FDNBUSM 03 · Ordinary Annuities and Loan Balances

One month before the first payment

The first payment is discounted one period inside the factor. If first payment is at month 3, the factor gives value at month 2.

R a(i,24)

Twenty-four future payments remain and the next is one period away. This is the prospective balance immediately after payment 12.

Before = after + the installment R

At one instant the installment reduces the obligation by R; no additional time passes between before and after.

20,000 + 6,000 a(.01,24)

The down payment is already valued now. Only the 24 future installments need discounting.

Discount the year-2 annuity value by (1 + i)^2

R a(i,4) is located at year 2, so only two additional periods separate its anchor from year 0.

PHP 1,800

The opening principal earns one month at i = .12/12 = .01. Interest = 180,000(.01).

R > Pi

R must exceed first-period interest to start reducing principal; otherwise the standard positive finite loan cannot amortize.

P = R a(i,n) + F(1 + i)^−n

Value the regular stream and the additional balloon separately at time 0, then add them.

The last payment earns zero periods of interest at that valuation date.

At the instant of the nth payment, that newest cash flow has not been in the account for a full interest period.

n

With no growth or discounting, n equal payments are worth their arithmetic sum nR; use the limit instead of division by zero.