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Corporate Finance Unit 3: Annuities, Loans and Amortization

Most real cash flows come in streams: loan payments, salaries, rents and dividends. This unit gives shortcut formulas for level, growing and never-ending streams and shows how loans are paid off.

Covers: Ordinary annuities: PV and FV, Annuities due, Perpetuities and growing perpetuities, Growing annuities, Loan payments and amortization schedules, Solving for rate, periods and balances

Study guide

An annuity is a series of equal payments for a fixed number of periods. Rather than discount each payment separately, use the annuity factor: PV = C × [1 − (1 + r)⁻ⁿ] ÷ r. For future value, FV = C × [(1 + r)ⁿ − 1] ÷ r. These formulas assume an ordinary annuity, with payments at the end of each period. When payments come at the start of each period, as with rent or leases, it is an annuity due, and both PV and FV are multiplied by (1 + r) because every payment arrives one period earlier. Always match the rate to the payment frequency: monthly payments need a monthly rate and the number of months.

A perpetuity is an annuity that never ends. Its present value is simply C ÷ r, which is finite because payments far in the future are worth almost nothing today. Preferred stock and some government bonds behave like perpetuities. A growing perpetuity pays C next period and then grows at rate g forever, with PV = C ÷ (r − g), valid only when r > g. Both formulas give the value one period before the first payment, so if the first payment is delayed you must discount the formula's result back the extra periods. A growing annuity is a growing perpetuity that stops after n periods.

Loans are annuities from the borrower's side. The payment that fully repays a loan of PV over n periods is C = PV × r ÷ [1 − (1 + r)⁻ⁿ]. Each payment has an interest part, equal to the rate times the balance at the start of the period, and a principal part, equal to the rest. Because the balance falls, the interest part shrinks and the principal part grows over time. An amortization schedule lays this out period by period. On a 30-year mortgage, early payments are mostly interest, so the balance falls slowly at first.

The outstanding balance of a loan at any point equals the present value of the remaining payments, discounted at the loan rate. This shortcut lets you find balloon payments, payoff amounts and refinancing savings without building the whole schedule. You can also solve annuity equations for other unknowns: the number of periods comes from logarithms, while the interest rate usually requires trial and error or a financial calculator. When a loan carries upfront fees, the borrower receives less than the face amount, so the true effective rate is higher than the quoted APR.

Many personal and business decisions reduce to comparing annuities. To compare a rebate with low-rate dealer financing, discount both payment streams at the same market rate. To plan retirement, first compute the lump sum needed at retirement as the PV of the withdrawals, then compute the savings annuity whose future value equals that sum. To decide on refinancing, compare the present value of the payment savings with the closing costs. In each case, the key is to put all cash flows at the same point in time before comparing.

Key ideas

• Annuity formulas value equal, evenly spaced payments in one step.
• An annuity due is worth (1 + r) times the matching ordinary annuity.
• A perpetuity's value is C ÷ r; a growing perpetuity's value is C ÷ (r − g) with r > g.
• Perpetuity formulas value the stream one period before the first payment.
• Each loan payment covers that period's interest first; the rest reduces principal.
• A loan's balance equals the PV of its remaining payments at the loan rate.
• Fees and rebates change the effective cost of borrowing, so compare deals by PV at one rate.

Formulas and rules

Common mistakes

• Using annual rates and years for monthly payments instead of monthly rates and months.
• Forgetting the (1 + r) adjustment for payments that start immediately.
• Discounting a delayed perpetuity by the year of the first payment instead of one year earlier.
• Using the current dividend instead of next period's dividend in a growing perpetuity.
• Finding a loan balance by subtracting payments from the original amount, ignoring interest.

Before the test, make sure you can

• Calculate the PV and FV of ordinary annuities and annuities due.
• Value perpetuities, growing perpetuities and delayed perpetuities.
• Calculate the PV of a growing annuity.
• Calculate a loan payment and build the first rows of an amortization schedule.
• Find a loan's remaining balance as the PV of remaining payments.
• Solve an annuity for the number of periods or the interest rate.
• Compare loan offers, rebates or refinancing choices using present value.

Words to know

Annuity

A series of equal cash flows at regular intervals for a fixed number of periods.

Ordinary annuity

An annuity whose payments occur at the end of each period.

Annuity due

An annuity whose payments occur at the beginning of each period.

Annuity factor

[1 − (1 + r)⁻ⁿ] ÷ r; multiplying it by the payment gives the annuity's PV.

Perpetuity

A level stream of cash flows that continues forever; PV = C ÷ r.

Growing perpetuity

A stream growing at rate g forever; PV = C₁ ÷ (r − g).

Growing annuity

A stream growing at rate g that ends after n periods.

Amortization

Paying off a loan through regular payments that cover interest and reduce principal.

Amortization schedule

A table showing each payment's interest, principal and remaining balance.

Outstanding balance

The amount still owed on a loan; equals the PV of the remaining payments.

Balloon payment

A large final payment that pays off the remaining balance of a loan early.

Interest-only loan

A loan where payments cover only interest and the full principal is repaid at maturity.

Pure discount loan

A loan repaid with a single lump sum at maturity, with no periodic payments.

Refinancing

Replacing an existing loan with a new one, usually at a lower rate.

Sample practice questions

30 more questions like these, from level 1 to 5, are in the practice set.