# Future value vs total amortized cost of a loan

My goal is to determine the total cost of a mortgage loan (principle + interest for 30 years) I assumed that the future value(FV) of a loan is the total amortized cost of the loan. Is that assumption incorrect?

For a loan amount of 480,000 at an annual interest rate of 4.414%,

My FV calculation: PV * (1 + r)n = `480,000 * (1 + 0.04414/12)^360` = `1800026.445`

Using an amortization chart in google sheets: ``` INTEREST PMT \$386,744.46 PRINCIPAL PMT \$480,000.00 TOTAL \$866,744.46 ``` For principal pmt. for every period, I used the following expression : `PPMT(4.414%/12,<period_no>,30*12,480000)`

For interest pmt. for every period, I used the following expression: `IPMT(4.414%/12,<period_no>,30*12,480000)`

As you can see, I ended up with two different values(1800026.445 and \$866,744.46). Wondering if there is a quicker way to compute the total cost of a loan if future value is not the right path?

## 2 Answers

With the following variables and values

``````s is the principal
r is the periodic rate
n is the number of periods
d is the periodic payment

s = 480,000
r = 4.414/100/12
n = 30*12
``````

The formula for the periodic payment is

``````d = r (1 + 1/(-1 + (1 + r)^n)) s = 2407.62

∴ total payments = n*d = 866,744.46

& total interest = n*d - s = 386,744.46
``````

Your calculation ...

``````s (1 + r)^n = 1,800,026.44
``````

gives the future value of the loan if no repayments were being made, but repayments decrease the loan amount throughout the term of the loan.

Loan formula derivation

It is usual and simpler to derive the formula by equating the sum of the present values of the payments to the present value of the principal The closed form is found by induction from the summation.

Future values can also be used In both cases, rearranging for the payment gives

``````d = r (1 + 1/(-1 + (1 + r)^n)) s
``````
• Is it correct that since the original value( or present value PV) is decreasing in a loan's case, compounding take place periodically on the decreased PV? Any idea if the formula you provided can be derived from the FV formula? – Vivek Oct 1 '18 at 1:01
• @Vivek Yes, in the future values calculation compounding is applied to each payment, from the first `d*(1 + r)^(n - 1)` to the last `d(1 + r)^0`. Derivation added to my answer. – Chris Degnen Oct 1 '18 at 9:47

The typical fixed rate mortgage has the feature that the monthly principal payment and the monthly interest portion when added together doesn't change during the life of the mortgage. That is why the early payments are heavy on the interest portion and the later payments are biased towards principal payment.

To get the total payment it is 360 times the monthly payment. Everything above the initial mortgage amount is the interest.

The first number you calculated would be what you would have owed if you paid \$0 per month and let the debt grow.