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* Arbitrage free valuation* is an approach that determines bond values based on the assumption that arbitrage opportunities do not exist. The arbitrage-free valuation model is based on the

An arbitrage opportunity is a deal that involves no cash outlay but generates risk-free profits.

There are two types of arbitrage opportunities: * value additivity* and

The concept of* value additivity* suggests that the value of the whole can differ from the value obtained by adding the values of parts. For example, the value of a bond could be less than the value of the sum of its individual cash flows. In these circumstances, an arbitrage opportunity exists because a trader could buy the bond and at the same time sell claims to the individual cash flows.

The following information relates to Asset A and Portfolio B.

$$ \begin{array}{c|c|c} & \textbf{Price Today} & \textbf{Payoff in One Year} \\ \hline \text{Asset A} & $0.9346 & $1 \\ \hline \text{Portfolio B} & $93 & $107 \end{array} $$

Asset A is a risk-free discount bond that pays off $1 in one year and is priced at $0.9346 today (1/1.07).

Portfolio B consists of 107 units of Asset A with a payoff of $107 in a year and the price of the asset today is $93.

In this case, a dealer can make a riskless profit by:

- Selling $107 units of Asset A for \(107\times$0.9346=$100\).
- Buying Portfolio B at $93.
**Arbitrage profit \(= $100-$93=$7\).**

* Dominance *is a situation where one security consistently offers a better yield than another security, despite both having the

To make a riskless profit when dominance is present, an investor could buy the dominant security and sell the dominated security.

Consider two risk-free zero-coupon bonds A and B that mature one year from today.

$$ \begin{array}{c|c|c} \textbf{Asset} & \textbf{Price Today} & \textbf{Payoff in One Year} \\ \hline A & 100 & 105 \\ \hline B & 300 & 330 \end{array} $$

Since both assets are risk-free, they have the same discount factor. Therefore, a dealer can generate arbitrage profit by:

- Selling three units of A (Proceeds = $300).
- Purchasing one unit of B with the sell proceeds for $300.
- This investment requires
**zero cash outlay today**.

- This investment requires

At maturity:

- Asset B will generate a cash inflow of $330.
- The two units of Asset A sold will produce a cash outflow of $315.
**Arbitrage profit generated = \($15 ($330-$315)\).**

The arbitrage-free model values a bond, assuming that it can be converted into a series of zero-coupon bonds. This gives rise to two possibilities: stripping and reconstitution.

* Stripping *is a process where periodic coupon payments of an existing security are converted into tradeable zero-coupon securities. The strips can be traded in the market at a discount and are redeemed at face value. For example, a three-year 2% coupon Treasury issue could be viewed as a package of seven zero-coupon instruments (6 semiannual coupon payments, one of which is made at maturity, and one principal value payment at maturity).

* Reconstitution *is the reverse of stripping, where the coupon strips and principal strips are reassembled into the original security.

A three-year bond with a 10% annual coupon and $1,000 face value is priced at $1,000. The Treasury spot rates are given in the following table:

$$ \begin{array}{c|c} \textbf{Year} & \textbf{Spot Rates} \\ \hline 1 & 7\% \\ \hline 2 & 8\% \\ \hline 3 & 7\% \end{array} $$

The discounted price of this bond is:

$$ \frac{100}{1.07}+\frac{100}{{1.08}^2}+\frac{1,100}{\left(1.07\right)^3}= 1,077.12 $$

Notice from above that a dealer can:

- Buy the security at the current market price of $1,000.
- Strip the security and earn a value of $1,077.12 using Treasury spot rates.
- There exists an arbitrage opportunity to make a riskless profit of $77.12 ($1,077.12-$1,000).

If the price of the bond were $1,100 rather than $1,000, the dealer would:

- Buy up the Treasury strips for a value of $1,077.12.
- Repackage as a synthetic Treasury security for $1,100.
- This process, known as reconstitution, provides the dealer with a profit of $22.88 (profit from reconstitution = \($1,100 – $1,077.12 = $22.88\)).
- Arbitragers would either bid prices up or down until such arbitrage profits could no longer be earned.

Arbitrage free valuation:

- Maintains the
**value additivity**principle.- Stripping and reselling bonds.
- Reconstitution of bonds.
- The value of strips and reconstituted bonds should be the same; otherwise, arbitrage opportunities would exist.

- Eliminates
**dominance**.- One bond cannot be priced more attractively as compared to an otherwise identical bond.

## Question

The yield of an 8% annual coupon bond that matures in five years is 5.2% in Montreal. The same bond sells for CAD 112.06 per CAD 100 face value in Ottawa.

Which of the following statements is

most likelycorrect?

- There is no arbitrage opportunity.
- An arbitrage opportunity can be utilized by selling the bond in Montreal and buying in Ottawa.
- An arbitrage opportunity can be exploited by buying the bond in Montreal and selling it in Ottawa.
## Solution

The correct answer is A.The price for the bond in Montreal is determined as follows:

$$ P=\frac{8}{1.052}+\frac{8}{{1.052}^2}+\frac{8}{{1.052}^3}+\frac{8}{{1.052}^4}+\frac{108}{{1.052}^5}=$112.06 $$

The price in Montreal matches the price in Ottawa. Therefore, no arbitrage opportunity exists in the market.

Reading 29: The Arbitrage-Free Valuation Framework

*LOS 29 (a) Explain what is meant by arbitrage-free valuation of a fixed-income instrument.*