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GNMA pass-through certificates
Journal of Bankin g and Finance 5 (1981) 547-556. North-Holland Publishing Company
GNMA PASS-THROUGH CERTIFICATES Behavior Through Time of the Mean Maturity and Duration Douglas W. MITCHELL* University of Texas at Austin, Austin, TX 787/2. USA
Received May 1980, final version received January 1981
Th is paper adds to the literature on the illogical p ricing of GNMA pass-through cert ificates. It is already known that the market incorrectly computes the mean capital gains yield on such a pool of mortg ages as the capital gains yield on the mortgage of mean maturity. The present paper shows the strong assumptions about the behavior through time of the mean maturity or duration of the pool inherent in the market's attempt to give identical yields to securities issued at different times in the past. This market behavior implies a very strong assumption about the frequency distribut ion of prepaym ent dates of the individual mortgages.
1. Introduction GNMA pass-through certificates are portions of a pool of FHA-insured or VA-guaranteed mortgages. The fixed monthly payments from the borrowers are 'passed through' to the investors by a financial intermediary which deducts a servicing fee. The Government National Mortgage Association (GNMA) guarantees the timely monthly payments to the investors. In addition, any prepayments of principal by the borrowers are passed through to the investors. Because of the two types of guarantees (those of GNMA and of the FHA or VA) there is no default risk. The value of the pass-throughs in the secondary market fluctuates inversely with interest rates, creating price uncertainty unless the investor plans to retain the certificate throughout its life. The life of the certificate equals the life of the mortgage which is fully prepaid last, unless GNMA decides to force early liquidation of the certificate on the grounds that, due to large numbers of prepayments, the remaining cash flows are not worth the bookkeeping costs. The mortgages in the pool typically have scheduled lives of thirty years, and the mean time from origination to full prepayment is about twelve years. *1 am grateful to Eric Bond and two referees for helpful comments.
0378-4266/81/0000-0000/S02.75
©
1981 North-Holland
548
D.W: Mitchell, GNMA pass-through certificates
An important novelty about pass-throughs is the uncertainty about when the principal will be repaid. This uncertainty is analogous to the uncertainty about the maturity of a callable bond, but there are two complicating factors. First, the pass-through as an investment in a pool of mortgages which will be called (prepaid) at various times. Second, unlike the principal of a callable corporate bond, the principal on a pool of mortgages is scheduled to be repaid over time at an increasing rate rather than entirely at the end of the life. This complicates maturity calculations. Curley and Guttentag (CG) (1974 and 1977) showed that the rules of thumb used by the market in computing yields are incorrect if the certificates are selling at a discount or a premium. Given a perceived distribution of prepayment dates, the market incorrectly assumes that the yield on the pool equals the yield on a mortgage with the mean prepayment date. For a certificate sold at a discount this generates downward bias in the computed yield because the capital gains portion of yield is not a linear function of the prepayment date. CG point out that the correct procedure for computing yield involves listing all cash flows, including prepayments, making the appropriate assumption about how the cash receipts are reinvested, and applying the present value formula to obtain the yield.' The above considerations show that actual yield is computed wrong for a pass-through certificate sold at a discount or a premium. The purpose of the remainder of this paper is to show that, for older mortgage pools sold at par or otherwise, the market incorrectly computes the maturity of the investment as equal to the original maturity of the mortgage pool, thereby placing it in the wrong spot in the term structure and assigning it the wrong appropriate yield based on term structure considerations. Hence even if yields were computed correctly, the market assurries the wrong maturity and thus the wrong appropriate yield, so the security is incorrectly priced.' Section 2 considers the behavior through time of the maturity of the average dollar of unpaid principal on a single mortgage, and of the duration of the mortgage. Section 3 discusses the average maturity and duration of the pool of mortgages; two countervailing influences of movement through time on average maturity and duration are identified. The market assumes these influences always exactly cancel each other out; this implies a strong assumption about the pattern of prepayments. Section 4 gives a numerical example, and section 5 summarizes the paper.
1 A further consideration dealt with in CG's papers is that, even if yields were computed correctly, there will still be uncertainty about the accuracy of our assumptions about the distribution of prepayments. This translates into yield uncertainty which CG show to be asymmetric; and this asymmetry makes it difficult to compare expected pass-through yields with those on ordinary securities with symmetric risk. 2While GNMA's are quoted on a price basis rather than a yield basis, the calculated yield relative to the yield deemed appropriate alTects buy and sell decisions and hence prices.
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For convenience, all notation in the paper is defined here: S R
= total continuous flow of payment on a non-prepaid mortgage, =flow of scheduled repayment of principal, I =flow of scheduled interest payments, P = principal scheduled to be outstanding at any time, r =coupon interest rate, = a point in time, r = index of time, D =date of prepayment of all outstanding principal on a mortgage (used to index mortgages), M (D ; t) = mean maturity of outstanding principal, at time t, of a mortgage with prepayment date D, IIl(D ;t) =duration at time t of a mortgage with prepayment date D, E(M; t) = mean maturity of the mortgage pool at time t, Y = predetermined life of the entire pool, feD) = frequency distribution of prepayment dates of mortgages in the pool, R(D) =one minus the cumulative distribution of prepayment dates, p =discount factor.
2. Individual mortgage The focus of this paper is the question of where in the term structure a particular mortgage pool falls, and how its place in the term structure varies through time. Two measures of 'maturity' will be employed. One is the maturity of the average dollar of unpaid principal. This will be referred to as 'average (or mean) maturity', and should not be confused with average maturity in the alternative and misleading sense of the life of the average mortgage in the pool.' The other maturity concept is 'duration' [see, e.g., Weil (1973)]. Duration focuses on the timing of interest receipts as well as principal repayments. If the pool consisted of conventional bonds, the maturity of every dollar of principal on a given bond would equal the term from now till the end of the bond's life (either the date it is called or the date it is scheduled to mature). In this case, oM(D; t)/ot= -1, where M(·) is the average maturity; that is, for every time unit we advance, the maturity date becomes one unit nearer. However, mortgages are more complicated because the principal is amortized. Repayment R of principal P is R(r)= -Per),
(1)
3This concept of average maturity is the same average maturity concept used by len and Wert (1966) in a related context.
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D.W: Mitchell, GNMA pass-through certificares
so that P(r)= -R(r)= -(S-I(r))=r'P(r)-S
(2)
where S is the total scheduled flow of repayment, I is interest, and r is the coupon rate, and where a dot indicates a time derivative. The solution of (2) is
S P(r)=Poert - - (ert -1),
(3)
r
where Po is the initial principal on the mortgage." We will assume that the outstanding principal follows the predetermined path given by (3) until time D, at which point the entire remaining principal is prepaid. Thus we do not allow for accelerated repayment of principal in a continuous fashion (which is rare in practice); the only early prepayment consists of complete prepayment at time D. (If D = Y, there is no prepayment.) Eq. (4) gives the average maturity of the principal on such a mortgage, computed at time t(t
(4)
In each half of (4) the term (-P(r)/P(t)) indicates the fraction of the principal outstanding at time t (at which time we are evaluating the maturity) which is scheduled to be repaid at time r. The first half of (4) indicates that prior to the time D of prepayment, any principal repaid on schedule at time r currently has term to maturity [r-t]. By the second half of (4), the principal scheduled for repayment at t z-D will be prepaid at time D, so it currently has term to maturity [D - t]. The change in the average maturity of the mortgage, as we move through time, is given by (5) for t < D : dM(D ;t) dt
p(t)
-1--·M(D·t» P(t) ,
-1 ,
(5)
where use has been made of the fact that the integral of (-P(r)/P(t)) from t to D, plus the same integral from D to Y, equals unity. It can be shown that the expression in (5) is between negative one and 4'rhe constraint that the mortgage be designed so that the principal is zero for that S = Tr"c'r I(en' -I).
t
= Y implies
D.W. Mitchell, GNMA pass-through certificates
551
zero." The intuition is this: As we move through time by 'one unit', each dollar of principal which is scheduled to remain unpaid becomes one time unit shorter-term. This is captured by the term negative unity in (5). However, as we move through time there is 'attrition' as some dollars are repaid according to schedule; this causes the short-term dollars to drop out of the averaging process, thereby lengthening the average maturity of those dollars which remain unpaid. The attrition effect is not the dominant effect. In the early years of the mortgage this latter effect will not be important because the attrition takes place very slowly. In the late years of the mortgage (just prior to either the prepayment date chosen by the borrower or the scheduled end of the mortgage's life) the attrition is relatively unimportant because there is not much difference between the maturities of the dollars which drop out of the outstanding principal and of those which remain. However, during the middle years the effect of attrition of shortterm dollars could be significant. Now consider the time behavior of duration. Duration m is given by (6):
m(D;t)
Jf Se-p(t-Il(r -t)dr+ P(D)e- P(D- I)(D - t) JfSe pIt Ildr+P(D)e p(D I)
(6)
The denominator in (6) is the present value, with discounting factor p, of all payments. The first term gives scheduled interest and principal payments (totaling S) until prepayment date D; the second term gives the unscheduled principal prepayment at time D. The numerator of (6) weights these flows by the term-length from the present time t until the time the payment occurs [r or D). The time derivative of duration, dm(D; t)/dt, is an extremely complicated expression which is not reported here. It is ambiguous in sign. It is worthwhile noting a simpler special case, however. If the discount factor p appearing in (6) can be approximated by zero, then we have dm(D,t) dt
S(D - t)P(D)+ S2(D - t)2/2
-1 +------'----------'------=----'---[S(D-t)+P(D)]l
(7)
This expression is between negative one and zero. The intuition is analogous to that for the other maturity concept: movement through time by one unit decreases by one time unit the term until payment of any future SBy using eqs. (2) and (3) and the expression for S in the previous footnote, we can evaluate the integrals in (4), yielding the following: M(D;t)=[ -exp(rD)+exp(rt)+rD ·exp(ry) -rt·exp(rY)]j[r·exp(rY)-r·exp(rt)]. Using this, (5) can alternatively be written as dMjdt= -1 +X . exp (rt)j [exp(rY)-exp (rtW , where X = [ -exp(rD)+exp(rt)+rD 'exp(rY) -rt'exp(rY))' dMjdt has the same sign as the following expression: [2+r(D-t)]-exp(rD - r Y) - exp (r Y - rt). Given the constraint that t;;i, D;;i, Y, this expression reaches its peak value of zero as D-. Y and t-s D, Since this expression cannot be positive, dMjdt cannot be positive.
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dollar; but there is attrition of the shortest-term dollars as they drop out of the averaging process. In this case the former influence dominates, so the expression in (7) is negative.
3. The pool of mortgages f(D) is the frequency distribution of prepayment dates of mortgages in the pool. H(t) is the fraction of the mortgages which have not been prepaid as of time t. Hencef(D)!H(t) is the frequency distribution of prepayment dates of mortgages outstanding at time t. For any D,o[f(D)/H(t)]/8t>O because as time goes on and attrition occurs the remaining mortgages each become a larger fraction of the pool. The discussion of this section is in terms of the mean maturity of principal in the pool (E(M; t»; However, the entire discussion is equally applicable to the duration the pool. The reader may simply substitute 'duration' for 'mean maturity of principal' everywhere, and substitute E(m; t) for E(M; r) everywhere. The mean term to maturity of a dollar of principal in the pool is a weighted average of the mean maturity of individual mortgages with alternative prepayment dates [each given by (4)]. The weights are the frequencies with which these prepayment dates occur. Thus
or
Y f(D) E(M;t)=J-(-) ·M(D;t)dD. rHt
(8)
We are primarily concerned with the time pattern of E(M; t). Taking the time derivative of (8), we have
dE(M;t) dt
f(t) E(M.t)+Sf(D).dM(D;t)dD H(t) , r H(t) dt Y
=( +)( +)+
J( +)( -
)dD~O,
(9)
r
where use has been made of the fact that M (D ; t) is zero when evaluated at D=t. Thus movement through time has an ambiguous effect on the mean maturity of the pool. On the one hand, movement through time decreases the average maturity of those mortgages remaining in the pool; this is the most obvious effect and is captured by the last half of (9). On the other hand, movement through time causes attrition as the shortest-term mortgages are prepaid and drop out of the pool; this increases the weight
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put on the longer-term mortgages and is captured by the positi ve first term in (9). When the pool is still young - that is, a short time after the mortgages were originated - there will not be much attrition [f(t) is near zero] so the latter factor should not be important. When the pool is old - as t approaches Y - there is not much difference in the mean maturities of the mortgages being prepaid and of those remaining [E(M; t) is near zero], so the latter factor again would not be too important. However, during the interim the mean maturity of the dollars of principal in the pool theoretically could actually be an increasing function of time. This is most likely to happen when we are near a mode in the distribution f(D) of prepayment times of all mortgages initially in the pool. The possibility that dE(' )/dt>O is just a theoretical oddity. The essential point, however, is that there is no reason to assume as the market does that dE( ' )/dt=O.6 In fact, most or all of the time it will be negative . Eq . (9) can be used to see the market's complicated implicit maturity assumption based on its pricing behavior. By pricing so as to give identical yields to pass-throughs with identical coupon rates and identical perceived prepayment distributions [f(D)J, but different ages, the market is attempting to give them the same yield. (As pointed out in section I, this is in fact the effect of that pricing mechanism unless the security is sold at a discount or premium, in which case the capital gain or loss is realized more quickly on the older security and the yields are not actually identical.) This effort to give securities of different ages the same yield, regardless of whether the term structure for financial markets in general is flat, implies a market assumption that average maturity is not a function of age. This is equivalent to assuming that the right-hand side of (9) equals zero, which obviously imposes a very strong restriction on the frequency distribution of prepayments. Only in a razor's edge case, and only for certain values of t, will this restriction in fact be met. 4. Numerical example We now turn briefly to a numerical example which illustrates the time behavior of average maturity and of duration. Consider a 30-year mortgage pool with $1000 principal initially and with an 8 % coupon rate. (Obviously the scale of the example does not matter.) Assume, following Cirillo (1979), that the prepayment dates of the individual mortgages are log-normally distributed. Prepayments are assumed to be made during the twelfth month 6This market assumption is quite aside from the fact that traders examine each old pool to see whether it is a 'fast payer'. Ob serving whether a pool is a fast payer is simply a way of formulat ing a frequency distribution of remaining prepayments. Given this j(D)/H(t) distribution, the market does not discriminate on the basis of pool age, and is therefore assum ing dE( . )dt = O.
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of each year." Assume also, for the duration calculations, that the current discount rate p is 10%. Under these assumptions, I have computed the month-by-month behavior of the mean maturity and the duration of the pool, for various parameterizations of the log-normal distribution. Since all parameterizations yielded similar results, I report only one, for which the mode is five years and the mean is a little less than fourteen years." Since all prepayments were assumed to be made in the twelfth month of any year, each twelfth month represents a 'mini-mode' of prepayments. As a result the mean maturity and duration rise by a few months. immediately after every twelfth month. Of course, this characteristic would not appear if we assumed prepayments are distributed smoothly over all months." Table 1 reports the mean maturity and duration for the first month of the pool's life and for every twelfth month thereafter. (In-between months are not reported in order to conserve space.) The numbers for duration are quantitatively very similar to those for average maturity. In each case the measure of maturity begins at a little less than ten years and declines by about four months for every year that passes. This is considerably less than the twelve-month-per-year decline that a conventional bond would show for average maturity. On the other hand, it is significantly greater than the zero decline assumed by the market pricing mechanism. Thus the market should discriminate by age of pool even among pools which are judged to be equally fast-paying, since pools of different ages have different maturities. Table 1 Mean maturity and duration computed at the ith month of the mortgage pool.
1 13 25 37 49 61 73 85 97 109 121 133 145 157 169
Mean
Duration
118.82 113.93 109.72 105.78 101.97 98.23 94.51 90.81 87.11 83.41 79.69 75.96 72.21 68.45 64.67
116.26 111.55 107.50 103.72 100.05 96.45 92.87 89.29 85.72 82.13 78.52 74.90 71.26 67.59 63.91
181 193 205 217 229 241 253 265 277 289 301 313 325 337 349
Mean
Duration
60.88 57.07 53.24 49.39 45.53 41.65 37.75 33.84 29.91 25.97 22.00 18.01 13.98 9.87 5.50
60.20 56.47 52.72 48.95 45.15 41.34 37.50 33.64 29.75 25.85 21.91 17.95 13.94 9.85 5.50
7Data on actual prepayment distributions are not released by HUD. 8The log-normal distribution isf(x)=A/B, where A=exp[ -(1nx-c)2/2a 2] and B=x(2rra 2)1. The mode is exp(c-a 2 ). The parameterization reported in the text is c=6 and a 2=4.3906. 9This more realistic assumption was not used because of computer core limitations.
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5. Conclusion The analogy between the offsetting influences on the behavior of the mean maturity or duration of an individual mortgage and those on a pool of mortgages is straight-forward. For the pool, moving forward through time alters the mean maturity and duration of each of the mortgages that make up the pool, tending to decrease the mean maturity and duration of the pool; attrition of the shortest-term mortgages tends to increase the mean maturity and duration of the pool. For an individual mortgage, moving forward through time alters the mean maturity of the dollars of unpaid principal and interest that make up the mortgage, tending to decrease the mean maturity and duration of the mortgage; attrition of the shortest-termto-maturity dollars of principal and interest tends to lengthen the duration and the mean maturity of the remaining principal. The mean maturity and duration of the mortgage pool (and thus of the pass-through certificates which comprise it) are most likely to increase over time, or decrease most slowly, when we are near a mode of the frequency distribution of prepayments. It should be pointed out, however, that predicting the f(D) distribution, for mortgage pools currently being formed, on the basis of past experience involves a rather large degree of uncertainty. Possibilities for structural shifts abound; for instance, if in the near future interest rates decline from historically high levels, the mean and mode of the frequency distribution of prepayments could be closer than anticipated to the origination time of high interest rate mortgages. Further, tracing the frequency distribution of prepayments through time would involve uncertainty even if the market assumption were more sophisticated than merely assuming constant maturity through time. Finally, the behavior of the mean maturity and duration over time should influence the behavior of both the price of the pass-through certificate on the secondary market and the interest sensitivity of that price. The more time the pass-through will spend as a relatively long-term security, the longer should its price remain sensitive to interest rate changes. For an investor who does not plan to hold the security throughout its entire life, this implies increased price risk as compared with a more conventional security.
References Cirillo, Susan Neary, 1979, Prepayment expectations for GNMA securities: Their impact on yield calculations, Federal Home Loan Bank Board, Invited Research Working Paper no. 25, June. Curley, Anthony and Jack Guttentag, 1974, The yield on insured residential mortgages, Explorations in Economic Research, no. 1, Summer (National Bureau of Economic Research) 114-161. Curley, Anthony and Jack Guttentag, 1977, Value and yield risk on outstanding insured residential mortgages, Journal of Finance, May, 403--412.
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D.lV. Mitchell, GNMA pass-through certificates
Jen, Frank and James Wert, 1966, Imputed yields of a sinking fund bond and the term structure of interest rates, Journal of Finance, Dec., 697-713. Weil, Roman, 1973, Macaulay's duration: An appreciation, Journal of Business, Oct., 589-592.
GNMA PASS-THROUGH CERTIFICATES Behavior Through Time of the Mean Maturity and Duration Douglas W. MITCHELL* University of Texas at Austin, Austin, TX 787/2. USA
Received May 1980, final version received January 1981
Th is paper adds to the literature on the illogical p ricing of GNMA pass-through cert ificates. It is already known that the market incorrectly computes the mean capital gains yield on such a pool of mortg ages as the capital gains yield on the mortgage of mean maturity. The present paper shows the strong assumptions about the behavior through time of the mean maturity or duration of the pool inherent in the market's attempt to give identical yields to securities issued at different times in the past. This market behavior implies a very strong assumption about the frequency distribut ion of prepaym ent dates of the individual mortgages.
1. Introduction GNMA pass-through certificates are portions of a pool of FHA-insured or VA-guaranteed mortgages. The fixed monthly payments from the borrowers are 'passed through' to the investors by a financial intermediary which deducts a servicing fee. The Government National Mortgage Association (GNMA) guarantees the timely monthly payments to the investors. In addition, any prepayments of principal by the borrowers are passed through to the investors. Because of the two types of guarantees (those of GNMA and of the FHA or VA) there is no default risk. The value of the pass-throughs in the secondary market fluctuates inversely with interest rates, creating price uncertainty unless the investor plans to retain the certificate throughout its life. The life of the certificate equals the life of the mortgage which is fully prepaid last, unless GNMA decides to force early liquidation of the certificate on the grounds that, due to large numbers of prepayments, the remaining cash flows are not worth the bookkeeping costs. The mortgages in the pool typically have scheduled lives of thirty years, and the mean time from origination to full prepayment is about twelve years. *1 am grateful to Eric Bond and two referees for helpful comments.
0378-4266/81/0000-0000/S02.75
©
1981 North-Holland
548
D.W: Mitchell, GNMA pass-through certificates
An important novelty about pass-throughs is the uncertainty about when the principal will be repaid. This uncertainty is analogous to the uncertainty about the maturity of a callable bond, but there are two complicating factors. First, the pass-through as an investment in a pool of mortgages which will be called (prepaid) at various times. Second, unlike the principal of a callable corporate bond, the principal on a pool of mortgages is scheduled to be repaid over time at an increasing rate rather than entirely at the end of the life. This complicates maturity calculations. Curley and Guttentag (CG) (1974 and 1977) showed that the rules of thumb used by the market in computing yields are incorrect if the certificates are selling at a discount or a premium. Given a perceived distribution of prepayment dates, the market incorrectly assumes that the yield on the pool equals the yield on a mortgage with the mean prepayment date. For a certificate sold at a discount this generates downward bias in the computed yield because the capital gains portion of yield is not a linear function of the prepayment date. CG point out that the correct procedure for computing yield involves listing all cash flows, including prepayments, making the appropriate assumption about how the cash receipts are reinvested, and applying the present value formula to obtain the yield.' The above considerations show that actual yield is computed wrong for a pass-through certificate sold at a discount or a premium. The purpose of the remainder of this paper is to show that, for older mortgage pools sold at par or otherwise, the market incorrectly computes the maturity of the investment as equal to the original maturity of the mortgage pool, thereby placing it in the wrong spot in the term structure and assigning it the wrong appropriate yield based on term structure considerations. Hence even if yields were computed correctly, the market assurries the wrong maturity and thus the wrong appropriate yield, so the security is incorrectly priced.' Section 2 considers the behavior through time of the maturity of the average dollar of unpaid principal on a single mortgage, and of the duration of the mortgage. Section 3 discusses the average maturity and duration of the pool of mortgages; two countervailing influences of movement through time on average maturity and duration are identified. The market assumes these influences always exactly cancel each other out; this implies a strong assumption about the pattern of prepayments. Section 4 gives a numerical example, and section 5 summarizes the paper.
1 A further consideration dealt with in CG's papers is that, even if yields were computed correctly, there will still be uncertainty about the accuracy of our assumptions about the distribution of prepayments. This translates into yield uncertainty which CG show to be asymmetric; and this asymmetry makes it difficult to compare expected pass-through yields with those on ordinary securities with symmetric risk. 2While GNMA's are quoted on a price basis rather than a yield basis, the calculated yield relative to the yield deemed appropriate alTects buy and sell decisions and hence prices.
D.lY. Mitchell, GNMA pass-through certificates
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For convenience, all notation in the paper is defined here: S R
= total continuous flow of payment on a non-prepaid mortgage, =flow of scheduled repayment of principal, I =flow of scheduled interest payments, P = principal scheduled to be outstanding at any time, r =coupon interest rate, = a point in time, r = index of time, D =date of prepayment of all outstanding principal on a mortgage (used to index mortgages), M (D ; t) = mean maturity of outstanding principal, at time t, of a mortgage with prepayment date D, IIl(D ;t) =duration at time t of a mortgage with prepayment date D, E(M; t) = mean maturity of the mortgage pool at time t, Y = predetermined life of the entire pool, feD) = frequency distribution of prepayment dates of mortgages in the pool, R(D) =one minus the cumulative distribution of prepayment dates, p =discount factor.
2. Individual mortgage The focus of this paper is the question of where in the term structure a particular mortgage pool falls, and how its place in the term structure varies through time. Two measures of 'maturity' will be employed. One is the maturity of the average dollar of unpaid principal. This will be referred to as 'average (or mean) maturity', and should not be confused with average maturity in the alternative and misleading sense of the life of the average mortgage in the pool.' The other maturity concept is 'duration' [see, e.g., Weil (1973)]. Duration focuses on the timing of interest receipts as well as principal repayments. If the pool consisted of conventional bonds, the maturity of every dollar of principal on a given bond would equal the term from now till the end of the bond's life (either the date it is called or the date it is scheduled to mature). In this case, oM(D; t)/ot= -1, where M(·) is the average maturity; that is, for every time unit we advance, the maturity date becomes one unit nearer. However, mortgages are more complicated because the principal is amortized. Repayment R of principal P is R(r)= -Per),
(1)
3This concept of average maturity is the same average maturity concept used by len and Wert (1966) in a related context.
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D.W: Mitchell, GNMA pass-through certificares
so that P(r)= -R(r)= -(S-I(r))=r'P(r)-S
(2)
where S is the total scheduled flow of repayment, I is interest, and r is the coupon rate, and where a dot indicates a time derivative. The solution of (2) is
S P(r)=Poert - - (ert -1),
(3)
r
where Po is the initial principal on the mortgage." We will assume that the outstanding principal follows the predetermined path given by (3) until time D, at which point the entire remaining principal is prepaid. Thus we do not allow for accelerated repayment of principal in a continuous fashion (which is rare in practice); the only early prepayment consists of complete prepayment at time D. (If D = Y, there is no prepayment.) Eq. (4) gives the average maturity of the principal on such a mortgage, computed at time t(t
(4)
In each half of (4) the term (-P(r)/P(t)) indicates the fraction of the principal outstanding at time t (at which time we are evaluating the maturity) which is scheduled to be repaid at time r. The first half of (4) indicates that prior to the time D of prepayment, any principal repaid on schedule at time r currently has term to maturity [r-t]. By the second half of (4), the principal scheduled for repayment at t z-D will be prepaid at time D, so it currently has term to maturity [D - t]. The change in the average maturity of the mortgage, as we move through time, is given by (5) for t < D : dM(D ;t) dt
p(t)
-1--·M(D·t» P(t) ,
-1 ,
(5)
where use has been made of the fact that the integral of (-P(r)/P(t)) from t to D, plus the same integral from D to Y, equals unity. It can be shown that the expression in (5) is between negative one and 4'rhe constraint that the mortgage be designed so that the principal is zero for that S = Tr"c'r I(en' -I).
t
= Y implies
D.W. Mitchell, GNMA pass-through certificates
551
zero." The intuition is this: As we move through time by 'one unit', each dollar of principal which is scheduled to remain unpaid becomes one time unit shorter-term. This is captured by the term negative unity in (5). However, as we move through time there is 'attrition' as some dollars are repaid according to schedule; this causes the short-term dollars to drop out of the averaging process, thereby lengthening the average maturity of those dollars which remain unpaid. The attrition effect is not the dominant effect. In the early years of the mortgage this latter effect will not be important because the attrition takes place very slowly. In the late years of the mortgage (just prior to either the prepayment date chosen by the borrower or the scheduled end of the mortgage's life) the attrition is relatively unimportant because there is not much difference between the maturities of the dollars which drop out of the outstanding principal and of those which remain. However, during the middle years the effect of attrition of shortterm dollars could be significant. Now consider the time behavior of duration. Duration m is given by (6):
m(D;t)
Jf Se-p(t-Il(r -t)dr+ P(D)e- P(D- I)(D - t) JfSe pIt Ildr+P(D)e p(D I)
(6)
The denominator in (6) is the present value, with discounting factor p, of all payments. The first term gives scheduled interest and principal payments (totaling S) until prepayment date D; the second term gives the unscheduled principal prepayment at time D. The numerator of (6) weights these flows by the term-length from the present time t until the time the payment occurs [r or D). The time derivative of duration, dm(D; t)/dt, is an extremely complicated expression which is not reported here. It is ambiguous in sign. It is worthwhile noting a simpler special case, however. If the discount factor p appearing in (6) can be approximated by zero, then we have dm(D,t) dt
S(D - t)P(D)+ S2(D - t)2/2
-1 +------'----------'------=----'---[S(D-t)+P(D)]l
(7)
This expression is between negative one and zero. The intuition is analogous to that for the other maturity concept: movement through time by one unit decreases by one time unit the term until payment of any future SBy using eqs. (2) and (3) and the expression for S in the previous footnote, we can evaluate the integrals in (4), yielding the following: M(D;t)=[ -exp(rD)+exp(rt)+rD ·exp(ry) -rt·exp(rY)]j[r·exp(rY)-r·exp(rt)]. Using this, (5) can alternatively be written as dMjdt= -1 +X . exp (rt)j [exp(rY)-exp (rtW , where X = [ -exp(rD)+exp(rt)+rD 'exp(rY) -rt'exp(rY))' dMjdt has the same sign as the following expression: [2+r(D-t)]-exp(rD - r Y) - exp (r Y - rt). Given the constraint that t;;i, D;;i, Y, this expression reaches its peak value of zero as D-. Y and t-s D, Since this expression cannot be positive, dMjdt cannot be positive.
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dollar; but there is attrition of the shortest-term dollars as they drop out of the averaging process. In this case the former influence dominates, so the expression in (7) is negative.
3. The pool of mortgages f(D) is the frequency distribution of prepayment dates of mortgages in the pool. H(t) is the fraction of the mortgages which have not been prepaid as of time t. Hencef(D)!H(t) is the frequency distribution of prepayment dates of mortgages outstanding at time t. For any D,o[f(D)/H(t)]/8t>O because as time goes on and attrition occurs the remaining mortgages each become a larger fraction of the pool. The discussion of this section is in terms of the mean maturity of principal in the pool (E(M; t»; However, the entire discussion is equally applicable to the duration the pool. The reader may simply substitute 'duration' for 'mean maturity of principal' everywhere, and substitute E(m; t) for E(M; r) everywhere. The mean term to maturity of a dollar of principal in the pool is a weighted average of the mean maturity of individual mortgages with alternative prepayment dates [each given by (4)]. The weights are the frequencies with which these prepayment dates occur. Thus
or
Y f(D) E(M;t)=J-(-) ·M(D;t)dD. rHt
(8)
We are primarily concerned with the time pattern of E(M; t). Taking the time derivative of (8), we have
dE(M;t) dt
f(t) E(M.t)+Sf(D).dM(D;t)dD H(t) , r H(t) dt Y
=( +)( +)+
J( +)( -
)dD~O,
(9)
r
where use has been made of the fact that M (D ; t) is zero when evaluated at D=t. Thus movement through time has an ambiguous effect on the mean maturity of the pool. On the one hand, movement through time decreases the average maturity of those mortgages remaining in the pool; this is the most obvious effect and is captured by the last half of (9). On the other hand, movement through time causes attrition as the shortest-term mortgages are prepaid and drop out of the pool; this increases the weight
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put on the longer-term mortgages and is captured by the positi ve first term in (9). When the pool is still young - that is, a short time after the mortgages were originated - there will not be much attrition [f(t) is near zero] so the latter factor should not be important. When the pool is old - as t approaches Y - there is not much difference in the mean maturities of the mortgages being prepaid and of those remaining [E(M; t) is near zero], so the latter factor again would not be too important. However, during the interim the mean maturity of the dollars of principal in the pool theoretically could actually be an increasing function of time. This is most likely to happen when we are near a mode in the distribution f(D) of prepayment times of all mortgages initially in the pool. The possibility that dE(' )/dt>O is just a theoretical oddity. The essential point, however, is that there is no reason to assume as the market does that dE( ' )/dt=O.6 In fact, most or all of the time it will be negative . Eq . (9) can be used to see the market's complicated implicit maturity assumption based on its pricing behavior. By pricing so as to give identical yields to pass-throughs with identical coupon rates and identical perceived prepayment distributions [f(D)J, but different ages, the market is attempting to give them the same yield. (As pointed out in section I, this is in fact the effect of that pricing mechanism unless the security is sold at a discount or premium, in which case the capital gain or loss is realized more quickly on the older security and the yields are not actually identical.) This effort to give securities of different ages the same yield, regardless of whether the term structure for financial markets in general is flat, implies a market assumption that average maturity is not a function of age. This is equivalent to assuming that the right-hand side of (9) equals zero, which obviously imposes a very strong restriction on the frequency distribution of prepayments. Only in a razor's edge case, and only for certain values of t, will this restriction in fact be met. 4. Numerical example We now turn briefly to a numerical example which illustrates the time behavior of average maturity and of duration. Consider a 30-year mortgage pool with $1000 principal initially and with an 8 % coupon rate. (Obviously the scale of the example does not matter.) Assume, following Cirillo (1979), that the prepayment dates of the individual mortgages are log-normally distributed. Prepayments are assumed to be made during the twelfth month 6This market assumption is quite aside from the fact that traders examine each old pool to see whether it is a 'fast payer'. Ob serving whether a pool is a fast payer is simply a way of formulat ing a frequency distribution of remaining prepayments. Given this j(D)/H(t) distribution, the market does not discriminate on the basis of pool age, and is therefore assum ing dE( . )dt = O.
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of each year." Assume also, for the duration calculations, that the current discount rate p is 10%. Under these assumptions, I have computed the month-by-month behavior of the mean maturity and the duration of the pool, for various parameterizations of the log-normal distribution. Since all parameterizations yielded similar results, I report only one, for which the mode is five years and the mean is a little less than fourteen years." Since all prepayments were assumed to be made in the twelfth month of any year, each twelfth month represents a 'mini-mode' of prepayments. As a result the mean maturity and duration rise by a few months. immediately after every twelfth month. Of course, this characteristic would not appear if we assumed prepayments are distributed smoothly over all months." Table 1 reports the mean maturity and duration for the first month of the pool's life and for every twelfth month thereafter. (In-between months are not reported in order to conserve space.) The numbers for duration are quantitatively very similar to those for average maturity. In each case the measure of maturity begins at a little less than ten years and declines by about four months for every year that passes. This is considerably less than the twelve-month-per-year decline that a conventional bond would show for average maturity. On the other hand, it is significantly greater than the zero decline assumed by the market pricing mechanism. Thus the market should discriminate by age of pool even among pools which are judged to be equally fast-paying, since pools of different ages have different maturities. Table 1 Mean maturity and duration computed at the ith month of the mortgage pool.
1 13 25 37 49 61 73 85 97 109 121 133 145 157 169
Mean
Duration
118.82 113.93 109.72 105.78 101.97 98.23 94.51 90.81 87.11 83.41 79.69 75.96 72.21 68.45 64.67
116.26 111.55 107.50 103.72 100.05 96.45 92.87 89.29 85.72 82.13 78.52 74.90 71.26 67.59 63.91
181 193 205 217 229 241 253 265 277 289 301 313 325 337 349
Mean
Duration
60.88 57.07 53.24 49.39 45.53 41.65 37.75 33.84 29.91 25.97 22.00 18.01 13.98 9.87 5.50
60.20 56.47 52.72 48.95 45.15 41.34 37.50 33.64 29.75 25.85 21.91 17.95 13.94 9.85 5.50
7Data on actual prepayment distributions are not released by HUD. 8The log-normal distribution isf(x)=A/B, where A=exp[ -(1nx-c)2/2a 2] and B=x(2rra 2)1. The mode is exp(c-a 2 ). The parameterization reported in the text is c=6 and a 2=4.3906. 9This more realistic assumption was not used because of computer core limitations.
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5. Conclusion The analogy between the offsetting influences on the behavior of the mean maturity or duration of an individual mortgage and those on a pool of mortgages is straight-forward. For the pool, moving forward through time alters the mean maturity and duration of each of the mortgages that make up the pool, tending to decrease the mean maturity and duration of the pool; attrition of the shortest-term mortgages tends to increase the mean maturity and duration of the pool. For an individual mortgage, moving forward through time alters the mean maturity of the dollars of unpaid principal and interest that make up the mortgage, tending to decrease the mean maturity and duration of the mortgage; attrition of the shortest-termto-maturity dollars of principal and interest tends to lengthen the duration and the mean maturity of the remaining principal. The mean maturity and duration of the mortgage pool (and thus of the pass-through certificates which comprise it) are most likely to increase over time, or decrease most slowly, when we are near a mode of the frequency distribution of prepayments. It should be pointed out, however, that predicting the f(D) distribution, for mortgage pools currently being formed, on the basis of past experience involves a rather large degree of uncertainty. Possibilities for structural shifts abound; for instance, if in the near future interest rates decline from historically high levels, the mean and mode of the frequency distribution of prepayments could be closer than anticipated to the origination time of high interest rate mortgages. Further, tracing the frequency distribution of prepayments through time would involve uncertainty even if the market assumption were more sophisticated than merely assuming constant maturity through time. Finally, the behavior of the mean maturity and duration over time should influence the behavior of both the price of the pass-through certificate on the secondary market and the interest sensitivity of that price. The more time the pass-through will spend as a relatively long-term security, the longer should its price remain sensitive to interest rate changes. For an investor who does not plan to hold the security throughout its entire life, this implies increased price risk as compared with a more conventional security.
References Cirillo, Susan Neary, 1979, Prepayment expectations for GNMA securities: Their impact on yield calculations, Federal Home Loan Bank Board, Invited Research Working Paper no. 25, June. Curley, Anthony and Jack Guttentag, 1974, The yield on insured residential mortgages, Explorations in Economic Research, no. 1, Summer (National Bureau of Economic Research) 114-161. Curley, Anthony and Jack Guttentag, 1977, Value and yield risk on outstanding insured residential mortgages, Journal of Finance, May, 403--412.
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Jen, Frank and James Wert, 1966, Imputed yields of a sinking fund bond and the term structure of interest rates, Journal of Finance, Dec., 697-713. Weil, Roman, 1973, Macaulay's duration: An appreciation, Journal of Business, Oct., 589-592.