Annuities and loans

Why it matters

Most financial promises are not single payments but series of payments: monthly mortgage instalments, yearly pension benefits, quarterly insurance premiums. A series of payments at regular intervals is called an annuity (in this lesson, an annuity-certain: the payments happen no matter what; in module 2 the payments depend on someone being alive). Annuity formulas are the workhorse of Exam FM and of every loan, lease and pension calculation.

Annuity-immediate and annuity-due

Take n payments of 1, one per period, at effective rate i per period, with v = 1/(1+i).

• Annuity-immediateA series of level payments made at the end of each period; its present value is a_n = (1 - v^n) / i.: payments at the end of each period (times 1, 2, ..., n). Its present value is

a_n = v + v^2 + ... + v^n = (1 - v^n) / i

• Annuity-dueA series of level payments made at the start of each period, worth (1 + i) times the annuity-immediate.: payments at the start of each period (times 0, 1, ..., n-1). Every payment arrives one period earlier, so

ä_n = a_n * (1 + i) = (1 - v^n) / d,   where d = i / (1 + i)

The accumulated valueThe value at the end of the term of all the payments with interest, s_n = ((1 + i)^n - 1) / i. at time n of an annuity-immediate is s_n = ((1 + i)^n - 1) / i. The formula for a_n is a geometric series, and it is worth deriving once by hand: write a_n and v * a_n, subtract, and almost every term cancels.

Worked example 1: valuing a payout

A lottery pays 50,000 at the end of each year for 20 years. At 6% effective:

v^20 = 1.06^-20 = 0.311805
a_20 = (1 - 0.311805) / 0.06 = 11.469921
PV = 50000 * 11.469921 = 573,496.06

So the 1,000,000 "headline" prize is worth about 573,496 today at 6%. If the payments were at the start of each year, the value would be 573,496.06 * 1.06 = 607,905.82.

Loans: finding the payment

A loan of L repaid by n level end-of-period payments P is fair when the present value of the payments equals the amount borrowed:

L = P * a_n    so    P = L / a_n

Example: 200,000 borrowed for 30 years at 6% nominal, compounded monthly. Use monthly periods: j = 0.005, n = 360.

a_360 at 0.5% = (1 - 1.005^-360) / 0.005 = 166.791614
P = 200000 / 166.791614 = 1,199.10 per month

The amortization schedule

Each payment first pays the interest due on the outstanding balance; the rest repays principal:

interest_k  = j * balance_(k-1)
principal_k = P - interest_k
balance_k   = balance_(k-1) - principal_k

A 4-year loan of 10,000 at 8% with annual payments (P = 10000 / a_4 = 3,019.21):

The table shows amounts rounded to cents, but the calculation keeps full precision, so the final balance is exactly zero. Notice that interest falls and principal rises each year. A useful shortcut: the balance after k payments equals the present value of the payments still to come, P * a_(n-k) (the prospectiveThe method that values an outstanding balance as the present value of the payments still to come. method). That same idea returns as the policy reserve in module 2.

def loan_payment(principal, rate, n):
    a_n = (1 - (1 + rate) ** -n) / rate
    return principal / a_n

print(round(loan_payment(10000, 0.08, 4), 2))  # 3019.21

Common mistakes

• Due versus immediate. Read the timing of the first payment carefully; the two values differ by a factor of 1 + i.
• Period mismatch. Monthly payments need the monthly effective rate and n in months.
• Forgetting the zero-rate case. If i = 0, a_n = n; the formula divides by zero, so code needs a special case.
• Rounding the payment before building the schedule. Lenders do round to cents, which leaves a small final adjustment; know which convention you are using.

Quick recap

• a_n = (1 - v^n)/i, ä_n = (1 + i) * a_n, s_n = ((1+i)^n - 1)/i.
• Level loan payment P = L / a_n.
• Each payment = interest on the balance + principal; balance after k payments = P * a_(n-k).

Try it

In the Do tab, code annuity_pv, loan_payment and a full amortization scheduleA table showing, for each loan payment, the interest, the principal repaid and the balance left., and check that the loan is paid off exactly.