Seven Dollars in Eighteen Months, Two Dollars Now, and Why the Cents Are Unknowable

Heads pays $7 in eighteen months, tails costs $2 today, and the curve gives 12% for one year and 18% for two. Averaging the amounts gives $2.50, which is 38.68% too high, because expectation and discounting only commute when every cash flow lands on the same date. The answer is about $1.80, and four defensible compounding conventions spread it from 1.7862 to 1.8381, so one decimal is honest and two are not.

A contract settles on one fair coin. Heads, you receive $\$7$ in eighteen months. Tails, you pay $\$2$ immediately. The one-year interest rate is $12\%$ , the two-year rate is $18\%$ , both quoted as annually compounded rates for money lent from today. What should you pay to hold the contract?

The fast answer averages the two amounts: $(7 - 2)/2 = 2.50$ . Every step of that arithmetic is right and the answer is off by more than a third. The contract is worth about $\$1.80$ , and the word “about” is load-bearing rather than modest.

Two amounts, two dates, and one operation applied too early

The averaging answer commits a specific error, and it is not carelessness about interest. It is a reordering. Expectation and discounting are both linear, so they commute when every cash flow lands on the same date. Here they do not: the payment happens now and the receipt happens in eighteen months. Averaging first collapses two different dates into one, and no later discounting can recover the information.

Write the contract as a random cash-flow schedule instead. On heads, the schedule is $+7$ at $t = 1.5$ and nothing today. On tails it is $-2$ at $t = 0$ and nothing later. Value each dated amount on its own, then average the values:

P = \tfrac{1}{2}\,\underbrace{7\,Z(1.5)}_{\text{heads, discounted}} \;+\; \tfrac{1}{2}\,\underbrace{(-2)}_{\text{tails, already today}}

(1)

where $Z(1.5)$ is the price today of one dollar delivered at eighteen months. The tails leg is not discounted at all, because it falls immediately. That asymmetry is the whole gap between $2.50$ and the real number.

Fig. 1 — The two legs live at different dates. Only the delayed one travels, and it travels through two different rates.

The eighteen-month rate is not quoted, so build it

The curve gives one and two years. Eighteen months sits between them, so the discount factor has to be assembled out of what is quoted. The route with the least guesswork is to strip the second year out of the two-year rate first.

Forward rate

The rate $f$ that money must earn from year one to year two if lending for two years at the two-year rate is to match lending for one year and reinvesting. It is defined by $(1+r_1)(1+f) = (1+r_2)^2$ , and it is implied by the quoted curve rather than chosen.

Solving that with $r_1 = 0.12$ and $r_2 = 0.18$ gives an exact rational:

f = \frac{(1.18)^2}{1.12} - 1 = \frac{681}{2800} = 24.3214\overline{285714}\%

(2)

No rounding enters anywhere. The consistency check runs in exact fractions too: $1.12 \times \left(1 + \tfrac{681}{2800}\right) = \tfrac{3481}{2500} = (1.18)^2$ . A steeply rising curve does this: two years at $18\%$ after one year at only $12\%$ forces the second year to carry more than $24\%$ on its own.

Discounting the delayed leg

Eighteen months is one full year at the spot rate followed by half of the forward year. Compounding the forward rate geometrically over that half year gives

\frac{7}{1.12\left(1+\frac{681}{2800}\right)^{1/2}} \;=\; \frac{875\sqrt{7}}{413} \;=\; 5.60540\ldots

(3)

A surd appearing in an interest calculation is a good sign, not a bad one: it is the square root of the forward growth factor, and it means the half year was handled by compounding rather than by halving a percentage. Averaging that against the immediate payment gives the price.

P = \frac{1}{2}\cdot\frac{875\sqrt{7}}{413} - \frac{1}{2}\cdot 2 = \frac{875\sqrt{7}}{826} - 1 = 1.8027\ldots

(4)

Compare that with the reflex figure. The ratio $2.50 / 1.8027 = 1.3868$ puts the averaging answer $38.68\%$ too high, so this is a structural error rather than a rounding quibble. Two effects push in the same direction: a receipt eighteen months out is worth substantially less than its face value, while a payment due today is worth exactly its face value and gets no such relief.

Why the cents are not knowable

Now the uncomfortable part, which is also the interesting part. Equation (4) prints four decimals and only one of them is defensible. Nothing in the problem states how a rate is supposed to act over a fraction of a year, and reasonable conventions disagree in the second decimal.

Price it four ways. Compounding the forward geometrically over the half year, as above, gives $1.8027$ . Treating eighteen months as a year and a half of the one-year rate, $3.5/(1.12)^2 - 1$ , gives $1.7902$ . Applying simple interest at the forward rate for half a year gives $1.7862$ . Reading an interpolated flat $15\%$ off the curve at the eighteen-month point gives $1.8381$ . The spread runs from $1.7862$ to $1.8381$ .

Fig. 2 — Every defensible convention lands inside the band that rounds to 1.8 at one decimal. The two-decimal figures are 1.79, 1.80 and 1.84, so no cent is unanimous.

The whole interval sits inside $[1.75,\, 1.85)$ , so every convention rounds to $1.8$ at one decimal place. That is exactly what an honest answer can claim here: one decimal, unanimous; two decimals, not. Quoting $\$1.80$ to the cent would be asserting a convention the problem never fixed.

Where this valuation stops being the right one

Averaging the two outcomes with weight one half each is a risk-neutral valuation, and it needs a defence. Here the defence is size. Amounts of a few dollars are small enough that nobody's tolerance for risk changes measurably, so the expected present value is the price a rational counterparty will trade at.

Scale the same contract by a factor of a million and that stops being true. A contract paying seven million or costing two million is a material exposure, a buyer will demand compensation for carrying it, and the price falls below the risk-neutral figure by whatever premium the market charges. The valuation in (4) is a limit for small amounts.

Two smaller assumptions are worth naming. The coin is taken to be independent of interest rates, which is what allows the probability weighting and the discounting to be applied separately; if the payoff were correlated with rates, each leg would need its own risk-adjusted measure. And the quoted numbers are treated as zero-coupon spot yields. If they were par yields on coupon-paying instruments, the discount factors would have to be bootstrapped from the coupons first and the forward would come out slightly different. That is a third source of movement in the cents, and it points the same way as the first two: the structure of the answer is exact, and the last digit belongs to a convention.

Sources and further reading

Wikipedia: Present value and Compound interest, for the discount factor and the compounding conventions compared above.
Wikipedia: Forward rate and Bootstrapping (finance), on extracting an unquoted rate from a quoted curve.
Wikipedia: Day count convention, which is the institutional answer to the question the four conventions above leave open.
Irving Fisher, The Theory of Interest (1930), for the original careful treatment of why a dated amount and its present value are different quantities.

The forward rate above was checked twice: against the closed form, and by two hundred steps of exact-fraction bisection on $(1+r_1)(1+f) - (1+r_2)^2$ , which lands on $681/2800$ with a residual of exactly zero.