LESSON 2 - GENERAL LENDING EQUATIONS

Ongoing re-writes, updates, and additional material are noted on the LATEST UPDATES page.

NOT FOR NON-MATHEMATICIANS - YOU CAN SKIP THIS LESSON

This lesson is more of an appendix

leading to some useful equations at the end

Other lessons are both simple and exciting with almost zero algebra.

LESSON 2
GENERAL EQUATIONS FOR MORTGAGE FINANCE

Thanks go to W J Waghorn who kindly worked out the general equation that determines the entry cost (the initial payments) for a mortgage  in which the repayments are given a specified escalation rate p.a. which is not necessarily zero. In fact it may even be negative and still repay the Mortgage.

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To work out the repayment period N for a loan L₁ with payments escalating at e% p.a. consider the repayments being invested, at the same interest rate, until the end of the loan (i.e. for N years), and only then used to pay it off.

Then on completion, the loan will have accumulated N years of compound interest. 
In other words, L_N = L₁ * R^N 

At the same time, each payment P_n (= P₁*E^n-1) will have accumulated (N-n) years of compound interest, and thus be worth P₁*E^n-1*R^N-n. The total value of these repayments, from the first in year 1 to the last in year N, will thus be

T  = P₁*E⁰*R^N-1 + P₁*E¹*R^N-2 + P₁*E²*R^N-3 + … + P₁*E^N-2*R¹ + P₁*E^N-1*R⁰. 

To calculate this total, note that

T*E  = P₁*E¹*R^N-1 + P₁*E²*R^N-2 + P₁*E³*R^N-3 + … + P₁*E^N-1*R¹ + P₁*E^N*R⁰ 

T*R  = P₁*E⁰*R^N + P₁*E¹*R^N-1 + P₁*E²*R^N-2 + … + P₁*E^N-2*R² + P₁*E^N-1*R¹ 

Subtracting these two (thus cancelling the un-emboldened areas), we get: -

T*R - T*E = P₁*E⁰*R^N - P₁*E^N*R⁰

whence T = P₁*(R^N - E^N) / (R – E)

The loan is paid off after N years if this accumulated value equals the loan value, which implies that P₁*(R^N - E^N) / (R – E) = L_N  = L₁ * R^N

Multiplying the two sides by (R - E) and dividing by P₁*R^N we get

      (1 – (E/R)^N) = L₁/P₁ * (R – E) 

That equation enables us to calculate the initial payment (the entry cost) required to achieve completion after a given period of N years at e% p.a. escalation in the annual payments: -

P₁  = L₁ * (R – E) / (1 – (E/R)^N) ………………………… W1.

It also enables us to calculate the termination period for a given initial payment. It gives us (E/R)^N = 1 – L₁/P₁*(R – E), whence: -

N = log(1 – L₁/P₁*(R – E)) / log(E/R) ……………………… W2.

Note that the above algebra does not work if R = E. But in that case the initial equation for T simply reduces to T = N*P₁*R^N-1, and the termination condition is thus

N*P₁*R^N-1 = L₁*R^N. This reduces to: -

P₁ = L₁*R/N …………………W3a

Or

N = L₁*R/P₁, ………………..W3b

which are the characteristic equations of the original ‘Ingram Sliced Mortgage’(See NOTE 1), which entails payment escalation at the interest rate (i.e. E = R).

E&OE, of course. 

NOTE 1: A reference to the early exploratory mathematics done by Edward in the 1980s or earlier. The sliced mortgage simply sliced an area representing the loan into N equal slices. The slices were allowed to grow as interest was added and one slice was paid off every year. e% was thus the same as the interest rate. Not a very good idea!

HOMEWORK

Homework to be sent to eingram@ingramsure.com

every week please.

Re-arrange the equations  W1 and W2 by substituting for e=0%.

What do you get?

1. How does it compare with the equation with which people calculate the cost of a Level Payments (LP) Mortgage? 

2. Assuming it does, would these W1 and W2 equations represent a more general equation for Mortgage repayments showing that it may be possible to switch from an LP Mortgage to a Mortgage with a smoother rate of increase in the payments than just a jump upwards?

3. Does the equation also work for negative values of e%?

Discuss.