Finance-Theory · Unit 6 · Video 8 · Interactive Practice

Four Theories of the Yield Curve, and the Arbitrage That Prices Every Coupon Bond

IKey Formulas

Formula	Name	What it says
$\mathrm{E}_0[R_k] = f_k$	Expectations Hypothesis	Today's forward rate is an unbiased forecast of the future one-year spot rate $R_k$
$\mathrm{E}_0[R_k] = f_k - \text{LP}_k$	Liquidity Preference (the rival theory)	Forward rates carry a premium that grows with maturity (none in year 1), so $\mathrm{E}_0[R_k] < f_k$ for $k \ge 2$
$P_{0,t} = \dfrac{1}{(1+r_t)^t}$	STRIPS price	One dollar at date $t$ , discounted at its own spot rate $r_t$
$P_0 = 50\,P_{0,1} + 50\,P_{0,2} + 1{,}050\,P_{0,3}$	Coupon bond = package of STRIPS	The video's bond: $961.36 at spot rates 5%, 6%, 6.5%

Key Insight: the theories disagree about why the curve slopes, and none fully explains it. The coupon-bond price needs none of them: if $P_0$ differed from the cost of its STRIPS, buying the cheaper side and short-selling the dearer one would be money today with nothing owed later.

IIOne Curve, Two Readings

Forward rates are expected future spot rates plus any liquidity premium, so a premium alone can tilt the curve upward.

Theory Premium per extra year 0.25% Expected path

💡 Preferred Habitat lets the premium differ by maturity instead of growing steadily, which can put bumps and dips into the curve; Market Segmentation treats each maturity range as a separate market, its rate set by supply and demand in that range alone.

IIIA Coupon Bond Is a Package of STRIPS

Date by date, a coupon bond's cash flows are exactly those of a package of STRIPS, so its price is the package's cost.

Coupon rate 5% r₁ 5% r₂ 6% r₃ 6.5%

IVThe Arbitrage That Fixes the Price

Any gap between the bond's price and its package's cost is money today with nothing owed later, so it cannot last.

Starting bond price Gap closed 0%

VQuiz Questions

Problem 1 · Price a Bond from Its STRIPS

Given: a two-year bond with $1,000 par and a 4% annual coupon; STRIPS prices $P_{0,1} = 0.97$ and $P_{0,2} = 0.93$ per dollar. Find the bond's price.

✅ Correct! 40 one-year STRIPS plus 1,040 two-year STRIPS deliver the bond's cash flows exactly, so the bond costs what they cost.

❌ Not quite. That prices only 1,000 two-year STRIPS. Year 2 pays the last coupon as well as the face value: $1,040, not $1,000.

❌ Not quite. A coupon bond is not automatically at par. Here the STRIPS prices imply spot rates below the 4% coupon, so the package costs more than $1,000.

❌ Not quite. The STRIPS prices are swapped: the year-2 cash flow must be discounted with the two-year price 0.93, not the one-year price 0.97.

❌ Not quite. Match each cash flow to the STRIPS maturing on its date.

Show solution

The bond pays $40 at year 1 and $1,040 (last coupon plus face value) at year 2. Buy 40 one-year STRIPS and 1,040 two-year STRIPS:

P_0 = 40\,P_{0,1} + 1{,}040\,P_{0,2} = 40(0.97) + 1{,}040(0.93) = 38.80 + 967.20 = 1{,}006.00

The bond is worth $1,006.00, above par.

Problem 2 · Reading a Rising Curve

Given: the yield curve slopes upward. Which conclusion does Liquidity Preference allow?

✅ Correct! With

\mathrm{E}_0[R_k] = f_k - \text{LP}_k

and a premium that grows with maturity, forwards sit above today's one-year rate even when no change is expected.

❌ Not quite. That is the Expectations Hypothesis reading, where the slope comes from expectations only. Under Liquidity Preference the premium alone can make the curve rise.

❌ Not quite. Unbiased forwards are the Expectations Hypothesis. Liquidity Preference says the forward overstates the expected spot rate by the premium.

❌ Not quite. The inequality runs the other way: the premium is built into the forward, so

\mathrm{E}_0[R_k] < f_k

❌ Not quite. Compare

\mathrm{E}_0[R_k] = f_k - \text{LP}_k

with the Expectations Hypothesis.

Show solution

Liquidity Preference: lenders must be paid to give up the use of their money for longer, so $f_k = \mathrm{E}_0[R_k] + \text{LP}_k$ with the premium growing in $k$ . If no change is expected, every $\mathrm{E}_0[R_k]$ equals today's one-year rate, yet the forwards rise with the premium and the curve slopes upward.

So a rising curve is consistent with rates expected to stay where they are; it can also reflect an expected rise plus the premium. The curve can still slope downward if the market expects rates to fall by more than the premium adds.

Problem 3 · From Spot Rates to an Expected Rate

Given: the video's spot rates $r_1 = 5\%$ and $r_2 = 6\%$ , and a year-2 liquidity premium $\text{LP}_2 = 0.50\%$ . Find the market's expectation $\mathrm{E}_0[R_2]$ of the one-year rate that will prevail in year 2, under Liquidity Preference.

✅ Correct! The forward rate is

7.01\%

; removing the

0.50\%

premium leaves an expected rate of

6.51\%

❌ Not quite. The premium is already inside the forward rate.

\mathrm{E}_0[R_2] = f_2 - \text{LP}_2

: subtract it, don't add it.

❌ Not quite.

r_2

is the two-year spot rate, an average over years 1 and 2. The rate for year 2 alone is the forward rate

f_2

❌ Not quite.

7.01\%

is the forward rate itself, which is the Expectations Hypothesis answer. Liquidity Preference subtracts the premium.

❌ Not quite. First find

f_2

from the two spot rates, then apply

\mathrm{E}_0[R_2] = f_2 - \text{LP}_2

Show solution

Step 1: forward rate for year 2. Two years at $r_2$ must equal one year at $r_1$ followed by one year at $f_2$ :

1 + f_2 = \frac{(1.06)^2}{1.05} = \frac{1.1236}{1.05} \approx 1.070095 \quad\Rightarrow\quad f_2 \approx 7.01\%

Step 2: remove the premium.

\mathrm{E}_0[R_2] = f_2 - \text{LP}_2 = 7.01\% - 0.50\% = 6.51\%

Under the Expectations Hypothesis the answer would be $7.01\%$ ; the premium is the whole difference between the two readings.

Problem 4 · A Bond Below Its Package

Given: the video's bond ($1,000 par, 5% coupon, 3 years) trades at $955.00, while its package of STRIPS costs $961.36. Which position is an arbitrage?

✅ Correct! Buy the cheap side, short the dear side: the bond's coupons and face value cover every STRIPS you owe.

❌ Not quite. That is the video's trade for a bond priced ABOVE the package. Here it pays $961.36 and takes in $955.00: −$6.36 today.

❌ Not quite. The bond you hold pays $50, $50 and $1,050 on exactly the dates the short STRIPS come due, so every later column nets to zero.

❌ Not quite. Being below par is not the issue: the package is below par too, at $961.36. The bond is below its own package, and that gap is free money.

❌ Not quite. Buy whichever side is cheaper and short-sell the dearer one.

Show solution

The bond is cheaper than its package, so buy the bond and short-sell the STRIPS (50 one-year, 50 two-year, 1,050 three-year).

Position	Year 0	Year 1	Year 2	Year 3
Long bond	−955.00	+50	+50	+1,050
Short STRIPS package	+961.36	−50	−50	−1,050
Net	+6.36	0	0	0

$6.36 today and nothing outstanding: an arbitrage. Traders buying the bond push its price up and traders short-selling the STRIPS push their prices down until the gap closes, so the bond cannot stay cheaper than its package either.

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