Finance-Theory · Unit 6 · Video 2 · Interactive Practice

Two Prices Today, One Rate Next Year: Spot Rates and the Term Structure

IKey Formulas

Formula	Name	What it says
$P_0 = \dfrac{F}{(1+R_1)(1+R_2)\cdots(1+R_T)}$	Price from the chain of one-year rates	A theory of the price: the one-year rates $R_t$ (date $t-1$ to date $t$ ) compound in sequence, and the face value $F$ is discounted by their product. The $R_t$ are not observed today
$r_{0,T} = \left[(1+R_1)(1+R_2)\cdots(1+R_T)\right]^{1/T} - 1$	$T$ -year spot rate	One rate that, applied every year, gives the same growth factor as the chain: a geometric average of the one-year rates, not the rate for year $T$ alone
$r_{0,T} = \left(\dfrac{F}{P_{0,T}}\right)^{1/T} - 1$	Spot rate from a price	Price and face value are observed, so the spot rate follows: $r_{0,5} = (1/0.797)^{1/5} - 1 \approx 4.64\%$ . Below a year the exponent $1/T$ exceeds 1: $r_{0,\frac12} = (1/0.983)^{2} - 1 \approx 3.49\%$ , an annual rate
$\dfrac{P_{0,1}}{P_{0,2}} = 1+R_2$	Forecast from two prices	With the future one-year rates taken as known today, $F$ and $(1+R_1)$ cancel. Both prices are observed today, so the ratio is their forecast of $R_2$ : $0.967/0.927 \approx 1.0431$ , $R_2 \approx 4.31\%$

Key Insight: The spot rates are observable, and inside them sit the unobservable one-year rates, or at least today's expectations of them. For a one-year bond the chain holds one year, so $r_{0,1} = R_1$ ; comparing the one-year and two-year prices then isolates the market's forecast of $R_2$ .

IIOne Day's Term Structure, Written as Prices

Every STRIPS price converts to a spot rate by the same inversion, maturity by maturity.

Maturity T 5 yr

IIITwo Prices, One Forecast

Dividing the one-year price by the two-year price leaves $1+R_2$ : today's forecast of next year's one-year rate.

One-year price P₀,₁ 0.967 Two-year price P₀,₂ 0.927

IVA Spot Rate Averages the Whole Chain

The five-year spot rate is a geometric average of five one-year rates, not the rate for year five.

R₁ 3.50% R₂ 4.25% R₃ 4.75% R₄ 5.25% R₅ 5.50%

💡 Only $P_0$ and $F$ are observed; the one-year rates in the chain are not, so $r_{0,5}$ is what the market's price reveals about them, as one average.

VQuiz Questions

Problem 1 · Read a Spot Rate off a Price

Given: on 1 August 2001 the ten-year STRIPS traded at 0.605 per $1 of face value. Find the ten-year spot rate $r_{0,10}$ .

✅ Correct!

0.605 = 1/(1+r_{0,10})^{10}

, so

r_{0,10} = (1/0.605)^{1/10} - 1 \approx 5.15\%

: one of the rates in that day's term structure.

❌ That spreads the total return evenly.

(1/0.605 - 1)/10 \approx 6.53\%

ignores compounding. The rate compounds for ten years, so take the tenth root of

1/0.605

instead of dividing by ten.

❌ That is the ten-year total return.

1/0.605 - 1 \approx 65.29\%

is growth over the whole decade. The spot rate is annual: raise

1/0.605

to the power

1/10

before subtracting 1.

❌ The price was not inverted.

1 - 0.605^{1/10} \approx 4.90\%

is a rate of discount on the price. The spot rate grows 0.605 into 1:

(1/0.605)^{1/10} - 1

Show solution

A STRIPS is a pure discount bond, so one dollar in ten years discounted back ten years gives the price:

0.605 = \frac{1}{(1+r_{0,10})^{10}} \quad\Longrightarrow\quad r_{0,10} = \left(\frac{1}{0.605}\right)^{1/10} - 1

Since $1/0.605 \approx 1.652893$ and $1.652893^{1/10} \approx 1.051537$ , the ten-year spot rate is about 5.15% a year.

Problem 2 · Below a Year, the Exponent Flips

Given: on the same day the three-month STRIPS traded at 0.991 per $1 of face value, so $T = \tfrac14$ . Find the annual spot rate $r_{0,\frac14}$ .

✅ Correct! The exponent is

1/T = 1/(1/4) = 4

, so the ratio is raised to the fourth power:

(1/0.991)^4 - 1 \approx 3.68\%

a year.

❌ The exponent went the wrong way.

(1/0.991)^{1/4} - 1

takes a fourth root. The exponent is

1/T

, and with

T = \tfrac14

that is

4

: below a year the ratio is raised to a power greater than 1.

❌ That is the three-month return.

1/0.991 - 1 \approx 0.91\%

is earned over a quarter of a year. The spot rate is an annual rate, so compound it over four quarters.

❌ Multiplying by four skips compounding.

4 \times 0.908\% \approx 3.63\%

adds four quarters of return without growth on growth. The spot rate compounds:

(1/0.991)^4 - 1

Show solution

The price of one dollar in a quarter of a year satisfies $0.991 = 1/(1+r_{0,\frac14})^{1/4}$ . Solving, the exponent becomes $1/T = 4$ :

r_{0,\frac14} = \left(\frac{1}{0.991}\right)^{4} - 1 \approx 1.036825 - 1

The three-month spot rate is about 3.68%, stated as an annual rate.

Problem 3 · Forecast, Then Average

Given: a one-year pure discount bond sells for $P_{0,1} = 0.960$ and a two-year one sells for $P_{0,2} = 0.910$ , both per $1 of face value. Take the future one-year rates as known today.

What forecast of $R_2$ do the two prices give?

What is the two-year spot rate $r_{0,2}$ ?

✅ Correct! The ratio gives

1 + R_2 = 0.960/0.910 \approx 1.0549

, and the two-year spot rate

\sqrt{1/0.910} - 1 \approx 4.83\%

sits between

R_1 \approx 4.17\%

and

R_2 \approx 5.49\%

❌ That subtracts the prices.

0.960 - 0.910 = 0.050

is not a rate. The forecast comes from the ratio:

P_{0,1}/P_{0,2} = 1 + R_2

❌ That is the two-year spot rate.

r_{0,2}

averages both years. The forecast is for year 2 alone:

0.960/0.910 - 1

❌ That is the two-year total return.

1/0.910 - 1 \approx 9.89\%

uses only the two-year price. Dividing by the one-year price removes the first year's factor.

❌ Halving the total return skips compounding.

(1/0.910 - 1)/2 \approx 4.95\%

. The spot rate is the square root:

\sqrt{1/0.910} - 1

❌ That is the rate for year 2 alone.

r_{0,2}

is not the rate for year

T

; it is the geometric average of

R_1

and

R_2

❌ That is the rate for year 1 alone.

R_1 = 1/0.960 - 1 \approx 4.17\%

equals

r_{0,1}

. The two-year spot rate averages it with

R_2

❌ Not quite. Divide the one-year price by the two-year price and subtract 1.

❌ Not quite. Solve

0.910 = 1/(1+r_{0,2})^2

for

r_{0,2}

Show solution

Forecast. With the same face value, $F$ and $(1+R_1)$ cancel in the ratio of the prices:

\frac{P_{0,1}}{P_{0,2}} = 1 + R_2 = \frac{0.960}{0.910} \approx 1.0549 \quad\Longrightarrow\quad R_2 \approx 5.49\%

Spot rate. The two-year price alone gives it:

r_{0,2} = \sqrt{\frac{1}{0.910}} - 1 \approx 1.048285 - 1 \approx 4.83\%

Check through the chain. $1 + R_1 = 1/0.960$ , and $\sqrt{(1/0.960)(0.960/0.910)} = \sqrt{1/0.910}$ , so both routes give the same $r_{0,2}$ : the geometric average of $R_1 \approx 4.17\%$ and $R_2 \approx 5.49\%$ .

Problem 4 · From Two Spot Rates to a Forecast

Given: the one-year spot rate is $r_{0,1} = 3\%$ and the two-year spot rate is $r_{0,2} = 4\%$ . Find the forecast of $R_2$ , the one-year rate from year 1 to year 2, that these rates imply.

✅ Correct!

(1.04)^2 = (1.03)(1+R_2)

, so

1 + R_2 = 1.0816/1.03 \approx 1.0501

and

R_2 \approx 5.01\%

❌ That treats $r_{0,2}$ as an arithmetic average.

2 \times 4\% - 3\% = 5\%

would make 4% the simple mean of 3% and 5%, but

\sqrt{(1.03)(1.05)} - 1 \approx 3.9952\%

, short of 4%. Compound:

1.04^2 / 1.03

❌ The two-year spot rate is not the rate for year 2.

r_{0,2}

averages

R_1

and

R_2

. With

R_1 = 3\%

below the average,

R_2

must sit above 4%.

❌ That is the gap between the spot rates.

4\% - 3\%

is not a one-year rate. Solve

(1.04)^2 = (1.03)(1 + R_2)

Show solution

For one year the chain holds one factor, so $R_1 = r_{0,1} = 3\%$ . The two-year spot rate compounds to the same growth as the chain of two one-year rates:

(1 + r_{0,2})^2 = (1+R_1)(1+R_2) \quad\Longrightarrow\quad 1 + R_2 = \frac{1.04^2}{1.03} = \frac{1.0816}{1.03} \approx 1.050097

The forecast is $R_2 \approx 5.01\%$ . The same number comes from prices per $1 of face value: $P_{0,1} = 1/1.03 \approx 0.970874$ , $P_{0,2} = 1/1.0816 \approx 0.924556$ , and their ratio is $1.0816/1.03$ again.

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