Finance-Theory · Unit 7 · Video 3 · Interactive Practice

Why Arbitrage Needs Short Selling: STRIPS, Frictions and the 2008 Ban

IKey Formulas

Formula	Name	What it says
$P = C\,P_{0,1} + C\,P_{0,2} + \cdots + (C+F)\,P_{0,T}$	Coupon bond = its STRIPS	Identical cash flows carry one price; the video's 3-year 5% bond costs $961.36
$P - \big[C\,P_{0,1} + \cdots + (C+F)\,P_{0,T}\big] > 0$	Arbitrage profit today	Short the bond, buy $C$ STRIPS of every maturity and $F$ more at $T$ ; every later date nets 0 (bond at $970.00: $8.64)
$\lvert P - \text{package cost} \rvert > k + b$	When the trade pays	$k$ = trading costs, $b$ = fee to borrow the side you short; a smaller gap survives, and a ban sets $b = \infty$
$\text{cash today} = P - \text{package cost}$	Owner's trade	A holder sells the bond it owns and buys the STRIPS: no short sale, same future payments ($110 − $100 = $10)

Key Insight: The identity is held in place by a trade, not by supply equalling demand. Take away the short sale and only holders of the dear side, who must notice the gap and be able to act, push the two prices together: closer, but not to exact equality.

IIThe Band That Frictions Leave

A gap between the bond and its STRIPS is an arbitrage only when it beats trading costs plus the borrowing fee.

Bond price P Trading costs k Borrowing fee b

💡 The fee levels are illustrative. Lo's argument needs only their ordering: a tripled or quadrupled fee still leaves a finite band, while a ban makes the fee, and the band, infinite. The ban setting, here and in the next panel, is hypothetical for this bond: the September 2008 order covered the shares of financial companies, not Treasuries. The logic is the same for any security.

IIIWithout Shorting, Only the Owner Can Sell

Under a ban, a gap closes only if someone who already owns the dear side notices it and can trade.

Bond's market value Trader Short sales knows the gap willing and able to trade

IVQuiz Questions

Problem 1 · The Arbitrage Trade

Given: a 3-year bond with $1,000 face value and a $60 annual coupon trades at $1,020.00. STRIPS prices per dollar are $P_{0,1} = 0.95$ , $P_{0,2} = 0.90$ , $P_{0,3} = 0.85$ . Find the net cash flows of the arbitrage.

✅ Correct! The package costs $1,012.00, so shorting the bond at $1,020.00 and buying the STRIPS keeps $8.00, and every later STRIPS payment goes straight to the bond's lender.

❌ Not quite. The bond is compared with the cost of its STRIPS package, $1,012.00, not with its $1,000 face value.

❌ Not quite. That is buying the bond and shorting the STRIPS: it pays the dear side and sells the cheap one. Sell the expensive side, buy the cheap one.

❌ Not quite. The STRIPS payments are not yours to keep: as a short seller you owe the bond's $60, $60 and $1,060 to the lender, and the STRIPS pay exactly that, so later dates net zero.

Show solution

The package that reproduces the bond is 60 one-year, 60 two-year and 1,060 three-year STRIPS:

60(0.95) + 60(0.90) + 1{,}060(0.85) = 57.00 + 54.00 + 901.00 = 1{,}012.00

The bond at $1,020.00 is the expensive side, so short it and buy the package:

Position	Date 0	Date 1	Date 2	Date 3
Short coupon bond	+1,020.00	−60	−60	−1,060
Buy STRIPS	−1,012.00	+60	+60	+1,060
Net	+8.00	0	0	0

$8.00 today, nothing owed later, none of your own money committed.

Problem 2 · Up to Transaction Costs

Given: the video's bond, whose STRIPS package costs $961.36, trades at $965.00. Carrying out the trade costs $5.00 per bond in all (spreads, fees and the borrowing fee). Is there an arbitrage?

✅ Correct! The pricing equation holds only up to transaction costs: a $3.64 wedge can survive because closing it would cost more than it pays.

❌ Not quite. $3.64 is the gap before costs. Put the $5.00 in and check whether money can still be made: 3.64 − 5.00 = −1.36.

❌ Not quite. A losing trade in one direction does not make the reverse profitable. Buying the dear bond and shorting the cheap STRIPS loses the gap and the costs: −3.64 − 5.00 = −8.64.

❌ Not quite. Exact equality is the frictionless ideal. With costs, the price can sit anywhere inside a band around $961.36 without offering a profitable trade.

Show solution

The bond is the dear side, so the candidate trade is short the bond, buy the STRIPS. Before costs it keeps

965.00 - 961.36 = 3.64

After the $5.00 of costs it nets

3.64 - 5.00 = -1.36

The price $965.00 lies inside the band $961.36 \pm 5.00$ , from $956.36 to $966.36, so no one can profit from closing it and the wedge survives.

Problem 3 · Price It, Don't Ban It

Given: a bond trades $12.00 above its STRIPS package. Trading costs are $2.50 per bond and the ordinary fee to borrow the bond is $1.50. A regulator quadruples the borrowing fee instead of banning short sales.

What does the short-sale arbitrage now net per bond?

What is now the widest gap that survives without a profitable short-sale trade?

✅ Correct! The quadrupled fee is $6.00, so the trade nets $3.50 and only gaps above $8.50 get closed. The cost is higher but finite, so the trade still happens.

❌ Check the net. Subtract both frictions from the gap, using the new fee: 4 × 1.50 = 6.00.

❌ Check the band. A gap survives while it is no bigger than trading costs plus the new borrowing fee.

❌ Not quite. That is what a ban does, by making the fee infinite. A quadrupled fee is finite, so the trade stays open to those willing to pay it.

Show solution

New borrowing fee: $b = 4 \times 1.50 = 6.00$ . Net of both frictions:

12.00 - 2.50 - 6.00 = 3.50

The trade pays only when the gap exceeds $k + b$ :

k + b = 2.50 + 6.00 = 8.50

Before the change the band was $2.50 + 1.50 = 4.00$ and the trade netted $8.00; after it the band is $8.50 and the trade nets $3.50. Under a ban, $b = \infty$ and no short-sale trade closes any gap.

Problem 4 · The Owner's Trade

Given: short sales of this bond are banned (a hypothetical: the 2008 order did not cover Treasuries). A pension fund holds a 2-year bond with $1,000 face value and a $70 annual coupon, trading at $1,015.00; STRIPS prices are $P_{0,1} = 0.94$ and $P_{0,2} = 0.88$ . What can still narrow the gap, and for how much?

✅ Correct! Selling something you own needs no short sale. If the fund notices the gap and is able to trade, its selling pushes the prices closer, though nothing guarantees exact equality.

❌ Not quite. The number is right, but that trade starts with a short sale, which is exactly what the ban forbids.

❌ Not quite. The fund replaces the bond with its STRIPS package, which costs $1,007.40, not the $1,000 face value.

❌ Not quite. The ban stops people who do not own the bond. Whoever already holds the expensive side can still sell it.

Show solution

The package that reproduces the bond's $70 and $1,070:

70(0.94) + 1{,}070(0.88) = 65.80 + 941.60 = 1{,}007.40

The fund sells the bond it owns for $1,015.00 and buys the package:

1{,}015.00 - 1{,}007.40 = 7.60

Its future receipts are unchanged: $70 at date 1 and $1,070 at date 2, now paid by STRIPS. No borrowing is involved. Whether it happens depends on the fund knowing the gap and being willing and able to trade, which is why a ban leaves prices closer but not pinned.

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