BASIC MATHEMATICS
OF ILS Mortgages
Kindly
Contributed by W J Waghorn
OF ILS Mortgages
I am indebted to Bill Waghorn for his guidance on terminology and the choice of mathematical symbols used throughout the mathematics of Mortgage Finance.
It should be noted that Mr. Waghorn uses the suffix ‘0’ to represent the values at the start of the first year whereas I have used the suffix ‘1’ in my mathematics.
DEFINITIONS
It should be noted that Mr. Waghorn uses the suffix ‘0’ to represent the values at the start of the first year whereas I have used the suffix ‘1’ in my mathematics.
DEFINITIONS
DEFINITIONS
'ILS Mortgages' are those which were originated by Edward C D Ingram and are referred to as Ingram Lending and Savings (ILS) Mortgages.
'Entry Cost' is the cost of the first year's annual payment 'P_{1}'
It is assumed that readers have already read 'MATHS STRAIGHT IN'
For those who may not have read that:
Most students of mortgage finance are told that the repayments are fixed unless or until the interest rate changes. The monthly or annual repayments are then calculated over the repayment period of N years for the given rate of interest r% p.a.
When the interest rate r% p.a. changes the new repayment calculation once more assumes that the interest rate will never change and then the new cost of the repayments of capital and interest is calculated over the remaining period.
With ILS Mortgages it is assumed that the rate of repayments, or more accurately, annual payments 'P' will rise (escalate) at e% p.a. over the whole period, and that everything else like the rate of Average Earnings Growth, AEG% p.a. and the rate of interest r% will remain fixed.
For more information on the definitions please read the definitions at the start of 'MATHS STRAIGHT IN'
And read the whole of that (later) for more on how to manage the repayments and the changing level of payments as conditions change.
This page finds the equations for:
(a) the total repayment period and
(b) the entry cost
for a given repayment period
Assume a loan of L_{0} at a fixed interest rate of r%,
with annual repayments starting at P_{1} and escalating at e%.
To work out the repayment period N 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.
To keep the algebra simple, let R=1+r/100 and let E=1+e/100.
Then on completion the loan will have accumulated N years of
compound interest. In other words, L_{N }= L_{0} *
R^{N}
At the same time, each payment P_{n} (= P_{1}*E^{n1})
will have accumulated (Nn) years of compound interest, and thus be worth P_{1}*E^{n1}*R^{Nn}. The
total value of these repayments, from the first in year 1 to the last in year
N, will thus be
T = P_{1}*E^{0}*R^{N1} + P_{1}*E^{1}*R^{N2} +
P_{1}*E^{2}*R^{N3} + … + P_{1}*E^{N2}*R^{1} +
P_{1}*E^{N1}*R^{0}.
To calculate this total, note that
T*E = P_{1}*E^{1}*R^{N1} + P_{1}*E^{2}*R^{N2} +
P_{1}*E^{3}*R^{N3} + … + P_{1}*E^{N1}*R^{1} +
P_{1}*E^{N}*R^{0}
T*R = P_{1}*E^{0}*R^{N} +
P_{1}*E^{1}*R^{N1} + P_{1}*E^{2}*R^{N2} +
… + P_{1}*E^{N2}*R^{2} + P_{1}*E^{N1}*R^{1}
Subtracting these two (thus cancelling the shaded areas), we get:

T*R  T*E = P_{1}*E^{0}*R^{N}  P_{1}*E^{N}*R^{0}
whence T = P_{1}*(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 P_{1}*(R^{N}  E^{N})
/ (R – E) = L_{N } = L_{0} * R^{N}
Multiplying the two sides by (R  E) and dividing by P_{1}*R^{N} we
get
(1 – (E/R)^{N}) = (R – E)*L_{0}/P_{1}
That equation enables us to calculate the initial payment required
to achieve completion after a given period of N years: 
P_{1} = (R – E) * L_{0} / (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 – (R –
E)*L_{0}/P_{1}, whence: 
N = log(1 – (R – E)*L_{0}/P_{1}) /
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*P_{1}*R^{N1}, and the termination condition is thus
N*P_{1}*R^{N1} = L_{0}*R^{N}. This
reduces to: 
P_{1} = L_{0}*R/N …………………W3a
Or
N = L_{0}*R/P_{1}, ………………..W3b
These are the characteristic equations of the original ‘Ingram
Sliced Mortgage’[1], which entails payment escalation at the
interest rate (i.e. E = R).
E&OE, of course.
'ILS Mortgages' are those which were originated by Edward C D Ingram and are referred to as Ingram Lending and Savings (ILS) Mortgages.
'Entry Cost' is the cost of the first year's annual payment 'P_{1}'
It is assumed that readers have already read 'MATHS STRAIGHT IN'
For those who may not have read that:
Most students of mortgage finance are told that the repayments are fixed unless or until the interest rate changes. The monthly or annual repayments are then calculated over the repayment period of N years for the given rate of interest r% p.a.
When the interest rate r% p.a. changes the new repayment calculation once more assumes that the interest rate will never change and then the new cost of the repayments of capital and interest is calculated over the remaining period.
With ILS Mortgages it is assumed that the rate of repayments, or more accurately, annual payments 'P' will rise (escalate) at e% p.a. over the whole period, and that everything else like the rate of Average Earnings Growth, AEG% p.a. and the rate of interest r% will remain fixed.
For more information on the definitions please read the definitions at the start of 'MATHS STRAIGHT IN'
And read the whole of that (later) for more on how to manage the repayments and the changing level of payments as conditions change.
This page finds the equations for:
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