# Chapter 13. The annuity mortgage Interest discrete versus continuous compounding. Reading material for this chapter: none recommended

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1 Chapter 13 The annuity mortgage Reading material for this chapter: none recommended We formulate a mathematical model for the simplest possible mortgage, the annuity mortgage. We assume that interest is compounded continuously, meaning that ODEs provide the necessary framework to write out the model. One feature of the annuity mortgage is that if the repayment level is too low, then not only is the capital not paid off, but the interest payments explode. We will see the dynamics of this undesirable process and others in this chapter Interest discrete versus continuous compounding Recall, in Chapter 2 we learned how the exponential function was discovered in the context of compound interest. Specifically, we looked at a sum of money P 0 placed on deposit at time t = 0 and compounded after every t periods of time at a rate r. We found a formula for the amount of money in the account after n periods, which we recall here as P (T ) = P 0 (1 + r t) n, T = n t. (13.1) We then looked at what happens when t 0, in such a way that T = n t remains finite. We do that again here. From Equation (13.1) we have P (T ) = P (T t) (1 + r t). Re-arranging gives P (T ) P (T t) t 119 = rp (T t). (13.2)

2 120 Chapter 13. The annuity mortgage We take the limit as t 0. We have, P (T ) P (T t) lim t 0 t We put these results into Equation (13.2) and obtain = dp dt, T lim P (T t) = P (T ). t 0 dp dt = rp. (13.3) Thus, in the limit as t 0, we pass from discrete compounding of interst to continuous compounding of interest, and Equation (13.1) can be recast as an ordinary differential equation, specifically, Equation (13.3). We use the model of continuously-compounded interest for the rest of this chapter. Equation (13.3) was related to a person saving money in the bank in this case the bank can be thought of as the debtor and the person saving as the creditor. For a mortgage the roles are reversed the bank is the creditor and the person in question is the debtor. In any case, the mathematics is the same, only we switch notation and use S for the amount of money owed by the debtor (S is the standard notation for this kind of work). After switching the notation, Equation (13.3) reads ds dt = rs, where r is the interest rate, S is the amount owed by the debtor, and t is time (in units of years). Let S 0 denote the sum advanced by the bank to the debtor at t = 0 (called the capital). Using S(t = 0) = S 0, the solution of the model ODE is S(t) = S 0 e rt. This model is similar to the Malthusian population model, except here, our population is the sum of money outstanding to the bank, and the pertinent rate is the interest rate payable on the loan Repayments After a time T = 1 month = 1 year/12, the debtor makes a repayment R. The amount owed is then reduced to S 0 e rt R.

3 13.3. Repayments schedule 121 The debtor continues to do this for N months, whereat the loan is paid off in its entirety. The first few actions by the debtor are summarized in Table The aim of the rest of this section is to find Action Time Sum outstanding to the bank Notation Zeroth repayment 0 S 0 = S(0) Time T elapsed T S 0 e rt = CP (1) First repayment T S 0 e rt R = S(1) Time 2T elapsed 2T S(1)e rt = e rt (C 0 e rt R) = CP (2) Second repayment 2T e rt (C 0 e rt R) R = S(2) Time 3T elapsed 3T S(2)e rt = e rt [e rt (C 0 e rt R) R] = CP (3) Third repayment 3T e rt [e rt (C 0 e rt R) R] R = S(2) Table 13.1: The actions taken by the debtor in relation to the bank up to and including the first two repayments. a formula for the monthly amount R in terms of S 0, r, and N, such that the sum outstanding to the bank is reduced to zero after N repayments (paying of a mortgage in this way, so that interest is paid and the outstanding amount is reduced to zero, is called amortization. 1 ) 13.3 Repayments schedule Observe that in lines 4 and 6 of the table, the exponential factor is e rt because only the interval is relevant for the compounding, i.e. 2T T = T. Thus, S(3) = e rt [e rt (C 0 e rt R) R] R = C 0 e 3rT Re 2rT Re rt R. Guessing the pattern, the sum outstanding to the bank just after the n th repayment will be S(n) = S 0 e nrt R ( e (n 1)rT + e (n 2)rT + e rt + 1 ), or n 1 S(n) = S 0 e nrt R e jrt. j=0 1 For the actuaries and the financially-inclined, EBITDA, or earnings before interest, tax, depreciation, and amortization is one of the key measures of a firm s financial health.

4 122 Chapter 13. The annuity mortgage But the sum here is a geometric progression, with multiplier ρ = e rt. In case you haven t seen the formula for summing such a series (I always forget it), here is a neat trick: R(n) = R ( 1 + ρ + + ρ n 1), ρr(n) = R ( ρ + + ρ n 1 + ρ n), ρr(n) R(n) = R (ρ n 1), R(n) = R (ρn 1), ρ 1. ρ 1 Hence, S(n) = S 0 e nrt R ( e nrt 1 ) e rt 1 We want to work out N, the term of the loan in months such that (13.4) S(N) = 0. This is easy now: we set Rearranging, as required. S 0 e NrT R ( e NrT 1 ) e rt 1 R = S 0e ( NrT e rt 1 ), e NrT 1 = 0. Quantitative example Freddy Mae finances the purchase of his first house with a fixed-rate mortgage with the following properties: Capital amount: S 0 = e 200, 000. Fixed rate at 4% per annum. Term: 30 years. Calculate the total sum that Freddy will pay to the bank at the end of the mortgage term.

5 13.4. Interest-only mortgage 123 Solution: Freddy makes N monthly payments, each of size R, so the amount he pays to the bank over the 30 years is NR. We have R = S 0e ( NrT e rt 1 ), e NrT 1 ( = (e 200, 000) e(30 12) 0.04 (1/12) e 0.04(1/12) 1 ), e (30 12) 0.04 (1/12) ( 1 = (e 200, 000) e e 0.04/12 1 ), e = e Hence, NR = e 344, 016. In this calculation, we have calculated the monthly repayment amount explicitly in the first instance, giving R = e rounded to the nearest cent. However, if one were to keep a large number of significant figures here, say R = e , and then multiply N R = (e ), one would obtain NR = e 344, , correct to the nearest cent. However, given the large (nominal) profit made by the bank in this transaction, quibbling over 10 cents at the end of 30 years would seem a bit silly Interest-only mortgage In an interest-only mortgage, the sum outstanding at the end of the mortgage term is still S 0, such that S(N) = S 0, or Hence, S 0 e NrT R ( e NrT 1 ) e rt 1 = S 0. ( R = S 0 e NrT 1 ) ( ) e rt 1, e NrT 1 ( = S 0 e rt 1 ) It is doubtful whether such an arrangement ever makes sense to a homeowner, although it may well work for a speculator hoping to secure a capital gain, as in the following example:

6 124 Chapter 13. The annuity mortgage Example: A certain G. Gecko finances the purchase of a commercial property with an interestonly mortgage with the following properties: Capital amount: S 0 = e 1, 000, 000. Fixed rate at 7.5% per annum. Term: 20 years, with monthly repayments. Gecko is able to rent out the property, for a net yield of 8%. Compute the monthly repayment. Will Gecko secure a net montly income from the transaction? Additionally, consider the following: The purchase of the property is financed entirely by the mortgage, i.e. it is a 100% mortgage. The market value of commercial property increases at a constant (continuous) rate of 2% per annum, averaged over the term of the loan. There is no inflation and Gecko is able to avoid all taxes and transaction costs. Will the intrepid Gecko be able to make a profit (capital gain) on this deal at the end of the 20 years? Solution: For the first part, use R = S 0 ( e rt 1 ) = (1, 000, 000) ( e (0.075 (1/12)) 1 ) hence R = e6, correct to the nearest cent. For the income part, the net monthly income is the rent minus the repayment: net monthly income = monthly rental income monthly mortgage repayment, = (0.08/12) (e1, 000, 000) e6, , = e 6, e 6, , = e For the capital gain part, the property owned by Gecko appreciates in value every year, and by the

7 13.5. Discussion 125 end of twenty years is worth (at market value) S 0 e = (e1, 000, 000)e = e1, 491, However, the net worth of Gecko s investment in the property is its market value, minus the outstanding value of any loan secured on the property, which is S 0, hence net worth of investment = S 0 e S 0, = e 1, 491, e 1, 000, 000, = e491, This is the capital gain made by Gecko on the deal. Gecko s advantage from the deal therefore comes in two parts: the monthly income over the duration of the loan and the capital gain at the end of the term Discussion In practice, interest rates in Ireland and the UK cannot be fixed over the course of a loan, so the model considered in this section is not realistic. In Ireland and the UK, fixed-rate mortgages have a rate fixed over a period of five years, typically. Such a rate is a synthetic rate, and is obtained when a bank purchases an interest-rate swap in a financial market. Here, a third party swaps a fixed rate for a floating rate, and gambles that the floating rate will be less than the fixed rate it has offered to the bank. It is called a swap because cash flows based on a notional principal are swapped between the parties. In the US, mortgages are originated by banks and sold on in bundles to investors in the form of bonds. The bonds are again synthetic, as they are built from component loans. Typically, they are guaranteed by a government agency. Thus, the creation of fixed-rate mortgages requires strong government intervention. More philosophically, lending money at interest (or usury ) has made many thinkers and cultures uneasy down through the ages. In the previous examples, the fictitious Freddy Mae paid back to the bank almost twice the initial purchase price of his house, while in a separate example, Gecko made a huge gain through his speculations. In all of these examples, one of the parties to the relevant transaction was able to make a large profit without having to produce anything or expend any physical effort. Of course, this criticism is simplistic but these observations have nevertheless been at the heart of the opposition of many cultures to usury down through the ages, an opposition which is summed up by the following passages from the holy scriptures of different religions:

8 126 Chapter 13. The annuity mortgage Take no usury or interest from him; but fear your God, that your brother may live with you. You shall not lend him your money for usury, nor lend him your food at a profit Leviticus 25: Also, O ye who believe! Devour not usury, doubling and quadrupling (the sum lent). Observe your duty to Allah, that ye may be successful Quran 3:130. One does not to accept such a simplistic characterization of interest, or to share all of these sentiments, to recognize that these passages contain some truth, namely that the compounding of interest is a powerful force that should be strongly regulated in the public interest.

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