Finance-Theory · Unit 7 · Video 4 · Interactive Practice

Fixed-Income Arbitrage: No Solution Means a Trade

IKey Formulas

Formula	Name	What you need
$P_i = C_{i1} P_{0,1} + C_{i2} P_{0,2} + \cdots + C_{iT} P_{0,T}$	One equation per bond, $i = 1, \ldots, n$	Known: the market price $P_i$ and the cash flows $C_{it}$ . Unknown: the $T$ discount-bond prices $P_{0,t}$ , shared by every bond
$n > T$	Over-determined system	More bonds than payment dates: 353 Treasury notes and bonds (31 August 2026) against at most 30 annual dates, or about 230 real payment dates
$(C_{31}, C_{32}) = w_1 (C_{11}, C_{12}) + w_2 (C_{21}, C_{22})$	Replication weights	Two bonds whose coefficient rows are not proportional; then weights always exist
$P_3 = w_1 P_1 + w_2 P_2$	Condition for a solution (linear dependence)	The same weights applied to the prices; if it fails, the system has no solution

Key Insight: If $P_3 > w_1 P_1 + w_2 P_2$ , sell bond 3 and buy $w_1$ of bond 1 and $w_2$ of bond 2: every future payment cancels and you keep $P_3 - (w_1 P_1 + w_2 P_2)$ today. If $P_3 < w_1 P_1 + w_2 P_2$ , reverse every position. No solution means a riskless trade.

IIThree Equations, Two Unknowns

Each bond is a line in the $(P_{0,1}, P_{0,2})$ plane; a solution is a point on every line.

System 6% bond quote 101.66

IIIBuilding the Riskless Trade

Which holdings of the three bonds put cash in hand today and cancel every future payment?

2% bond 0 10% bond 0 6% bond −2 6% quote

Position ($, face $100)	Today	Year 1	Year 2



Net

💡 Challenge: at the quote 100.66 the 6% bond is too cheap; find the position that collects 1.00 today with nothing owed in either year.

IVQuiz Questions

Problem 1 · Pricing a Bond from the Discount Prices

Given: The 2% and 10% bonds of the video imply $P_{0,1} = 0.96$ and $P_{0,2} = 0.90$ . Find the price of a 4% two-year bond (face $100, annual coupons) that keeps the three equations consistent.

✅ Correct!

4(0.96) + 104(0.90) = 3.84 + 93.60 = 97.44

: each cash flow is valued at the discount-bond price of its own date.

❌ Not quite.

4(0.90) + 104(0.96) = 103.44

pairs the year-1 coupon with

P_{0,2}

and the year-2 payment with

P_{0,1}

. The payment at date

t

takes

P_{0,t}

❌ Not quite.

4(0.96) + 100(0.90) = 93.84

leaves out the year-2 coupon. The bond's final payment is coupon plus face,

4 + 100 = 104

❌ Not quite.

104(0.90) = 93.60

values only the final payment. The year-1 coupon of 4 is worth

4(0.96) = 3.84

today and must be added.

Show solution

The 4% bond pays 4 in year 1 and $104$ in year 2, so its row of the system reads

P_4 = 4\,P_{0,1} + 104\,P_{0,2} = 4(0.96) + 104(0.90) = 3.84 + 93.60 = 97.44

The same price comes from replication: $(4, 104) = \tfrac{3}{4}(2, 102) + \tfrac{1}{4}(10, 110)$ , so the price must be $\tfrac{3}{4}(93.72) + \tfrac{1}{4}(108.60) = 70.29 + 27.15 = 97.44$ . Any other quote makes the system inconsistent.

Problem 2 · Which Way Is the Trade?

Given: The 2% and 10% bonds trade at $93.72 and $108.60, as in the video, but the 6% bond is quoted at $100.76. Which position is an arbitrage?

✅ Correct! The first two bonds value the 6% bond at 101.16, so at 100.76 it is 0.40 too cheap. Buying 2 costs 201.52, selling the pair brings in 202.32: you keep 0.80 today, and the year-1 flows (

12 - 2 - 10

) and year-2 flows (

212 - 102 - 110

) are both 0.

❌ That is the video's trade, but the mispricing has flipped. At 100.76 the 6% bond is below its value of 101.16. Selling 2 brings in 201.52 while the pair costs 202.32, so you pay 0.80 today. Buy the cheap bond, sell the expensive pair.

❌ Not quite. The direction is right but the amounts are not: year 1 nets

6 - 2 - 10 = -6

and year 2 nets

106 - 102 - 110 = -106

, so you still owe money later. The 6% bond's cash flows are the average of the other two, so it takes 2 of it to match 1 of each.

❌ Not quite. Market prices are data, not a homework answer to be corrected. When the three equations have no common solution, a combination of the bonds costs nothing, pays cash today and owes nothing later: that is the arbitrage.

Show solution

From the first two bonds, $P_{0,1} = 0.96$ and $P_{0,2} = 0.90$ , so the 6% bond is worth $6(0.96) + 106(0.90) = 101.16 = \tfrac{1}{2}(93.72 + 108.60)$ . Its quote of 100.76 is 0.40 too low.

With weights $w_1 = w_2 = \tfrac{1}{2}$ doubled, buy 2 of the 6% bond and sell 1 each of the others:

Position	Today	Year 1	Year 2
buy 2 of the 6%	$-201.52$	$+12$	$+212$
sell 1 of the 2%	$+93.72$	$-2$	$-102$
sell 1 of the 10%	$+108.60$	$-10$	$-110$
Net	$+0.80$	$0$	$0$

The profit is $2 \times 0.40 = 0.80$ , with no payment owed in either year. Selling the 2% and 10% bonds you do not own is a short sale, so the trade needs short selling to be allowed.

Problem 3 · Solve Two, Test the Third

Given: Three two-year bonds, face $100, annual coupons: bond A pays 2% and trades at $91.66, bond B pays 8% and trades at $102.64, bond C pays 5% and is quoted at $97.50. Find $P_{0,2}$ from bonds A and B, then decide whether bond C is mispriced.

What is $P_{0,2}$ ?

What does bond C's quote imply?

✅ Correct!

P_{0,1} = 0.95

P_{0,2} = 0.88

value C at

5(0.95) + 105(0.88) = 97.15

, so its quote is 0.35 too high. Selling 2 of C and buying A and B keeps

195.00 - 194.30 = 0.70

today with every future payment cancelled.

❌ Not quite.

0.95

P_{0,1}

, the price of 1 paid in year 1. Eliminate the year-1 terms first:

4 \times

(A's row)

-

(B's row) leaves only

P_{0,2}

❌ Not quite.

0.90

is the video's value for different bonds. Solve this system:

4(91.66) - 102.64 = 264

and

4(102) - 108 = 300

❌ Not quite.

91.66 / 102

uses bond A alone and ignores its year-1 coupon of 2. One equation cannot pin down two unknowns; combine A and B.

❌ Check $P_{0,2}$ . Multiply A's equation by 4 and subtract B's, so the year-1 terms cancel.

❌ Check the direction. C is worth 97.15 but quoted at 97.50: the quote is above value, so C is the one to sell.

❌ Check the amounts. Selling 1 of C leaves year 1 at

-5 + 2 + 8 = 5

and year 2 at

-105 + 102 + 108 = 105

: the future flows do not cancel. C's cash flows are half of A's plus half of B's, so sell 2 of C per pair, and the gap per bond is

97.50 - 97.15 = 0.35

❌ Not quite. A coupon between A's and B's only means the replicating weights are positive. Consistency needs C's quote to equal

\tfrac{1}{2}(91.66) + \tfrac{1}{2}(102.64) = 97.15

, and it is 97.50.

Show solution

Bonds A and B give two equations:

2\,P_{0,1} + 102\,P_{0,2} = 91.66, \qquad 8\,P_{0,1} + 108\,P_{0,2} = 102.64

Four times the first minus the second cancels the year-1 terms:

(408 - 108)\,P_{0,2} = 366.64 - 102.64 \;\Rightarrow\; 300\,P_{0,2} = 264 \;\Rightarrow\; P_{0,2} = 0.88

P_{0,1} = \frac{91.66 - 102(0.88)}{2} = \frac{91.66 - 89.76}{2} = 0.95

Bond C's value: $5(0.95) + 105(0.88) = 4.75 + 92.40 = 97.15$ . Equivalently $(5, 105) = \tfrac{1}{2}(2, 102) + \tfrac{1}{2}(8, 108)$ , so C must trade at $\tfrac{1}{2}(91.66 + 102.64) = 97.15$ . The quote 97.50 is 0.35 too high, so the system has no solution.

Sell 2 of C for $195.00$ and buy 1 each of A and B for $194.30$ . Today: $+0.70$ . Year 1: $-10 + 2 + 8 = 0$ . Year 2: $-210 + 102 + 108 = 0$ .

Problem 4 · Replicating a Bond Outside the Range

Given: The video's 2% bond ( $P_1 = 93.72$ ) and 10% bond ( $P_2 = 108.60$ ). A 12% two-year bond (face $100, annual coupons) pays $(12, 112)$ . Find the weights $w_1$ on the 2% bond and $w_2$ on the 10% bond that replicate it, and the price that keeps the system consistent.

Which weights replicate $(12, 112)$ ?

What price keeps the three equations consistent?

✅ Correct!

-0.25(2, 102) + 1.25(10, 110) = (12, 112)

, so the 12% bond is 1.25 of the 10% bond with 0.25 of the 2% bond sold short, worth

-0.25(93.72) + 1.25(108.60) = 112.32

❌ Not quite. Half of each gives

(6, 106)

, the video's 6% bond. The year-1 row needs

2w_1 + 10w_2 = 12

❌ Not quite. The two coefficient rows are not proportional, so weights always exist; outside the range one of them is negative, which is a short sale.

❌ Check the weights. Year 2 minus year 1 gives

100(w_1 + w_2) = 100

, so

w_1 + w_2 = 1

; then

2w_1 + 10w_2 = 12

❌ Not quite.

12(0.90) + 112(0.96) = 118.32

swaps the two discount prices. The year-1 coupon takes

P_{0,1} = 0.96

❌ Not quite.

1.25(108.60) = 135.75

prices only the long leg. The short 0.25 of the 2% bond brings in

0.25(93.72) = 23.43

, which lowers the cost.

❌ Not quite.

159.18

adds

0.25(93.72)

instead of subtracting it. The weight on the 2% bond is negative.

❌ Check the price. Apply the weights to the prices:

w_1(93.72) + w_2(108.60)

Show solution

Match both dates:

2w_1 + 10w_2 = 12, \qquad 102w_1 + 110w_2 = 112

Subtracting the first from the second gives $100w_1 + 100w_2 = 100$ , so $w_1 + w_2 = 1$ . Then $2(1 - w_2) + 10w_2 = 12$ , so $8w_2 = 10$ : $w_2 = 1.25$ and $w_1 = -0.25$ .

The price the system requires:

-0.25(93.72) + 1.25(108.60) = -23.43 + 135.75 = 112.32

Check with the discount prices: $12(0.96) + 112(0.90) = 11.52 + 100.80 = 112.32$ . A quote above 112.32 means sell the 12% bond, buy 1.25 of the 10% bond and short 0.25 of the 2% bond; a quote below means the reverse.

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