Sunflower Intelligence
Finance & Accounting

F04 · Introductory

Mathematics of Finance

9 lessons

F04 · Module 3, lesson 2 · Lesson 6 of 9

Perpetuities and growth

A payment that never stops still has a finite value today. Learn to value level and growing perpetuities, including ones that start now or later, price a preferred share, size an endowment, and see why anything valued as growing forever depends so heavily on the gap between the discount rate and the growth rate.

About 75 minutes

Before you start

You will be able to

  • Calculate the present value of a level perpetuity, a perpetuity due and a deferred perpetuity.
  • Calculate the present value of a growing perpetuity, and explain why it needs r > g and next period’s payment.
  • Calculate the value of a preferred share and the gift needed to endow a payment forever, with or without growth.
  • Analyse how the value of a growing perpetuity responds to small changes in the discount rate and the growth rate.
  • Explain what a terminal value is, and estimate one for a simple forecast.

Review first

  • Annuities — A perpetuity is an annuity with no last payment; the timing rules (end of period, in advance, deferred) and the growing annuity carry over.
  • Present and future value — You need discount factors and why a present value falls as the rate or the wait increases.
  • Interest and inflation — The growing scholarship keeps its purchasing power by growing with inflation, Ana’s 2.8% assumption.

Fictional case A fictional coffee business founded by Ana Lim. It starts as a catering and barista-training service, opens its own cafés and an online shop for beans, and plans a canned cold brew.

The idea

Fictional case

In 2027 the Bayside Community College Foundation, which raises money for a local college, invites businesses to endow named scholarships. Ana would like to create a Lantern Barista Scholarship: $2,000 a year to a student on the college’s hospitality course, for as long as the course runs, which she hopes is for ever. The foundation invests every gift and pays awards from the income; it expects to earn 5% a year. The first award would be made a year after the gift. How much would Lantern have to give? The payments never stop, so they add up to an infinite number of dollars. Yet the answer is a modest, finite sum. (The foundation, the scholarship and its figures are fictional.)

Payments that never stop

A perpetuity is a stream of equal payments at equal intervals that continues for ever. True perpetuities are rare, but several important things come close: a scholarship or professorship paid from an endowment, a preferred share whose fixed dividend has no end date, the rent from land, and, in valuation, the cash a business is expected to generate long after any forecast stops.

The value of a perpetuity is easiest to see from the endowment’s side. Suppose the foundation holds $40,000 and earns 5% a year. Each year the fund earns 40,000×0.05=2,00040{,}000 \times 0.05 = 2{,}000 dollars, which it pays out as the award, leaving the $40,000 untouched to earn the same again next year, and the year after, for ever. So $40,000 today is exactly enough to pay $2,000 a year in perpetuity: the payment divided by the rate.

PV=Cr=2,0000.05=40,000\text{PV} = \frac{C}{r} = \frac{2{,}000}{0.05} = 40{,}000

The infinite stream has a finite value because distant payments are worth almost nothing today. The table splits the $40,000 into the value contributed by each stretch of years.

Where the value of $2,000 a year for ever comes from, at 5% a year
Payments in yearsNumber of paymentsPresent valueShare of the total
1 to 1010$15,443.4738.61%
11 to 2515$12,744.4231.86%
26 to 5025$8,323.9620.81%
51 to 10050$3,183.977.96%
101 onwardsfor ever$304.180.76%
All yearsfor ever$40,000.00100.00%

Each row is the difference between two annuity values: for example, years 11 to 25 are a 25-year annuity minus a 10-year annuity. Rounded to the cent and to two decimal places of a percentage.

The first ten awards supply almost two-fifths of the value, and everything after year 100 adds less than one percent. That is why a perpetuity costs only a little more than a long annuity: a scholarship for 50 years would cost $36,511.85, against $40,000 for ever.

Payments that grow

A fixed $2,000 has a weakness. If prices rise by 2.8% a year, as Ana assumes, the award buys less every year; after 25 years it buys only about half what it does today. To keep the scholarship’s purchasing power, the award must grow with inflation: $2,000 in the first year, $2,056 in the second, and so on. A stream that grows at a constant rate gg for ever is a growing perpetuity, and its value is the first payment divided by the gap between the rate and the growth rate:

PV=C1r−g=2,0000.05−0.028=2,0000.022≈90,909.09\text{PV} = \frac{C_1}{r - g} = \frac{2{,}000}{0.05 - 0.028} = \frac{2{,}000}{0.022} \approx 90{,}909.09

The endowment view explains the formula again. A fund of $90,909.09 earns 5%, or $4,545.45, in the first year. It pays out $2,000, which is 2.2% of the fund, and keeps $2,545.45, which is exactly 2.8% of the fund. So the fund grows by 2.8%, and next year it can pay 2.8% more, for ever. Out of every year’s 5% return, the fund spends r−gr - g and reinvests gg.

Growth makes the scholarship cost more than twice as much, and it moves the value further into the future. With growth, the first ten awards supply only about 19.1% of the value, and the awards after year 100 about 12.0%, against 0.8% without growth. The further the value lies in the future, the more it depends on assumptions about the far future.

Fictional case

Ana now has three prices. A level $2,000 award for ever needs $40,000. An award that keeps pace with inflation needs about $90,909. A level award for ten years, an annuity, needs 2,000×(1−1.05−10)÷0.05≈15,443.472{,}000 \times (1 - 1.05^{-10}) \div 0.05 \approx 15{,}443.47 dollars. A year after Lantern opened, $90,909 is far beyond what it can give; the In practice section returns to her decision.

In depth

From annuity to perpetuity

A perpetuity is the limit of an annuity as the number of payments grows without end. In the annuity formula, PV=C[1−(1+r)−n]÷r\text{PV} = C[1 - (1 + r)^{-n}] \div r, the term (1+r)−n(1 + r)^{-n} is the present value of $1 paid in period nn, and it shrinks towards zero as nn grows. What is left is C÷rC \div r. In the growing-annuity formula the corresponding term is [(1+g)÷(1+r)]n[(1 + g) \div (1 + r)]^n. It shrinks to zero only if g<rg \lt r; if g≥rg \ge r it never shrinks, each payment is worth at least as much today as the one before, and the sum has no limit. That is the mathematical reason for the condition r>gr \gt g.

Turning the argument round gives a useful way to value annuities, used in the annuities lesson: an annuity is a perpetuity starting now minus a perpetuity that starts after the last payment.

When the payments start: perpetuity due and deferred perpetuity

Like the annuity formula, C÷rC \div r values the stream one period before the first payment. Every timing variation follows from that rule.

  • A perpetuity due pays its first amount today. It is today’s payment plus an ordinary perpetuity: C+C÷r=C(1+r)÷rC + C \div r = C(1 + r) \div r. If Bayside paid the first award on the day of the gift, Lantern would need 2,000+40,000=42,0002{,}000 + 40{,}000 = 42{,}000 dollars.
  • A deferred perpetuity starts later. If the first payment is at the end of period k+1k + 1, the formula gives its value at time kk; discount that by kk periods. If the course, and so the first award, began in four years’ time, the gift would be 40,000÷1.053≈34,553.5040{,}000 \div 1.05^3 \approx 34{,}553.50 dollars.
  • The same rules apply to a growing perpetuity: C1÷(r−g)C_1 \div (r - g) is its value one period before the payment C1C_1.

Preferred shares

A preferred stock (preference share in the UK) is a share that pays a fixed dividend, usually stated as a percentage of its par value, and ranks ahead of common stock: preferred holders are paid their dividends before common shareholders receive any, and have priority if the company is liquidated, though they usually have no vote. Most preferred shares have no maturity date. If the dividend is paid for ever, the share is a level perpetuity and its value is the dividend divided by the return investors require on similar shares:

P=DrpP = \frac{D}{r_p}

A fictional utility’s preferred share with a $100 par value and a 5.5% dividend pays $5.50 a year. If investors require 7%, it is worth 5.50÷0.07≈78.575.50 \div 0.07 \approx 78.57 dollars; if the required return rises to 8% the value falls to $68.75, and if it falls to 6% the value rises to $91.67. A fixed payment stream moves in price opposite to the rates investors demand, and a level perpetuity is very sensitive because all its payments are fixed for ever. Rearranging, a market price tells you the required return: rp=D÷Pr_p = D \div P.

Two cautions. Dividends are often paid quarterly: then use the quarterly dividend and the quarterly rate. If 7% is an APR compounded quarterly, the quarterly rate is 1.75% and 1.375÷0.01751.375 \div 0.0175 gives the same $78.57; if 7% is an effective annual rate, the quarterly rate is 1.071/4−1≈1.7059%1.07^{1/4} - 1 \approx 1.7059\% and the share is worth about $80.60, because quarterly dividends arrive sooner than one annual payment. And a fixed dividend does not keep pace with inflation, so its purchasing power falls year after year.

Growing perpetuities and share prices

Applied to a company’s common shares, the growing perpetuity becomes the constant-growth dividend discount model, also called the Gordon growth model: if the next dividend is D1D_1 and dividends are expected to grow at gg for ever, the share is worth P0=D1÷(r−g)P_0 = D_1 \div (r - g). Two practical points matter. The formula needs next period’s dividend: if a company has just paid D0D_0, then D1=D0(1+g)D_1 = D_0(1 + g). And rearranging gives r=D1÷P0+gr = D_1 \div P_0 + g: the return a shareholder can expect is the dividend yield plus the growth rate, a common way to estimate the return investors require from a market price. The model suits mature companies whose dividends grow steadily; it says nothing useful about a company that pays no dividends. F06 Investment Analysis develops share valuation, and F05 Corporate Finance the cost of equity.

How sensitive is the value to r and g?

The table shows the value of a growing perpetuity whose first payment is $100, for three discount rates and three growth rates.

Value of a growing perpetuity with a first payment of $100
Discount rate rg = 2%g = 3%g = 4%
8%$1,666.67$2,000.00$2,500.00
9%$1,428.57$1,666.67$2,000.00
10%$1,250.00$1,428.57$1,666.67

Each value is 100 ÷ (r − g). Cells with the same r − g have the same value, along each diagonal.

Raising gg from 3% to 4% at an 8% discount rate raises the value by 25%, from $2,000 to $2,500; raising rr from 8% to 9% with 3% growth cuts it by about 16.7%. The smaller the gap, the more violent the effect: with r−gr - g at 1%, narrowing the gap by half a percentage point doubles the value, and narrowing it by a full point makes it infinite. Small, uncertain differences in assumptions become large differences in value, which is why analysts present a range rather than a single figure.

Terminal value

No one can forecast a business’s cash flows year by year for ever. A standard approach forecasts them in detail for a few years, then assumes that from the end of the forecast the cash flows grow at a steady rate for ever. The value at the end of the forecast of everything that comes after it is the terminal value. With a forecast that ends in year NN:

TVN=CFN+1r−g,PV of the terminal value=TVN(1+r)N\text{TV}_N = \frac{\text{CF}_{N+1}}{r - g}, \qquad \text{PV of the terminal value} = \frac{\text{TV}_N}{(1 + r)^N}

The terminal value sits at year NN, because the formula values a stream one period before its first payment, CFN+1\text{CF}_{N+1}. It must be discounted NN periods, like any single sum at year NN. Terminal values often supply most of a business’s estimated value, as the worked example shows, so the choice of gg and rr deserves more scrutiny than any single year of the forecast. F05 and F07 build full discounted cash flow valuations.

Assumptions and limits

  • The rate and the growth rate are constant for ever. Neither is literally true; the model is a summary of the long run.
  • The growth rate must be below the discount rate. A long-run growth rate above the growth of the whole economy cannot last for ever: a business growing faster than the economy for ever would eventually be larger than the economy.
  • The payments must actually continue. A preferred share that the company may retire, a scholarship course that may close, or a business that may fail is worth less than the formula suggests.
  • Rates and payments must be consistent: nominal payments (which include inflation, like the growing scholarship) with a nominal rate, real payments with a real rate. The inflation-and-real-returns lesson in Module 4 shows the two give the same value when done consistently.
  • The payment period and the rate period must match, as for annuities.

Formulas

Present value of a level perpetuity

PV=Cr\text{PV} = \frac{C}{r}
PV\text{PV}
Value one period before the first payment$
CC
Payment per period, for ever$
rr
Interest (discount) rate per period, as a decimalper period

Assumes. Equal payments at the end of each period, for ever; a constant rate.

Rearranged: r = C ÷ PV, the return implied by a price.

Present value of a growing perpetuity

PV=C1r−g\text{PV} = \frac{C_1}{r - g}
PV\text{PV}
Value one period before the first payment$
C1C_1
The first payment, one period from now (C₁ = C₀(1 + g) if C₀ has just been paid)$
rr
Discount rate per period, as a decimalper period
gg
Constant growth rate of the payment per period, as a decimalper period

Assumes. r > g; the payment grows at g for ever.

For shares this is the Gordon growth model, P₀ = D₁ ÷ (r − g). Rearranged: r = C₁ ÷ PV + g.

Value of a preferred share

P=DrpP = \frac{D}{r_p}
PP
Value of one preferred share, one period before the next dividend$
DD
Fixed dividend per period (par value × dividend rate, per period)$
rpr_p
Return investors require on similar preferred shares, per period, as a decimalper period

Assumes. The fixed dividend is paid for ever; the share is not retired.

A level perpetuity. Rearranged: r_p = D ÷ P, the return implied by a market price. Quarterly dividends need a quarterly rate.

Perpetuity due (first payment today)

PVdue=C+Cr=C(1+r)r\text{PV}_{\text{due}} = C + \frac{C}{r} = \frac{C(1 + r)}{r}
PVdue\text{PV}_{\text{due}}
Value today, including today’s payment$
CC
Payment per period$
rr
Rate per period, as a decimalper period

Assumes. Equal payments at the start of each period, for ever.

Deferred perpetuity (first payment at the end of period k + 1)

PV0=Cr×1(1+r)k\text{PV}_0 = \frac{C}{r} \times \frac{1}{(1 + r)^{k}}
PV0\text{PV}_0
Value today$
CC
Payment per period$
rr
Rate per period, as a decimalper period
kk
One less than the period of the first paymentperiods

Assumes. A level perpetuity whose first payment is at the end of period k + 1.

For a deferred growing perpetuity replace C ÷ r with C₁ ÷ (r − g), where C₁ is the first payment.

Terminal value with constant growth

TVN=CFN+1r−g\text{TV}_N = \frac{\text{CF}_{N+1}}{r - g}
TVN\text{TV}_N
Value at the end of year N of all cash flows after year N$
CFN+1\text{CF}_{N+1}
Cash flow in the first year after the forecast, = CF_N × (1 + g)$
NN
Last year of the explicit forecastyears
rr
Discount rateper year
gg
Long-run growth rate of the cash flows, below rper year

Assumes. From year N + 1 on, cash flows grow at a constant g < r for ever.

Discount TV_N by (1 + r)^N to bring it to today.

Worked examples

A first example: Pricing a preferred share

Fictional figures

Ferncastle Utilities has preferred shares with a par value of $100 and a dividend rate of 5.5%, paid once a year, with no maturity date. Investors require 7% a year on similar shares. What is one share worth? What happens to its value if the required return rises to 8%?

  1. Dividend: par × dividend rate = 100×0.055=5.50100 \times 0.055 = 5.50 dollars a year, for ever: a level perpetuity.
  2. Value at 7%: P=D÷rp=5.50÷0.07≈78.57P = D \div r_p = 5.50 \div 0.07 \approx 78.57.
  3. Value at 8%: 5.50÷0.08=68.755.50 \div 0.08 = 68.75.
  4. Change: 68.75−78.57=−9.8268.75 - 78.57 = -9.82 dollars, a fall of 9.82÷78.57≈12.5%9.82 \div 78.57 \approx 12.5\%.

Answer. About $78.57 a share at 7%. If the required return rises to 8%, the value falls to $68.75, about 12.5% lower.

What it means. The share trades below its $100 par because its 5.5% dividend is below the 7% investors require. Because a level perpetuity’s payments are fixed for ever, a one-point rise in required returns cuts its value by an eighth; holders of fixed-dividend shares carry a lot of interest-rate risk.

A harder example: An award that starts later and grows

Fictional figures

A donor wants to endow an annual prize at a university. The first prize, $5,000, will be awarded in four years’ time, when a new programme produces its first graduates. After that the prize will grow by 2.5% a year for ever to keep up with costs. The endowment earns 6% a year. How much must the donor give today?

  1. This is a growing perpetuity with C1=5,000C_1 = 5{,}000, r=0.06r = 0.06, g=0.025g = 0.025, deferred: the first payment is at the end of year 4.
  2. The formula values the stream one year before its first payment, at year 3: 5,000÷(0.06−0.025)=5,000÷0.035≈142,857.145{,}000 \div (0.06 - 0.025) = 5{,}000 \div 0.035 \approx 142{,}857.14.
  3. Discount that single amount three years: 1.063=1.1910161.06^3 = 1.191016, so 142,857.14÷1.191016≈119,945.61142{,}857.14 \div 1.191016 \approx 119{,}945.61.
  4. Check the judgement on timing: discounting four years instead would give $113,156.24, too little to pay the first prize on time.

Answer. The donor must give about $119,945.61 today.

What it means. For three years the gift earns 6% and pays nothing, growing to $142,857.14 by year 3. From then on it earns $8,571.43 a year, pays out the prize (3.5% of the fund) and reinvests 2.5%, so the fund and the prize grow together for ever. The two judgements that matter are the timing (discount from year 3, not year 4) and the growth rate: at 3% growth the gift would be about $139,936.55.

Applied: a business decision: A terminal value for a battery-maintenance business

Fictional figures

An investor is valuing Tamarack Battery Services, a small business that maintains battery-storage systems. She forecasts the cash it will generate for its owners as $400,000 next year, $440,000 in year 2 and $468,000 in year 3. After that she expects the cash flows to grow at 2.5% a year for ever. She discounts at 10% a year. What is the business worth, and how much of the value comes from the years after the forecast?

Tamarack Battery Services: forecast and terminal value at 10%
YearCash flowDiscount factorPresent value
1$400,000.000.909091$363,636.36
2$440,000.000.826446$363,636.36
3$468,000.000.751315$351,615.33
3 (terminal value)$6,396,000.000.751315$4,805,409.47
Total$5,884,297.52

Discount factors 1 ÷ 1.10^t to six decimal places; present values computed unrounded and shown to the cent.

  1. Discount the three forecast years: 400,000÷1.1+440,000÷1.12+468,000÷1.13≈363,636.36+363,636.36+351,615.33=1,078,888.05400{,}000 \div 1.1 + 440{,}000 \div 1.1^2 + 468{,}000 \div 1.1^3 \approx 363{,}636.36 + 363{,}636.36 + 351{,}615.33 = 1{,}078{,}888.05.
  2. Cash flow in year 4: 468,000×1.025=479,700468{,}000 \times 1.025 = 479{,}700. Terminal value at year 3: 479,700÷(0.10−0.025)=479,700÷0.075=6,396,000479{,}700 \div (0.10 - 0.025) = 479{,}700 \div 0.075 = 6{,}396{,}000.
  3. Discount the terminal value three years: 6,396,000÷1.331≈4,805,409.476{,}396{,}000 \div 1.331 \approx 4{,}805{,}409.47.
  4. Total: 1,078,888.05+4,805,409.47=5,884,297.521{,}078{,}888.05 + 4{,}805{,}409.47 = 5{,}884{,}297.52. The terminal value supplies 4,805,409.47÷5,884,297.52≈81.7%4{,}805{,}409.47 \div 5{,}884{,}297.52 \approx 81.7\% of it.
  5. Sensitivity: with growth of 2% or 3% instead of 2.5%, the total is about $5,561,983.47 or $6,252,656.43; with a discount rate of 9% or 11% (growth 2.5%), about $6,797,407.63 or $5,186,171.86.

Answer. About $5.88 million ($5,884,297.52), of which about 81.7% comes from the terminal value: the cash flows after year 3.

What it means. Most of the value rests on one assumption about the distant future. Half a percentage point on the growth rate moves the value by $0.3–0.4 million, and one point on the discount rate by up to $0.9 million. A careful investor reports the range (here roughly $5.2 million to $6.8 million) and asks whether 2.5% growth for ever is plausible, rather than quoting $5,884,297.52 as if it were precise.

Explore

Start from the scholarship: $2,000 a year at 5% with no growth. The curve adds up the present values of the payments year by year; the flat line is the value of the perpetuity. Set growth to 2.8% and see how much longer the curve takes to approach its limit. Then move the growth rate towards the discount rate.

Loading the interactive…

Each line shows how the value of a $100 growing perpetuity rises with the growth rate, for one discount rate. Notice how the lines steepen as g approaches r.

Loading the interactive…

Common mistakes

  • Using this year’s payment instead of next year’s

    The growing-perpetuity formula needs C₁, the payment one period from now. If a company has just paid a $2.00 dividend and dividends grow 3%, the next dividend is $2.06, not $2.00. Using $2.00 understates the value by the factor (1 + g).

  • Applying C₁ ÷ (r − g) when g is not below r

    With g equal to or above r the formula returns infinity or a negative number. Neither is a value: the assumption of constant growth at or above the discount rate for ever is impossible. Revisit the growth rate, which should not exceed long-run economic growth.

  • Getting the date of the value wrong

    C ÷ r values the stream one period before the first payment. A perpetuity paying today is worth C + C ÷ r; a perpetuity whose first payment is in year 5 is worth (C ÷ r) at year 4, discounted four years, not five.

  • Mixing periods

    A quarterly dividend needs a quarterly rate. Dividing an annual dividend by a quarterly rate, or a quarterly dividend by an annual rate, is off by a factor of four.

  • Treating a terminal value as precise

    A terminal value often supplies most of a valuation, and it depends on r − g. Report a range over plausible rates and growth rates, and check the result against other evidence, such as the valuation multiples of similar businesses.

  • Mixing nominal and real figures

    A payment that grows with inflation is a nominal stream: discount it at a nominal rate. Discounting it at a real rate, or a level real payment at a nominal rate, double-counts or ignores inflation. Module 4 develops this.

Guided practice

Work through these with hints when you need them. Every hint and every worked solution is there to be used.

Loading four questions…

On your own

No hints this time. Answer, check, and read the feedback; the solution appears once you have answered or if you ask for it. Your first answers count toward mastery.

Loading six questions…

In practice

Perpetuity formulas are short, which makes them easy to misuse; professionals rely on them for their clarity and then test them hard.

Pricing preferred shares. The treasurer of a fictional regional bank plans to issue preferred shares to strengthen its capital. Investors require about 6.5% on similar issues, so to sell $100 shares at par the bank must offer a dividend of about $6.50 a share. If rates rise before the issue, the bank must raise the dividend or accept a lower price. An investor holding such shares watches the same arithmetic in reverse: when required returns rise, the market value of a fixed dividend for ever falls.

Running an endowment. The investment committee of a fictional university endowment expects a long-run return of 6% and inflation of about 2.5%. If it wants each year’s spending to keep its purchasing power for ever, it can spend about r−g=3.5%r - g = 3.5\% of the fund each year and must reinvest the rest. Spending more pleases today’s students at the expense of future ones; spending less does the opposite. The growing perpetuity turns that trade-off between generations into a single number the committee can debate.

Valuing a business. An analyst valuing a fictional chain of hotels forecasts five years of cash flows and adds a terminal value. Because the terminal value is most of the answer, she shows a grid of values for discount rates from 8% to 10% and growth rates from 1.5% to 3%, and compares the implied value with what similar hotel chains sell for. If her central case needs growth above the long-run growth of the economy to justify the price, she treats that as a warning, not a conclusion.

Fictional case

Ana decides that Lantern cannot yet endow a scholarship for ever. She asks the foundation whether Lantern could fund a ten-year award instead, which at 5% costs about $15,443.47, and plans to revisit a permanent, inflation-linked award, about $90,909 at today’s assumptions, once the business is established. If she did endow it, she would ask the foundation how it decides what to pay out each year, because its long-run return, not hers, sets the r in the formula.

Knowledge check

Six questions, one attempt each, no hints. Score 80% or more to complete the lesson; you can take it again at any time.

Loading six questions…

Summary

  • A perpetuity pays the same amount every period for ever. Its value is C ÷ r, one period before the first payment: the sum that, invested at r, pays C for ever without running down.
  • A growing perpetuity pays C₁ next period, growing at g for ever. Its value is C₁ ÷ (r − g), and it exists only when r > g.
  • A perpetuity due (first payment today) is worth C + C ÷ r; a deferred perpetuity is valued one period before its first payment, then discounted to today.
  • A preferred share with a fixed dividend for ever is worth D ÷ r_p; its price falls when required returns rise. The Gordon growth model values a share as D₁ ÷ (r − g).
  • Distant payments add little to a level perpetuity, but much more to a growing one: the value of anything growing for ever depends heavily on r − g.
  • A terminal value, CF(N+1) ÷ (r − g) at year N, captures everything after a forecast and often most of a business’s value, so it should be reported as a range.
  • Lantern’s $2,000 scholarship would need $40,000 at 5% as a level award, about $90,909 growing with 2.8% inflation, and about $15,443 for ten years only.

Key formulas

Terms introduced

Perpetuity
A stream of equal payments at equal intervals that continues for ever. Its present value is the payment divided by the rate per period, C ÷ r.
Growing perpetuity
A stream of payments that continues for ever and grows at a constant rate g each period. Its present value is the next payment divided by (r − g), and it exists only when r > g.
Perpetuity due
A perpetuity whose first payment is made today. Worth the payment plus an ordinary perpetuity: C + C ÷ r.
Endowment
A fund whose capital is invested so that its income pays for a purpose, such as a scholarship or a professorship, indefinitely.
Preferred stock
A class of shares that pays a fixed dividend, usually stated as a percentage of par value, and ranks ahead of common stock for dividends and in a liquidation, usually without voting rights. Called preference shares in the UK.
Gordon growth model
The constant-growth dividend discount model: a share is worth next year’s dividend divided by (required return − growth rate), P₀ = D₁ ÷ (r − g).
Terminal value
The value, at the end of an explicit forecast period, of all the cash flows expected after it, often estimated as a growing perpetuity: CF(N+1) ÷ (r − g).

Sources

  1. Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 8.1, “Perpetuities”. OpenStax. openstax.org/books/principles-finance/pages/8-1-perpetuities
  2. Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 11.2, “Dividend Discount Models (DDMs)”. OpenStax. openstax.org/books/principles-finance/pages/11-2-dividend-di…
  3. Dahlquist, J. & Knight, R. (2022). Principles of Finance, section 11.4, “Preferred Stock”. OpenStax. openstax.org/books/principles-finance/pages/11-4-preferred-s…
  4. US Securities and Exchange Commission, Investor.gov. Stocks – FAQs (common and preferred stock). investor.gov/introduction-investing/investing-basics/investm…