# Arbitrage-Free Valuation of a Fixed Income Security

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 law of one price, which states that two goods, which are perfect substitutes, must have equivalent market prices in the absence of transaction costs; otherwise, an arbitrage opportunity would exist.

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

## Types of Arbitrage Opportunities

There are two types of arbitrage opportunities: value additivity and dominance.

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 $$107\times0.9346=100$$. • Buying Portfolio B at$93.
• Arbitrage profit $$= 100-93=7$$.

### Dominance

Dominance is a situation where one security consistently offers a better yield than another security, despite both having the same characteristics, including the buying price. A security X is said to be dominant over another security Y when both can be purchased at the same price at t = 0, but X will yield a higher return in every state of the world.

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

#### Example: Dominance

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.

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 = $$15 (330-315)$$.

## Implications of Arbitrage-Free Valuation

### Stripping and Reconstitution

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.

#### Example: Stripping and Reconstitution

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.

### Key Takeaways

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 likely correct?

1. There is no arbitrage opportunity.
2. An arbitrage opportunity can be utilized by selling the bond in Montreal and buying in Ottawa.
3. An arbitrage opportunity can be exploited by buying the bond in Montreal and selling it in Ottawa.

#### Solution

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.

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