Inflation adjusted total formula for cost of mortgage

Question

True cost of a mortgage when considering inflation has an answer by a Chris Degnen with the relevant formula(s): adjusted = (p - (1 + inf)^-n*p)/inf;

pv = Σ p2 (1 + r2)^-k for k = 1 to n ∴ pv = (p2 (1 + r2)^-1) + (p2 (1 + r2)^-2) + (p2 (1 + r2)^-3).

My first question is whether or not these are performing the same calculations. Unlike 'real interest' (which simply subtracts rate of inflation from rate of interest) this method is supposed to account for the effect of inflation on each payment on the amortization schedule, which to me seems like the right way to do it.

In my mind the inflation adjusted value of the 2nd month's payment should be derived by dividing the first month's payment by the monthly inflation rate (I'll come back to this). So if we had an arbitrary monthly installment of $100 we would divide that by a monthly inflation rate of say 1.05 (5%). We would then divide the product by 1.05 for the third month and so on.

If this is not what's happening in the above formulas please explain why this would not be the correct method. If this is indeed correct what I would really like is an in depth illustration of how the individual parts of the formula(s) are accomplishing what's playing out in my head. If that's going to be too verbose to post I'd appreciate some useful links instead.

As an aside is it convention to show annual inflation as a nominal rate as Chris does? Nominal inflation = 5%; 5%\12 = 0.41666% monthly inflation; (1+0.0041666)^12 - 1 = 0.05116 = 5.116% for effective rate. If 5% is actually the effective rate then you'd need the 12th root of that to get the monthly rate. Or maybe they add every month's rate and divide by 12 (mean?), I don't know.

Answer 1

Re: "My first question is whether or not these are performing the same calculations."

adjusted = (p - (1 + inf)^-n*p)/inf

pv = Σ p2 (1 + r2)^-k for k = 1 to n

Notwithstanding the change of variable names, these are indeed performing the same calculation steps, as shown below (image from the original answer).

The formula is derived from the summation by induction. See Induction of closed form of summation.

2nd question

"In my mind the inflation adjusted value of the 2nd month's payment should be derived by dividing the first month's payment by the monthly inflation rate ... If this is not what's happening in the above formulas please explain why this would not be the correct method."

When a loan is agreed the fixed payments are set depending on the bank's loan-interest rate. Inflation doesn't come into it (except the bank might consider it in setting their interest rates). So your payments are set depending only on the bank's interest rate. Let's say the payments are $10 per month. The first $10 cost $10 present value. If inflation works in your favour the last payment may only cost $8 present value, (although it will cost $10 future value). Inflation simply reduces the buying power of your cash in the future.

Thirdly, Re: "As an aside is it convention to show annual inflation as a nominal rate?"

Whether rates are specified as nominal or effective rates depends on country standards. Generally (as far as I know) the US uses nominal rates and Europe uses effective rates.

The monthly rate calculated from a nominal annual rate compounded monthly isn't so much a mean rate. Rather, the nominal annual rate is calculated by multiplying the actual monthly rate by twelve. The nominal annual rate isn't much good for anything except comparing against other nominal rates (of the same compounding interval), and dividing by twelve, which is what it's intended for. It can't be used for annual compounding. You need the effective annual rate for that.

See Effective / Nominal Interest Rates

The nominal rate system is a hang-over from the days before people had calculators on their iPhones. Dividing by twelve is simpler than taking the twelfth root.

See also. Annual percentage rate - References

reference 3. US-Federal-Reserve-R1314

The "Truth in Lending Act" passed in 1968 did not incorporate the mathematically-true annual percentage rate, because the true calculation used compounding (sometime fraction compounding), which was not readily available. The result on expression of the APR on credit cards uses a Nominal (simple interest) method ... which can far from the truth. The Truth in Lending Act should be changed to the mathematically-true (EFFECTIVE) APR from the untrue (NOMINAL) APR, merely by changing the word in act from "multiplied by" to "compounded for".

(APR in Europe uses the Effective APR - ref. link)