Mortgage and Investment Mathematics

Mortgage and Investment Mathematics

Start With a Cash-Flow Timeline

A dollar available now can earn a return before a later dollar arrives. Present value discounts a future receipt to the valuation date; future value compounds an amount forward. Let $i$ be the effective interest rate per period and $n$ the number of matching periods. For a single amount, $PV=FV/(1+i)^n$ and $FV=PV(1+i)^n$.

An annual nominal mortgage rate of 6 percent with monthly payments ordinarily uses $i=0.06/12=0.005$ and 360 periods for a 30-year loan. An annual effective investment yield of 6 percent is a different convention: its equivalent monthly effective rate is $(1.06)^{1/12}-1$. Do not substitute one for the other without checking the problem's convention.

For ordinary annuities, equal payments occur at the end of each period. An annuity due pays at the beginning and has one additional period of earning or discounting. Write the timing explicitly before selecting a calculator mode. Rent paid in advance and mortgage installments paid in arrears can require different treatment even if both are monthly.

Amortizing a Residential Mortgage

For level payments that fully amortize principal $P$, the payment is:

$$PMT=P\frac{i}{1-(1+i)^{-n}}$$

Assume a hypothetical $240,000 mortgage, 6 percent nominal annual interest, monthly payments, and a 30-year amortization. Then $i=0.005$, $n=360$, and the payment is approximately $1,438.92 per month, before taxes, insurance, or other escrow items. The first month's interest is $240,000 times 0.005, or $1,200. The principal reduction is approximately $238.92.

The next month's interest is calculated on the reduced balance. This explains why early payments are interest-heavy even though the total scheduled payment stays level. The annual debt service is approximately $17,267.04 using the rounded monthly payment. The annual mortgage constant is annual debt service divided by original principal, approximately 0.071946, or 7.1946 percent. It exceeds the 6 percent interest rate because the payment also repays principal.

After $k$ payments, an equivalent balance formula is:

$$B_k=P(1+i)^k-PMT\frac{(1+i)^k-1}{i}$$

Using unrounded payments, the balance after 60 payments is approximately $223,330.46. Alternatively, discount the remaining 300 payments at the contract monthly rate. Both methods should agree apart from rounding. Multiplying 300 by the payment produces the total remaining payments, not the outstanding principal, because it includes future interest.

Cash Equivalency Is a Separate Question

A seller-financed note can have a contract rate different from the market-required rate. First determine the actual contractual payment and any balloon balance. Then discount those cash flows at a supported market yield appropriate to their risk. Add cash paid at closing to obtain the present value of the total consideration.

Do not discount every scheduled payment through full amortization and also add a balloon before the loan would amortize; that double counts payments after the balloon. If the note is paid off after five years, the cash-flow timeline contains the first 60 payments and the balance due at that time. The sales-comparison financing lesson works this case numerically. A mathematical financing benefit still needs market support before it becomes a comparable-sale adjustment.

Net Present Value of an Investment

Net present value subtracts the initial investment from the discounted future benefits. Assume an all-cash investment costing $100,000 now, annual end-of-year net receipts of $8,000 for three years, and $110,000 net resale proceeds at the end of year three, in addition to the third year's income. Assume an 8 percent annual effective required return.

$$NPV=-100{,}000+\frac{8{,}000}{1.08}+\frac{8{,}000}{1.08^2}+\frac{118{,}000}{1.08^3}$$

The three discounted receipts are approximately $7,407.41, $6,858.71, and $93,672.20. Total present value is $107,938.32, so NPV is positive $7,938.32. Under these assumptions, the price is below the investor's indicated investment value. That is not automatically a market-value conclusion: verify that the income, resale, risks, and required return represent market participants if market value is the assignment.

If the resale figure is gross rather than net, deduct selling expenses before discounting. If cash flows are after debt service, account for the mortgage payoff and use an equity yield. If they are before debt service, use a property-level yield and do not deduct loan payments halfway through the model. Combining property income with an equity discount rate changes the investment being measured.

Diagnose Before Rounding

Use a consistent number of decimal places only at the final reporting step unless a problem explicitly directs intermediate rounding. A one-dollar difference can be rounding; a large difference often signals an incorrect payment mode, missing balloon, wrong sign, or mismatched periods. Confirm that a positive interest rate produces a present value below the corresponding future single receipt. For a fully amortizing loan, the balance should approach zero after the final payment.

A zero-interest level-payment loan requires the special case $PMT=P/n$; the interest formula has a zero-over-zero limit at $i=0$. This conceptual check prevents treating a calculator error as proof that no payment exists. The cash-flow model remains the governing explanation even when a memorized formula needs adjustment.