Finance-Theory · Unit 6 · Video 7 · Interactive Practice

What the Treasury Yield Curve Plots, How Far Rates Have Moved, and What Its Shape Says

IKey Formulas

Formula	Name	What you need
$P_0 = \dfrac{C}{1+y} + \dfrac{C}{(1+y)^2} + \cdots + \dfrac{C+F}{(1+y)^T}$	Yield to maturity $y$ : the one rate the published curve plots, a point per maturity	A coupon bond's price $P_0$ , coupon $C$ , face value $F$
$P_0 = \dfrac{C}{1+r_{0,1}} + \dfrac{C}{(1+r_{0,2})^2} + \cdots + \dfrac{C+F}{(1+r_{0,T})^T}$	The same price from the spot rates, one per payment date	The term structure $r_{0,1}, \ldots, r_{0,T}$
$C = 0 \;\Rightarrow\; y = r_{0,T}; \quad C > 0 \;\Rightarrow\; y$ blends $r_{0,1}, \ldots, r_{0,T}$	Why a Strips curve plots spot rates and a coupon curve only approximates them	Whether the bond pays coupons
$\dfrac{18.63\%}{6.95\%} \approx 2.68$	A year's interest on the same balance, Oct 1981 peak against Sep 2026 (30-year fixed mortgage)	Two rates on the same balance

Key Insight: the published Treasury curve plots coupon-bond yields $y$ , not spot rates $r_{0,T}$ ; with coupons that are similar and not extreme, it is a good proxy for the Strips curve, and read as a forecast its shape gives the market's expectations of future rates.

IISame Maturity, Different Coupon, Different Yield

A five-year bond's yield $y$ blends five spot rates, weighted by its own payments, so the coupon moves $y$ .

Coupon rate 4.5%

IIIHow Far Rates Have Moved

Against the calendar, the 1-year Treasury yield has run from above 17% to close to zero.

IVWhat the Shape Says

Read as a forecast, each rising segment says rates are expected to rise, each falling one that they will come down.

Segment

💡 In March 2000 the Fed was still raising rates, and it began cutting them in January 2001. Expectations may not be the whole explanation: in 2000 the Treasury was also buying back long-term bonds, which reduced their supply.

VQuiz Questions

Problem 1 · Then and Now

Given: the 30-year fixed mortgage rate peaked at $18.63\%$ in October 1981 and averaged $6.95\%$ in the week ending 17 September 2026 — find how many times as large a year's interest on the same $200,000 balance was at the 1981 peak.

✅ Correct! The balance cancels:

18.63 / 6.95 \approx 2.68

, so a year's interest at the 1981 peak was about 2.68 times as large.

❌ Upside down.

6.95 / 18.63 \approx 0.37

is the 2026 interest as a fraction of the 1981 interest; the question puts the 1981 peak on top.

❌ That is how much more, not how many times.

(18.63 - 6.95)/6.95 \approx 1.68

is the extra interest as a multiple of the 2026 interest; add the 2026 interest itself back to get the ratio.

❌ That is a difference in percentage points.

18.63 - 6.95 = 11.68

points; a ratio divides the two rates.

Show solution

A year's interest is rate times balance: at the peak $0.1863 \times 200{,}000 = 37{,}260$ dollars, in September 2026 $0.0695 \times 200{,}000 = 13{,}900$ dollars.

\frac{37{,}260}{13{,}900} = \frac{18.63}{6.95} = 2.6806\ldots \approx 2.68

The balance cancels, so the ratio of the rates is the answer for any balance.

Problem 2 · Same Maturity, Two Coupons

Given: spot rates rise with maturity, and two 10-year Treasury notes trade today, one with a $2.3\%$ coupon rate and one with a $4.5\%$ coupon rate — which statement about their yields to maturity is right?

✅ Correct! Each yield blends the spot rates

r_{0,1}, \ldots, r_{0,10}

in proportions set by the note's own payments; the bigger coupons tilt the blend toward the lower short rates.

❌ Only a zero-coupon bond has $y = r_{0,T}$ . A coupon note's yield compresses one spot rate per payment date into a single number, so it is not

r_{0,10}

❌ The coupon rate is not the yield. A higher coupon puts more weight on early payments; with a rising spot curve those are discounted at lower rates, which pulls the yield down, not up.

❌ The yields sit much closer than the coupons. Both yields are blends of the same spot rates; the coupons only shift the weights, so the yields differ by a small amount, which is why the coupon curve is a reasonable proxy for the Strips curve.

Show solution

From the spot rates, each note's price is $P_0 = \sum_{t=1}^{10} \dfrac{C}{(1+r_{0,t})^t} + \dfrac{F}{(1+r_{0,10})^{10}}$ , and its yield $y$ is the single rate that reproduces that price.

A zero-coupon bond has all its value at year 10, so its $y$ is exactly $r_{0,10}$ . Adding coupons moves part of the value to earlier dates, which the rising curve discounts at lower rates; the single rate that matches the price is pulled below $r_{0,10}$ , and more so the larger the coupon.

So the $4.5\%$ note's yield is a little below the $2.3\%$ note's: same maturity, different coupon rate, different yield. Visualization II shows the same effect for a five-year bond: at a $2.3\%$ coupon $y \approx 4.37\%$ , at $4.5\%$ $y \approx 4.35\%$ .

Problem 3 · From Spot Rates to a Yield

Given: spot rates $r_{0,1} = 3\%$ and $r_{0,2} = 4\%$ , and a 2-year Treasury note with face value $100 paying a $5\%$ annual coupon ($5 at year 1, $105 at year 2) — find its price and its yield to maturity.

What is the price?

What is the yield to maturity?

✅ Correct! The price is $101.93, and the single rate that reproduces it is

y \approx 3.98\%

: close to

r_{0,2} = 4\%

but pulled below it by the year-1 coupon, discounted at

3\%

❌ Check the price. Discount each payment at the spot rate for its own date: $5 at

r_{0,1}

, $105 at

r_{0,2}

❌ Both payments discounted at $4\%$ . The year-1 coupon takes the 1-year spot rate

r_{0,1} = 3\%

, not

r_{0,2}

❌ The year-2 coupon is missing. The final payment is the $5 coupon plus the $100 face value, $105, not $100.

❌ Both payments discounted at $3\%$ . The year-2 payment takes the 2-year spot rate

r_{0,2} = 4\%

❌ Check the yield.

y

is the one rate for which

\frac{5}{1+y} + \frac{105}{(1+y)^2}

equals the price.

❌ That is $r_{0,2}$ , the zero-coupon answer. At

y = 4\%

the formula gives about 101.89, below the price; the coupon's weight on the

3\%

date pulls

y

a little under

4\%

❌ A plain average of the spot rates ignores the weights. Almost all of the price comes from the $105 at year 2, so

y

sits near

r_{0,2}

, not halfway.

❌ That is the coupon rate.

y

equals the coupon rate only for a bond priced at face value; this note trades above $100, so its yield is below

5\%

Show solution

Price from the spot rates:

P_0 = \frac{5}{1.03} + \frac{105}{1.04^2} = 4.8544 + 97.0784 = 101.9328 \approx 101.93

Yield: solve $\dfrac{5}{1+y} + \dfrac{105}{(1+y)^2} = 101.93$ . At $y = 4\%$ the left side is $4.8077 + 97.0784 = 101.8861$ , too low, so $y$ is a little below $4\%$ ; solving gives $y = 3.977\%$ , and $\dfrac{5}{1.03977} + \dfrac{105}{1.03977^2} = 101.930$ .

So $P_0 \approx$ $101.93 and $y \approx 3.98\%$ : one number standing in for the two spot rates, weighted toward $r_{0,2}$ because the $105 dominates the price.

Problem 4 · Reading a Shape

Given: a Treasury curve with $3$ -month $4.00\%$ , $2$ -year $4.60\%$ , $10$ -year $4.30\%$ and $30$ -year $4.10\%$ — which reading is right?

✅ Correct! The curve rises from

4.00\%

4.60\%

at the short end and falls to

4.30\%

and

4.10\%

further out, the same humped shape as 22 March 2000; expectations are the forecast reading, but not necessarily the whole explanation.

❌ The endpoints hide the shape. From

2

years on the curve falls (

4.60\% \to 4.30\% \to 4.10\%

), so it turns down at the long end.

❌ The short end rises. From

3

months to

2

years it goes

4.00\% \to 4.60\%

, upward.

❌ Supply can matter too. In 2000 the Treasury was buying back long-term bonds, reducing their supply, alongside whatever the market expected; explaining the shape is the job of term-structure models.

Show solution

Short end: $4.00\% \to 4.60\%$ , up $0.60$ points; read as a forecast, rates are expected to rise over the next couple of years.

Long end: $4.60\% \to 4.30\% \to 4.10\%$ , down $0.30$ then $0.20$ points; a curve that turns down at the long end says the market expects rates to come down in the long run.

Up at the short end and down further out is a humped curve, like 22 March 2000. The forecast reading is one explanation; supply, as with the Treasury's 2000 buybacks of long bonds, can also shape the long end.

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