Finance-Theory · Unit 3 · Video 8 · Interactive Practice

The Annuity Discount Factor: Mortgage Payments and the Lottery 'Millionaire'

IKey Formulas

Formula	Name	What it gives
$\text{PV} = \dfrac{C}{r} - \dfrac{C}{r}\dfrac{1}{(1+r)^T}$	Annuity present value	A perpetuity starting now, minus the perpetuity that takes over after date $T$
$\text{ADF}(r,T) \equiv \dfrac{1}{r}\left[1 - \dfrac{1}{(1+r)^T}\right]$	Annuity discount factor	Today's value of $1 a period for $T$ periods: one number, fixed by $r$ and $T$ alone
$\text{PV} = C \times \text{ADF}(r,T)$	Value from a payment	The lottery prize: multiply the payment by the factor
$C = \dfrac{\text{PV}}{\text{ADF}(r,T)}$	Payment from a value	The mortgage payment: divide the loan by the factor

Key Insight: The factor separates the rate from the cash flow, leaving three quantities — $\text{PV}$ , $C$ and $\text{ADF}(r,T)$ — so any two of them give the third. And since $1/(1+r)^T \to 0$ , the factor climbs toward $1/r$ , the perpetuity factor, without ever reaching it.

IIOne Number per Rate and Term

Every entry is one number the payment multiplies: rows rise with the term, columns fall with the rate.

Payment per period

💡 The printed rate books carried whole and half percents, so a rate such as $8\tfrac{3}{4}\%$ a year had no row at all. The table is computed from the formula, not the other way round, so any missing entry is one evaluation away.

IIIIs the Winner a Millionaire?

Twenty payments of $100,000 total $2,000,000, and today they are worth less than $1,000,000.

Years of payments T 20

IVThe Mortgage Payment

Run the factor backwards: the loan divided by $\text{ADF}(r,T)$ is the payment the bank asks each month.

Quoted rate Term Your guess $900

VQuiz Questions

Problem 1 · Value of a Payment Stream

Given: a prize pays $20,000 a year for 10 years, the first payment one year from today, at $r = 8\%$ , where $\text{ADF}(8\%, 10) = 6.71008$ — find the present value of the prize.

✅ Correct! $20,000 × 6.71008 = $134,201.60, so the prize is worth $134,202 today — about two-thirds of the $200,000 it pays out.

❌ That divides by the factor. Dividing turns a present value into a payment, which is the mortgage direction. Here the payment is known and the value is wanted, so multiply.

❌ $200,000 is what the prize pays out, not what it is worth. Ten payments arriving at ten different dates cannot simply be added; each is worth less than its face amount.

❌ $250,000 is $C/r$ — the value if the payments never stopped. Ten years is far short of forever:

\text{ADF}(8\%,10) = 6.71008

, well under the ceiling

1/r = 12.5

Show solution

The payment is known and the value is wanted, so multiply:

\text{PV} = C \times \text{ADF}(r,T) = 20{,}000 \times 6.71008 = 134{,}201.60

The prize is worth $134,202 today. Two checks on the size of that number:

It must be below the undiscounted total, $200,000 — the discounting can only shrink each payment.
It must be below the perpetuity value $C/r = 20{,}000/0.08 = 250{,}000$ — ten years of payments is worth less than payments that never stop.

Problem 2 · A Mortgage Payment

Given: a $300,000 mortgage with 30 years of monthly payments, quoted at 7.2% a year — so the rate per month is $0.6\%$ and $\text{ADF}(0.006, 360) = 147.32136$ — find the monthly payment.

✅ Correct! $300,000 ÷ 147.32136 = $2,036.36 a month. The period is the month, so the rate is the monthly rate and

T

counts months.

❌ $833.33 is $300,000 ÷ 360. That is the payment if the bank charged no interest at all; it ignores the factor entirely.

❌ $1,800 is one month's interest, $0.006 \times 300{,}000$ . Paid forever it is the perpetuity payment

C = \text{PV} \times r

and never repays a cent of principal; a 30-year mortgage must pay it off.

❌ That is a yearly payment split into twelfths. With

\text{ADF}(7.2\%, 30) = 12.16375

the annual payment is $24,663.45, or $2,055.29 a month — but monthly payments are discounted monthly, so

r

and

T

must both be in months.

Show solution

The loan is the present value and the payment is wanted, so divide:

C = \frac{\text{PV}}{\text{ADF}(r,T)} = \frac{300{,}000}{147.32136} = 2{,}036.36

The payments are monthly, so the period is the month: $r = 7.2\%/12 = 0.6\%$ per month and $T = 30 \times 12 = 360$ payments. Evaluating the factor directly,

\text{ADF}(0.006, 360) = \frac{1}{0.006}\left[1 - \frac{1}{1.006^{360}}\right] = 147.32136

The answer is $2,036.36 a month. Working in years instead would give $2,055.29 — close enough to look right, and wrong.

Problem 3 · Millionaire or Not

Given: a lottery prize pays $60,000 a year for 25 years, the first payment one year from today, at $r = 8\%$ , where $\text{ADF}(8\%, 25) = 10.67478$ .

What is the prize worth today?

What would the prize be worth if the payments never stopped?

✅ Correct! The 25 payments are worth $640,487; the same payments running forever would be worth $750,000. Twenty-five years already captures 85% of an infinite stream — and neither figure reaches a million.

❌ Check the present value. The payment is known, so multiply by the factor: $1,500,000 is the undiscounted total, $750,000 is

C/r

, and $5,621 divides where it should multiply.

❌ Check the perpetuity. Payments forever are worth

C/r

, not

C \times r

and not an unbounded amount: the discounting shrinks each payment fast enough for the sum to converge.

Show solution

Step 1 — the 25-year prize. Multiply the payment by the factor:

\text{PV} = 60{,}000 \times 10.67478 = 640{,}486.80

Step 2 — the same payments forever. As $T$ grows, $1/(1+r)^T \to 0$ and the factor climbs to its ceiling $1/r = 12.5$ :

\text{PV}_\infty = \frac{C}{r} = \frac{60{,}000}{0.08} = 750{,}000

So the prize is worth $640,487, and $750,000 if it ran forever. The winner is not a millionaire on either count, even though the 25 payments add up to $1,500,000.

Problem 4 · How Much House?

Given: your budget is $1,500 a month for 360 months. At $0.5\%$ a month the factor is $166.79161$ , so you can borrow $250,187. The rate then rises to $0.75\%$ a month, where the factor is $124.28187$ — find what the same $1,500 a month will borrow.

✅ Correct! $1,500 × 124.28187 = $186,423. A quarter of a point a month costs $63,764 of borrowing power on an unchanged budget.

❌ $540,000 is 1,500 × 360. That adds 360 payments made at 360 different dates as though they were all made today.

❌ The budget is unchanged, but what it buys is not. Each dollar of monthly payment now supports less loan, because the factor fell from 166.79161 to 124.28187.

❌ That scales the loan by the ratio of the rates, $0.005/0.0075$ . It would be right if the factor were exactly

1/r

, but the bracket

1 - 1/(1+r)^T

rises with

r

too, so the true fall is smaller.

Show solution

Here the payment and the factor are known, so multiply — the same direction as a lottery prize, read as a borrowing capacity:

\text{PV} = C \times \text{ADF}(r,T) = 1{,}500 \times 124.28187 = 186{,}422.81

The budget buys $186,423 of loan instead of $250,187, a fall of $63,764.

Why not the ratio of rates? The factor is $\tfrac{1}{r}$ only in the limit $T \to \infty$ . Scaling the $0.5\%$ borrowing power by $0.005/0.0075$ assumes that limit and gives $166,792, which understates what you can borrow: over a finite 360 months the bracket $1 - 1/(1+r)^{T}$ is nearer to 1 at the higher rate ( $0.9321$ against $0.8340$ ), which offsets part of the fall in $1/r$ .

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