Case Study 1 — What Is a Bond Worth? Geometric Series and the Price of Future Money
Field: Economics and finance Calculus used: Geometric series — partial sums and infinite sums (§21.3, §21.10)
A Trader's Question
Imagine you are sitting across from a colleague who slides a single sheet of paper toward you. It describes a bond: the issuer promises to pay you \$60 every year for the next ten years, and then hand back \$1,000 at the end. "Current market yield on comparable bonds is 5%," she says. "What's it worth today? Don't guess — show me the arithmetic." That question — what is a stream of future payments worth right now? — is the central question of finance, and its answer is a geometric series. This case study works the whole calculation by hand, beginning from the simplest instrument and building to the bond on the page.
The obstacle is that money has a time dimension. A dollar in your pocket today is not the same thing as a dollar promised next year, because today's dollar can be invested to earn interest. If the prevailing rate is $r$ per year, then \$1 today becomes $\$(1+r)$ next year. Running that logic backward: a dollar arriving one year from now is worth only $\dfrac{1}{1+r}$ today, and a dollar arriving $n$ years from now is worth $\dfrac{1}{(1+r)^n}$. That shrinking factor, $\dfrac{1}{(1+r)^n}$, is called the discount factor, and it is the engine of everything that follows. The present value of a future cash flow $C_n$ arriving in year $n$ is $\dfrac{C_n}{(1+r)^n}$, and the present value of an entire stream is the sum
$$PV = \sum_{n=1}^{\infty} \frac{C_n}{(1+r)^n}.$$
Notice the structure already: each term carries one more power of $\dfrac{1}{1+r}$ than the last. That is the fingerprint of a geometric series, and it is exactly the object §21.3 taught us to sum.
The Simplest Instrument: A Perpetuity
Begin with the cleanest case, where the payment never changes and never stops. A perpetuity pays a fixed amount $C$ every year, forever. The British Treasury actually issued such things — "consols," perpetual bonds that paid a fixed coupon with no maturity date, some of which traded for over two centuries before being redeemed in 2015. The present value is
$$PV = \sum_{n=1}^{\infty} \frac{C}{(1+r)^n} = \frac{C}{1+r} + \frac{C}{(1+r)^2} + \frac{C}{(1+r)^3} + \cdots.$$
This is geometric. Read off the actual first term — being careful, as §21.3 warns, not to grab a coefficient out of a formula — it is $a = \dfrac{C}{1+r}$, and the common ratio is $r_{\text{geo}} = \dfrac{1}{1+r}$. Because $r > 0$, we have $\left|\dfrac{1}{1+r}\right| < 1$, so the series converges, and the sum-formula $\dfrac{a}{1 - r_{\text{geo}}}$ gives
$$PV = \frac{C/(1+r)}{1 - \dfrac{1}{1+r}} = \frac{C/(1+r)}{\dfrac{(1+r) - 1}{1+r}} = \frac{C/(1+r)}{\dfrac{r}{1+r}} = \frac{C}{r}.$$
The $(1+r)$ factors cancel and we are left with a result of astonishing simplicity: a perpetuity is worth its annual payment divided by the interest rate. A consol paying \$50 a year, when comparable yields are 4%, trades near $\dfrac{50}{0.04} = \$1{,}250$; an endowed chair funding a \$200,000 salary forever at a 5% return requires a principal of $\dfrac{200{,}000}{0.05} = \$4{,}000{,}000$.
It is worth pausing on the paradox here. The perpetuity pays out an infinite total amount of money — \$50 a year, forever — yet its value today is a modest, finite \$1,250. There is no contradiction: the partial sums of the discounted payments climb toward \$1,250 and never pass it, because each successive payment is discounted more steeply than the last. The geometric decay of the discount factor wins the race against the relentless arrival of new payments — Zeno's runner in a pinstripe suit, infinitely many contributions summing to a finite whole.
The discount rate does the work. The smaller $r$ is, the larger $C/r$ becomes — low interest rates inflate the present value of long-dated payments. This is why falling rates send bond and stock prices upward, and why a perpetuity's price is exquisitely sensitive to the rate. At 4% our consol is worth \$1,250; at 2% the *same* \$50 coupon is worth \$2,500. Half the rate, double the price.
Finitely Many Payments: The Annuity
Real bonds, of course, do not pay forever. They pay for a fixed number of years $N$ and then stop. A stream of $N$ equal payments is an annuity, and now we need the partial sum of a geometric series rather than its infinite limit — exactly the closed form $S_N = a\dfrac{1 - r_{\text{geo}}^{\,N}}{1 - r_{\text{geo}}}$ derived in §21.3. With first term $a = \dfrac{C}{1+r}$ and ratio $\dfrac{1}{1+r}$ as before,
$$PV_{\text{annuity}} = \sum_{n=1}^{N} \frac{C}{(1+r)^n} = \frac{C}{r}\left[\,1 - \frac{1}{(1+r)^N}\,\right].$$
The bracket is the only difference from the perpetuity. As $N \to \infty$, the term $\dfrac{1}{(1+r)^N} \to 0$ (it is a power of a number less than one — §21.3 again), the bracket approaches $1$, and the annuity formula collapses back to the perpetuity's $C/r$. The annuity is just a perpetuity with its distant tail chopped off — and that tail, discounted across many years, is worth surprisingly little, which is why a 30-year stream already captures most of a perpetuity's value.
The Bond on the Page
Now we can answer the trader's question. The bond is a package of two cash-flow streams: an annuity of \$60 coupons for ten years, plus a single **lump sum** of \$1,000 (the face value) returned in year ten. We value each piece and add, which is legitimate because present value is linear — the present value of a sum of streams is the sum of their present values, exactly the linearity property of convergent series in §21.7.
The coupons. Ten payments of $C = \$60$ at yield $r = 0.05$:
$$PV_{\text{coupons}} = \frac{60}{0.05}\left[\,1 - \frac{1}{(1.05)^{10}}\,\right] = 1200\big[\,1 - 0.61391\,\big] = 1200 \times 0.38609 \approx \$463.31.$$
(Here $(1.05)^{10} \approx 1.62889$, so its reciprocal is about $0.61391$.)
The face value. A single \$1,000 arriving in year ten, discounted once:
$$PV_{\text{face}} = \frac{1000}{(1.05)^{10}} = 1000 \times 0.61391 \approx \$613.91.$$
The bond. Adding the pieces:
$$PV_{\text{bond}} = 463.31 + 613.91 \approx \$1{,}077.22.$$
So the bond is worth about \$1,077 today — more than its \$1,000 face value. That makes sense: the bond pays a 6% coupon (\$60 on \$1,000 face) in a market that only demands 5%, so investors will pay a premium for the above-market income. Had the market yield instead been 6%, every discount factor would have used $1.06$, the two streams would have summed to exactly \$1,000, and the bond would trade "at par." Had yields risen above 6%, the bond would trade at a discount. This inverse dance between yields and prices — the entire logic of the bond market — falls straight out of the geometric-series formula.
Growing Payments: One More Turn of the Crank
A final variation shows the reach of the same idea. Suppose payments grow at rate $g$ each year — a model for a stock's dividends. Then $C_n = C(1+g)^{n-1}$, and
$$PV = \sum_{n=1}^{\infty} \frac{C(1+g)^{n-1}}{(1+r)^n} = \frac{C}{r - g}, \qquad (r > g).$$
This is the Gordon growth model of equity valuation. The series is still geometric — its ratio is now $\dfrac{1+g}{1+r}$ — and it converges precisely when that ratio is less than one, i.e. when $g < r$. A stock paying a \$2 dividend, growing at 3%, discounted at 8%, is worth $\dfrac{2}{0.08 - 0.03} = \$40$. And the convergence condition is not a technicality: if dividends grew as fast as the discount rate ($g = r$), the series would diverge and the "value" would be infinite — the market's way of saying that a payment stream growing forever at the discount rate cannot be priced. The divergence we studied abstractly in §21.6 is, here, an economic impossibility made visible.
Why It All Reduces to One Series
Step back and the unity is striking. Bond pricing, mortgage payments, stock valuation, real-estate cap rates, pension funding, the net present value of a factory — every one is, at bottom, the sum of a geometric series whose common ratio is the discount factor $\dfrac{1}{1+r}$. The single most useful series in this chapter is also the single most used calculation in quantitative finance: recognize $\sum a r^n$ in a column of discounted cash flows, and you can write the answer in one line.
Discussion Questions
- A perpetuity pays an infinite total amount of money, yet its present value $C/r$ is finite. Explain in terms of partial sums why this is not a contradiction. Which property of the discount factor makes the difference?
- Our bond was worth \$1,077 at a 5% yield but exactly \$1,000 at a 6% yield. Recompute its value at a 7% yield (you may estimate $(1.07)^{10} \approx 1.967$) and confirm it trades at a discount. What general rule about yields and prices does this illustrate?
- The annuity formula $\frac{C}{r}[1 - (1+r)^{-N}]$ becomes the perpetuity formula $\frac{C}{r}$ as $N \to \infty$. Which term vanishes, and why is it a power of a number less than one? (Connect to §21.3.)
- The Gordon growth model requires $r > g$. Interpret the failure $g \ge r$ both as a series-divergence statement (§21.6) and as an economic statement about un-priceable growth.
- Why is the present value of a long-dated payment so sensitive to the interest rate? Use the perpetuity to argue that halving $r$ doubles the value.
A Short Annotated Reading
- Brealey, Myers & Allen, Principles of Corporate Finance (13th ed.). The standard treatment of present value; Chapters 2–3 derive the perpetuity and annuity formulas exactly as above, then apply them to bonds and projects.
- Bodie, Kane & Marcus, Investments (11th ed.). The bond-pricing chapter formalizes the premium/par/discount logic we computed and connects price to yield-to-maturity.
- Gordon, M. J. (1959), "Dividends, Earnings, and Stock Prices," Review of Economics and Statistics. The original growing-perpetuity model; reading it shows how far a single geometric series can be stretched.
Every present-value calculation in finance is a geometric series in disguise. Master $a/(1-r)$ and its partial-sum cousin, and the price of future money is a one-line computation.