P V (−bN , r, 16 ). The value of the mortgage is then calculated at time N − 1 for all levels of the interest rate. This exercise is repeated until we reach time zero. 3 We
are here assuming that all debtors are of the same tax clientelles and that the bond prices are determined on the market such that the mortgage has a net present value of zero. That is, bonds are priced by the after-tax payments as seen from the debtor’s point of view.
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PETER LØCHTE JØRGENSEN, KRISTIAN R. MILTERSEN, AND CARSTEN SØRENSEN
To determine the value of the callable mortgage at a given notification date the sum of the remaining debt and the call costs is first calculated. The present value of the callable mortgage is calculated by discounting remaining debt and costs back to the notification date. Then the value of waiting is calculated. This value depends on at the value at the succeeding notification date and is found by discounting the future value taking account of interest rate uncertainty. In this step the risk-adjustment parameter, λ, enters the calculations.4 In practice the calculations are done numerically. The technique is described in the appendix. 4. Estimation of the CIR-model In order to apply the CIR model the spot rate process parameters κ, θ, σ and the risk-adjustment parameter, λ, must be estimated. The spot rate process parameters are estimated using the GMM procedure described in Chan et al. (1992). We use data for the spot interest rate (’day-to-day interest rate’ as published by Danmarks Nationalbank). By choosing a very short interest rate as the basis for the estimation procedure we are loyal to the CIR model’s original specification. On the other hand it can be argued that the day-today interest rate hardly is the interest rate with the largest explanatory power towards variations in bond prices. Having mentioned this problem we do not discuss this issue further in the present paper. For concrete estimations we use data from the period January 1st, 1993 to November 30, 1995—a total of 739 observations. The obtained spot rate process parameter estimates are as follows: Parameter-estimater κ
0.3421
θ σ
0.0752 0.1185
Using these parameter estimates in the CIR expression for zero-coupon bond values we then performed an implicit estimation of the risk-adjustment parameter λ. More precisely, for a handful of trading dates in December of 1995 we compared market and model prices of the ten most actively traded Danish government bonds at the Copenhagen stock exchange. These bonds are non-callable and have maturity dates which are evenly spread over the time interval from the year 1996 to 2006. We found very small estimates for λ which were never significantly different from zero. For this reason as well as for the sake of simplicity we have set λ = 0 in the calculations. 5. Optimal Call Strategies In the present section we demonstrate the implementation of the model via a presentation of some illustrative examples. As a representative example we choose to analyze a problem where the debtor has a 30-year annuity mortgage (kontantlaan) based on issuing of 9% annuity bonds. With a market price for the annuity bond of 102 this is equivalent to a bond yield of 8.8235%. The principal is set to DKK 500,000.00 and the fixed call costs is set at DKK 5,000.00 whereas the variable cost is 0.5% of the remaining debt. As discussed in a previous section, at any notification date there will be a critical interest rate, r n , identifying that state where debtor’s gain from call and from postponing the call, respectively, are identical. If the present interest rate is lower than the critical interest rate, the mortgage is called, and vice versa. 4 For
a comprehensive description of the assumption and results of the CIR model, cf. the original paper, Cox, Ingersoll, and Ross (1985).
A COMPARISON OF CALL STRATEGIES FOR CALLABLE ANNUITY MORTGAGES
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Using the above described numerical procedure we establish the sequence of critical interest rates. Below we have chosen to illustrate the sequence of critical interest rates. It is not the sequence of critical spot rates, however. Instead we have chosen to plot the yield to maturity of a ten year non-callable bullet loan as implied by the critical spot interest rate in combination with the CIR-model. The calculations were carried out for three different tax rates: 0%, 34% and 50%. We also varied the interest rate volatility parameter, σ, which is of great importance for the volatility of bond prices. We tried σ = 0.05, σ = 0.1185 and σ = 0.2.
Kritisk effektiv rente på tiårig obl.
8 7,5 7 6,5 6 5,5 5
0
5
10
15 Restløbetid
0% skat
34% skat
20
25
30
50% skat
Figure 2. Critical interest rate depicted as the yield of a ten year non-callable buttet loan for three different tax rates. Perhaps a bit surprisingly from Figure 2, it is seen that critical bond yield curves are almost identical for the three different tax rates. Only just before the maturity date of the mortgage are the optimal strategies clearly different. It is seen that tax exempt individuals should always call before individuals paying tax at a higher rate (the critical bond yield curve for the tax exempt individual lies above the other curves and will therefore be reached first when interest rates fall). The reason behind this result is that a tax paying individual—ceteris paribus—must demand a higher gain from call than a tax exempt (taxed at a lower rate) investor. Hence, he will have to wait longer. Furthermore, it is seen that the critical bond yield curves are interrupted 2–3 years before maturity of the mortgage. This phenomenon occurs since during the final years of the life of the mortgage it is not possible to obtain gains which are large enough to pay for the costs in relation to the call. Note finally, that the fact that the critical bond yield curves are almost identical for the three types of individuals for the major part of the time to maturity does not imply that individuals are indifferent towards
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PETER LØCHTE JØRGENSEN, KRISTIAN R. MILTERSEN, AND CARSTEN SØRENSEN
taxation. What it merely means is that their call behavior is almost identical. Their gains, and hence the values of their mortgages, may still be very different. We return to this later.
Kritisk effektiv rente på tiårig obl.
8 7,5 7 6,5 6 5,5 5
0
5
10
15 Restløbetid
vol. 0,05
vol. 0,1185
20
25
30
vol. 0,20
Figure 3. Critical interest rate depicted as the yield of a ten year non-callable buttet loan for three different interest rate volatility levels. Figure 3 shows critical bond yield curves for a tax exempt debtor at three different levels of the volatility parameter, σ. It is seen that the curves move downwards as the volatility is increased. This is a general result that has to do with the fact that when spot interest rate volatility is increased, the bond price volatility increases which again increases the likelihood of high gains from call. Therefore rational debtors should be willing to wait longer compared to a situation with a lower volatility. 6. Simple Call Strategies The purpose of this section is to find and to be able to evaluate potential simple call strategies for call of callable mortgages. First of all, we have let us inspire by the key figures for mortgages as function of time to maturity at the critical interest rate level where call is optimal as depicted in Figure 2–3 in the previous section. An alternative criterion would be the key figure critical percentage gain as depicted in the Figures 4–5. The key figure, critical percentage gain, is almost linear as function of time to maturity as we have shown in Figures 4–5. The key figure critical percentage gain, denoted P G(n), at the n’th notification date is calculated as P G(n) = 100%
K(rn , n) − C(r n , n) . K(r n , n)
A COMPARISON OF CALL STRATEGIES FOR CALLABLE ANNUITY MORTGAGES
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Kritisk procentvis gevinstkrav
20
15
10
5
0
0
5
10
0% skat
15 Restløbetid
34% skat
20
25
30
50% skat
Figure 4. Critical percentage gain at three different tax rates.
In this formula K(r n , n) denotes the value of the mortgage if it is not called (that is, the value of the corresponding non-callable mortgage), and C(r n , n) = P V (−(RG(n) + O(n)), r, 16 ) can be interpreted as the value of a new mortgage initiated at that date on the market to refinance the old called mortgage. Therefore, the denominator in the formula can be interpreted as the gain of call at the notification date n and hence P G can be interpreted as the critical percentage gain. Because of the almost linearity of P G(·) depicted in the Figures 4–5, we are inspired to try with the following simple call strategy: 1. Calculate by the CIR model P G(0) when the mortgage is initiated. 2. Draw a straight line from P G(0) at the initial date of the mortgage and down to zero at the maturity of the mortgage. 3. Use this straight line to decide when to call: That is, calculate at any notification date the gain of calling the mortgage in percentage of the value of the mortgage if this is not called, P G(r, n). If this value, P G(r, n), is greater than or equal to P G(n), call the mortgage imediately. In order to be able to evaluate this (simple) call strategy relative to the optimal strategy as described earlier in the paper we can calculate the present value of the mortgage using the optimal call strategy and compare that value to the present value of the mortgage using the simple stategy. We can then calculate the difference in percentage of the value using the optimal strategy for a set of mortgages based on different underlying bonds and different interest rate levels of the mortgages.
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PETER LØCHTE JØRGENSEN, KRISTIAN R. MILTERSEN, AND CARSTEN SØRENSEN
Kritisk procentvis gevinstkrav
25
20
15
10
5
0
0
5
10
vol. 0,05
15 Restløbetid
vol. 0,1185
20
25
30
vol. 0,20
Figure 5. Critical percentage gain at three different interest rate volatility levels. bond init. mort. int. rate int. rate 10% 13% 10% 12% 10% 11% 9% 12% 9% 11% 9% 10% 8% 11% 8% 10% 8% 9% 7% 10% 7% 9% 7% 8% 6% 9% 6% 8% 6% 7%
P G(0) 20.09% 20.15% 20.09% 15.49% 15.53% 15.47% 10.38% 10.40% 10.33% 5.00% 4.94% 4.86% 0.39% 0.36% 0.31%
NPV of mort., opt. strategy 125.98 117.22 108.50 127.75 118.35 109.05 129.75 119.56 109.62 131.88 120.91 110.27 134.14 122.37 110.98
NPV of mort., perc. loss using simple strategy simple strategy 126.01 0.03% 117.24 0.02% 108.53 0.02% 127.78 0.02% 118.38 0.02% 109.07 0.02% 129.78 0.02% 119.58 0.02% 109.64 0.02% 131.92 0.03% 120.95 0.03% 110.30 0.03% 134.21 0.05% 122.42 0.05% 111.03 0.04%
Table 1. Evaluation of optimal versus simple call strategies and P G-values of some mortgages at a tax rate of 0%.
In the Tables 1–3 we have presented examples of this type of calculations for the tax rates s = 0, s = 0.34 og s = 0.50. As we can observe from the tables the difference between using the optimal call stategy and
A COMPARISON OF CALL STRATEGIES FOR CALLABLE ANNUITY MORTGAGES
bond init. mort. int. rate int. rate 10% 13% 10% 12% 10% 11% 9% 12% 9% 11% 9% 10% 8% 11% 8% 10% 8% 9% 7% 10% 7% 9% 7% 8% 6% 9% 6% 8% 6% 7%
P G(0) 14.91% 15.46% 16.10% 10.15% 10.79% 11.49% 5.07% 5.83% 6.68% 0.57% 1.16% 1.94% — — —
NPV of mort., opt. strategy 126.03 117.19 108.52 127.75 118.34 109.08 129.72 119.55 109.67 131.85 120.90 110.29 128.25 118.74 109.58
9
NPV of mort., perc. loss using simple strategy simple strategy 126.03 0.00% 117.19 0.00% 108.53 0.01% 127.75 0.00% 118.34 0.00% 109.08 0.01% 129.72 0.00% 119.56 0.00% 109.68 0.01% 131.85 0.00% 120.91 0.01% 110.31 0.02% 128.25 0.00% 118.74 0.00% 109.58 0.00%
Table 2. Evaluation of optimal versus simple call strategies and P G-values of some mortgages at a tax rate of 34%.
bond init. mort. NPV of mort., P G(0) opt. strategy int. rate int. rate 10% 13% 10.70% 126.02 10% 12% 11.81% 117.20 10% 11% 13.03% 108.50 9% 12% 6.05% 127.78 9% 11% 7.30% 118.35 9% 10% 8.70% 109.03 8% 11% 1.53% 129.71 8% 10% 2.74% 119.56 8% 9% 4.27% 109.64 7% 10% — 130.52 7% 9% — 120.92 7% 8% 0.43% 110.26 6% 9% — 122.38 6% 8% — 114.53 6% 7% — 107.00
NPV of mort., perc. loss using simple strategy simple strategy 126.04 0.02% 117.20 0.00% 108.50 0.00% 127.82 0.03% 118.36 0.01% 109.03 0.00% 129.72 0.01% 119.57 0.00% 109.64 0.00% 130.52 0.00% 120.92 0.00% 110.27 0.01% 122.38 0.00% 114.53 0.00% 107.00 0.00%
Table 3. Evaluation of optimal versus simple call strategies and P G-values of some mortgages at a tax rate of 50%.
the proposed simple strategy, measured as the percentage loss of present value of the mortgage, is neglisable. Moreover, for a set of callable mortgages we have in the tables given the correct P G(0). “—” in Tables 2–3 indicates that it is not optimal to call these mortgages at any time. For a tax rate of zero, s = 0, P G(0) should be independent of the interest rate on the mortgage because then there are no tax gain to the debtors. That this is not exactly the case in Table 1 is due to minor numerical inaccurazities. If we would have depicted the critical interest rate, r n together with P G(n) for the chosen set of mortgages it would be the case that the lower P G(0) is, the lower the r n would be, cf. the Figures 2 with 4 and 3 with 5. That is, the lower P G(0)
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PETER LØCHTE JØRGENSEN, KRISTIAN R. MILTERSEN, AND CARSTEN SØRENSEN
is, the lower must the short rate be before a call is optimal, and hence, the lower the probability that a call will be made at all. This may at first glace seems counter intuitive, but the fact is that the potential gain by calling is declining heavily as the interest rate of the bond decreases and the tax rate increases. That is, eventhough you compensate by lowering the critical percentage gain the probability of call will still decline. With this little exercise we have shown that in a CIR model a key figure as gain by call (that is, the gain from calling the mortgage compared to continuing with the old mortgage to maturity would be an extremely good figure to determine the optimal call date for a callable mortgage.
7. Conclusion In this paper, we got of the ground by observing that a callable mortgage can be interpreted as a portfolio of a corresponding non-callable mortgage and an American call option. Given this interpretation we showed how the recent option and term structure of interest rate theory can be exploited to derive the optimal call stategy for mortgages. Because of this structure of the mortgages this is equivalent to finding the optimal exercise strategy of an American call option. As the model for the stochastic evolution of the term structure of interest rates over time we have chosen the well-known CIR model. The results that we have acheved is, however, by no means dependent on this choice. Independent of the choice of term structure model the problem of finding the optimal call strategy in terms of finding the critical interest rate level will require a numerical procedure implemented on a computer. This, however, did not raise any new problems and we have in our implementation not suffered badly by neither precision problems nor computer time problems. We illustrated the proposed method using a concrete example and this was the inspiration for constructing the special percentage critical gain key figure. We showed how it was possible with this key figure to approximate the optimal call strategy with a simple linear strategy. A comparison of the value of the callable mortgage using the optimal and the simple stategy showed approximately no difference, that is, the loss from using the simple strategy to call the mortgage is negliable. This key figure is, in fact, already being used in practice. That is, it is this number that most financial institutions use in their programs monitoring callable mortgages today. However, it is our impression that these monotoring programs use a fixed absolute value as there trigger value. In this paper we have—in a model based sense—proven that critical gain is the right key figure to use as trigger value, however, we argue furthermore, that one should use a relative critical gain, that is percentage critical gain, and that this percentage critical gain value should decline linearly as function of time to maturity of the mortgage. Another question—and a more deficult one to answer—is to find the optimal initial nevel of the percentage critical gain. We have in Tables 1-3 calculated this optimal level for a set of callable mortgages, however, it must be emphasized that these results do strongly depent on the chosen model (CIR) and the specificly chosen parameter estimates to the CIR model.
Appendix In the CIR model, the value of a financial security—as seen from the debtors point of view—fulfill the following partial differential equation (PDE) 1 2 σ rt Vrr + (κ(θ − rt ) + λrt )Vr + Vt − (1 − s)rt V = 0, 2
A COMPARISON OF CALL STRATEGIES FOR CALLABLE ANNUITY MORTGAGES
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where λ is a risk premium. To find the net present value of a given financial security you simply solve this PDE with a set of boundary conditions characterizing the given financial asset. To find C(r, n) the PDE has been solved numerically initiated using the boundary condition assuring that the value at the final N ’th notification date is equal to C(r, N ). The numerical method is described in, e.g., Duffie (1992, pp. 204–210). References Bhattacharya, S. and G. M. Constantinides, editors (1989): Theory of Valuation, volume 1 of Frontiers of Modern Financial Theory. Rowman & Littlefield Publishers, Inc., Totowa, New Jersey, USA. Chan, K. C., G. A. Karolyi, F. A. Longstaff, and A. B. Sanders (1992): “An Empirical Comparison of Alternative Models of the Short-Term Interest Rate,” The Journal of Finance, XLVII(3):1209–1227. Cox, J. C., J. E. Ingersoll, Jr., and S. A. Ross (1985): “A Theory of the Term Structure of Interest Rates,” Econometrica, 53(2):385–407. Reprinted in Bhattacharya and Constantinides (1989, p. 129–151). Cox, J. C. and M. Rubinstein (1985): Options Markets, Prentice-Hall, Inc., Englewood Cliffs, New Jersey 07632, USA. Duffie, J. D. (1992): Dynamic Asset Pricing Theory, Princeton University Press, Princeton, New Jersey, USA.
Dept. of Management, School of Economics, University of Aarhus, Universitetsparken 350, DK–8000 ˚ Arhus C, Denmark E-mail address:
[email protected] Dept. of Management, School of Business and Economics, Odense Universitet, Campusvej 55, DK–5230 Odense M, Denmark E-mail address:
[email protected] Institute of Finance, Copenhagen Business School, Rosenørns All´ e 31, DK–1970 Frederiksberg C, Denmark E-mail address:
[email protected]
