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Hi everyone, here are my promised formulas. Sorry it’s a bit long winded. They mightn’t be strictly accurate from a purist economist point of view, but I’m sure they are a reasonable approximation. Hopefully I haven’t miscued the calculator again.
Define the following variables:
A = repayment Amount per compounding period
P = Present worth
F = Future worth
R = Residual amount owing after n periods if A is not enough to pay off P in that time.
r = interest per compounding period (usually interest pa/12) (interest is expressed as a ratio, not a percentage, eg in the example above, r = 0.07/12 = .00583333
The term (1+r) to the power n occurs often so I’ll call it “Q” to tidy up the formulas, ie Q = (1+r) to the nth power. (I can’t do superscripts)
I’ll use * as a multiply sign and / as a divide sign.Formulas used:
Capital Recovery Factor {A/P} = (r*Q)/(Q-1) ; multiply this by P to get the monthly repayment amount needed to pay off P after n compounding periods.
Single Payment Present Worth Factor {P/F} = 1/Q ; The present worth of a single future payment F in n compounding period’s time.
Number of periods required to pay off the principal P with a payment A and interest r per period is given by reverse engineering {A/P} to extract n as follows:
n = LOG(1+r/[A/P – r])/LOG(1+r) ; The LOG function is to base 10 or natural log, it doesn’t matter, so long as you use the same for both.The amount of loan still owing R after n periods if A is too small to pay off the principal P is given by R = Q*[P – A*(Q-1)/(r*Q)]
Applying these to the example:
P = $200,000, r = .07/12, n = 120 gives: {A/P} = .0116108, so A = (200k * {A/P}) = $2322.17 as shown in RentMaster’s P&I example. This will pay off the loan in 10 years, ie n = 120 months.Finding the present worth of these payments is basically a reversal of this formula, but using the inflation rate r = .035/12 (assuming a nominal inflation rate of 3.5% pa). Above, P was the initial loan principal, so I’ll call the real present worth of the repayments with inflation taken into account as P(real). So {A/P} now is .00988859 which gives P(real) = $2322/{A/P} = $234816.
Similarly P1(real) of A = $1166 for the I/O loan is $1166/{A/P} = $117913
P2(real) of the future payment F of 200k in 120 months time is given by P2(real) = 200k*{P/F} = $141009 (for r = .035/12).
Now total P(real) = P1(real) + P2(real) = $258922To find the effect with tax at 48.5%, firstly interest only is easy, as each payment is purely interest, and identical, so the net payment per month after tax refund is 0.515 * 1166 = $600.49 (Note that I’ve simplified the tax refund by assuming it is paid monthly, but this will be a reasonable assumption if you have put in a “withholding variation” to the ATO, ie you can get your tax reduced if you are negatively geared, so your fortnightly pay is correspondingly higher.) The P1(real) is now $600.49/{A/P} = $60726, which added to the same P2(real) above gives total P(real) after tax effect for I/O = $201,735.
To find the tax effect on P&I, I only know how to do uniform series, ie where A is always the same. In reality, most of your early payments are interest, and most of the final payments are principal, so you get more tax refund at the start then at the end. After the tax refund “each month” you effectively pay a smaller net amount per month. The net payment increases as the loan diminishes. Eg, at the end of period 1, interest payable = .07/12 * 200k = $1166, reduced by refund to 0.515*1166 = $600.49. At the end of period 60 (half way in time), residual R = q*[P – A*(q-1)/(r*q)] where q = (1+r) to the 60th power, ie R = $117286 which is the amount of principal still owing. Interest on this amount in the 61st period would be $684, which after tax refund would be $352.
To simplify the maths, I assumed the tax refund would be used to pay off principal (or sit in an offset account which give the same result). This would be equivalent to a P&I loan with an interest rate equal to .07*.515 or .03605 (3.605%). The loan would be paid off earlier if this were the case, and to calculate the reduced n:
n = LOG(1+r/[A/P – r])/LOG(1+r) The A/P terms refer to the actual payment per period divided by the principal, which is 2322/200000 = .01161 and r = .03605/12 which results in n = 99.82, so assume n = 100.
Now P(real) of the 100 payments of $2322 is $201150, calculated as before, with r = .035/12.Hope all that makes sense. I’ve shortcutted some of the detail, as it is long enough already.
Jim. (about to jump into my fox hole to dodge the bullets from real economists
])It works fine Sach. The effect isn’t that dramatic though, as it depends fully on how much you have in the offset account. There are lots of con jobs out there that fool people into thinking they are dramatically shortening their loans, but in reality they just end up paying a lot more principal.
I prefer the offset account arrangement, because it puts a “firewall” between your private and non private finances. (careful to not say “parts”!)[]
J.
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