Finance-Theory · Unit 5 · Video 4 · Interactive Practice

Any Two Give the Third: Zero-Coupon Bonds, STRIPS, and the Forecast Inside a Bond Price

IKey Formulas

Formula	Name	What you need
$P_0 = \dfrac{F}{(1+r)^T}$	Price of a pure discount bond	Face value $F$ and rate $r$
$r = \left(\dfrac{F}{P_0}\right)^{1/T} - 1$	Rate implied by a price	Price $P_0$ and face value $F$
$F = P_0\,(1+r)^T$	Face value from price and rate	Rate $r$ and price $P_0$
$r > 0,\; T > 0 \;\Rightarrow\; (1+r)^T > 1 \;\Rightarrow\; P_0 < F$	Why the price is a discount	A positive rate

Key Insight: with the maturity $T$ known, $(P_0,\ r,\ F)$ is over-determined: any two pin down the third, so a market price can be run backwards into the interest rate it implies.

IIStripping a Coupon Bond

Stripping makes every payment of a coupon bond its own zero; priced at one rate, the zeros add up to the bond.

Bond ↔ zeros bond Market rate r

IIIWhy a Discount, and How Deep

With $r > 0$ , $(1+r)^T > 1$ , so a zero's price sits below face value, and further below as $r$ or $T$ grows.

Rate r 5%

IVAny Two Give the Third

With $T$ known, any two of $P_0$ , $r$ and $F$ pin down the third.

Solve for Maturity T 20 yr

Price P₀ Rate r Face F

💡 Run on market prices, the price → rate direction yields the rates those prices imply, and implicit in them is a forecast of future rates. On Monday 15 September 2008 Lo read Treasury prices and fed funds futures as a Fed cut the next day with 99.5% confidence; on Tuesday the Fed held its target at 2%, and the forecast missed.

VQuiz Questions

Problem 1 · Pricing a Zero

Given: a pure discount bond with face value $1,000 that matures in $T = 5$ years, and an interest rate of $r = 4\%$ a year — find its price today.

✅ Correct!

(1.04)^5 = 1.216653

, and $1,000 divided by

1.216653

is $821.93. The $178.07 gap to face value is the holder's interest over the five years.

❌ That discounts for one year only. The payment is five years away, so the face value is divided by

(1.04)^5

, not by

1.04

❌ That grows $1,000 forward five years. The face value arrives at maturity, so today's price divides by the growth factor:

F/(1+r)^T

, not

F\,(1+r)^T

. With a positive rate the price has to come out below $1,000.

❌ That is simple interest.

1 + 5 \times 0.04 = 1.20

ignores interest on interest. Compounded, the divisor is

(1.04)^5 = 1.2167

, which is larger, so the price is lower.

Show solution

Price the zero with the one-line formula, in dollars:

P_0 = \frac{F}{(1+r)^T} = \frac{1{,}000}{(1.04)^5} = \frac{1{,}000}{1.216653} = 821.93

The zero costs $821.93 today. Since $r > 0$ , $(1.04)^5 > 1$ , and the price sits below the $1,000 face value; the $178.07 difference is the interest the holder earns by waiting five years.

Problem 2 · Reading the Rate from a Price

Given: a principal strip that pays $1,000 in $T = 10$ years trades today at $675.56 — find the annual interest rate its price implies.

✅ Correct!

\left(\dfrac{1{,}000}{675.56}\right)^{1/10} - 1 = 0.0400

, and pricing back,

\dfrac{1{,}000}{(1.04)^{10}} = 675.56

: the rate and the price are the same fact about the strip.

❌ That is the return over all ten years,

\dfrac{1{,}000}{675.56} - 1 = 48.03\%

. An annual rate takes the tenth root of the ratio before subtracting one.

❌ That divides the $324.44 of interest by the face value and by ten. The rate is earned on the price paid, and it compounds:

r = (F/P_0)^{1/T} - 1

❌ Dividing the $48.03\%$ total by ten ignores compounding. Ten years at

4.80\%

would grow the price past face value:

(1.048)^{10} = 1.598

, and $675.56 × 1.598 is about $1,080, not $1,000.

Show solution

Given the price and the face value, the rate is the middle direction of the triple:

r = \left(\frac{F}{P_0}\right)^{1/T} - 1 = \left(\frac{1{,}000}{675.56}\right)^{1/10} - 1 = (1.480253)^{0.1} - 1 = 0.0400

so the strip's price implies $4.00\%$ a year.

Check by pricing back: $\dfrac{1{,}000}{(1.04)^{10}} = \dfrac{1{,}000}{1.480244} = 675.56$ , the quoted price.

The tempting shortcut, $324.44 of interest over $675.56 is $48.03\%$ , then divided by ten, gives $4.80\%$ . That treats interest as simple: the $48.03\%$ total already contains interest on interest, so a tenth of it overstates the annual rate.

Problem 3 · Strip a Two-Year Note

Given: a two-year note with face value $1,000 that pays a $4\%$ coupon once a year is stripped into zeros, and every zero is priced at the market rate $r = 3\%$ — find the price of the principal strip, then the price of the whole note.

What is the principal strip worth?

What is the whole note worth?

✅ Correct! The three strips cost $38.83 + $37.70 + $942.60 = $1,019.13, which is the note's price at

3\%

. It sits above par because the market rate is below the

4\%

coupon rate.

❌ Not the principal strip. It is a zero that pays $1,000 in two years:

1{,}000/(1.03)^2

❌ That discounts one year only. The principal is paid at year 2, so the divisor is

(1.03)^2 = 1.0609

❌ That bundles the year-2 coupon into the principal strip. Stripping separates them: the $40 coupon at year 2 is its own coupon strip, and the principal strip pays only the $1,000.

❌ $1,000 is what the strip pays in two years, not its price today. At a positive rate, a zero's price sits below its face value.

❌ Not the note's price. Add the prices of all three strips: two coupon strips and the principal strip.

❌ Par is the price only when the market rate equals the coupon rate. Here the market rate is

3\%

and the coupon rate

4\%

, so the note is worth more than $1,000.

❌ That adds the payments without discounting them. $40 + $40 + $1,000 is what the note pays in total; each payment must first be priced as a zero.

❌ One strip is missing. $38.83 + $942.60 leaves out the year-2 coupon strip, worth

40/(1.03)^2 = 37.70

dollars.

Show solution

Step 1 — the pieces. The note pays $40 at year 1, and $40 plus $1,000 at year 2. Stripping makes three zeros, each priced with $P_0 = F/(1+r)^T$ (in dollars):

\text{coupon strip, year 1: } \frac{40}{1.03} = 38.83 \qquad \text{coupon strip, year 2: } \frac{40}{(1.03)^2} = 37.70

\text{principal strip, year 2: } \frac{1{,}000}{(1.03)^2} = \frac{1{,}000}{1.0609} = 942.60

Step 2 — the whole note. Priced at one rate, the zeros add up to the note:

38.83 + 37.70 + 942.60 = 1{,}019.13

Check with the coupon-bond formula: $\dfrac{40}{1.03} + \dfrac{1{,}040}{(1.03)^2} = 38.83 + 980.30 = 1{,}019.13$ , the same $1,019.13. Stripping repackages the note's payments without changing them, so, priced at one rate, the pieces cost what the note costs.

Problem 4 · A Price Above Face Value

Given: euro interest rates were negative for years (two-year German government yields sat below zero through 2020). A two-year zero with face value €1,000 trades at €1,010.08 — find the annual rate its price implies.

✅ Correct!

\left(\dfrac{1{,}000}{1{,}010.08}\right)^{1/2} - 1 = -0.0050

, a rate of

-0.50\%

a year; pricing back,

\dfrac{1{,}000}{(0.995)^2} = 1{,}010.08

❌ That is the two-year total,

\dfrac{1{,}000}{1{,}010.08} - 1 = -1.00\%

. Take the square root of the ratio to get the rate per year.

❌ The ratio is upside down. The rate solves

F = P_0(1+r)^T

, so it is

(F/P_0)^{1/T} - 1

. Paying more today than you get back at maturity means a negative rate.

❌ The formula forces $P_0 < F$ only when $r > 0$ . With

r < 0

(1+r)^T < 1

, and dividing

F

by a number below one gives a price above face value.

Show solution

Read the rate from the price and the face value:

r = \left(\frac{F}{P_0}\right)^{1/T} - 1 = \left(\frac{1{,}000}{1{,}010.08}\right)^{1/2} - 1 = (0.990021)^{1/2} - 1 = 0.994998 - 1 = -0.005002

so the price implies about $-0.50\%$ a year.

Check by pricing back: $(1 - 0.005)^2 = 0.990025$ , and $\dfrac{1{,}000}{0.990025} = 1{,}010.08$ .

The argument that a zero trades at a discount runs through $(1+r)^T > 1$ , which needs $r > 0$ . Here $(1+r)^T = 0.990025 < 1$ , so the price lands above face value and the holder pays €10.08 more than the bond returns.

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