# Arbitrage Free Value

Bonds can be valued either using the traditional valuation approach or the arbitrage-free valuation approach.

Under the traditional valuation approach, a single interest rate is used to discount all of a bond’s cash flows. In this approach, all cash flows of a bond are considered the same, regardless of their timing. In other words, each cash flow is viewed as just a token from the same package, and therefore the same discount rate is applied to all cash flows. This is a fundamental flaw because each individual cash flow of a bond is always unique. The use of a single discount rate in valuation may result in mispricing, thereby creating arbitrage opportunities.

Under the arbitrage-free valuation approach, each cash flow is discounted at its own discount rate that takes into account the shape of the yield curve and the timing of the cash flow.

## Question

An option free, 3-year 6% annual coupon bond priced at $100 has similar liquidity and risk to a Treasury bond whose par curve is shown in the table below. $$\textbf{Treasury Par Curve} \\ \begin{array}{c|c} \textbf{Term to Maturity (Years)} & \textbf{Par Rate} \\ \hline 1 & 3.00\% \\ \hline 2 & 4.00\% \\ \hline 3 & 5.00\% \end{array}$$ The arbitrage-free price for the bond is closest to: 1.$102.69.
2. $102.76. 3.$102.94.

#### Solution

We first calculate the implied one-year spot rates given the above term structure by bootstrapping.

The one-year implied spot rate is 3%, as it is simply the one-year par yield.

We can bootstrap the two-year implied spot rate, $$r(2)$$, as follows:

\begin{align*} 1 &=\frac{0.04}{1.03}+\frac{(1+0.04)}{\left(1+r\left(2\right)\right)^2} \\ r\left(2\right)&=4.02\% \end{align*}

Similarly, the three-year spot rate can be bootstrapped by solving the equation:

\begin{align*} 1 &=\frac{0.05}{1.03}+\frac{0.05}{{1.0402}^2}+\frac{1+0.05}{\left(1+r\left(3\right)\right)^3} \\ r\left(3\right) &=5.07\% \end{align*}

The implied spot rate curve:

$$\begin{array}{c|c|c} \textbf{Term to Maturity} & \textbf{Par Rate} & \textbf{Spot Rate} \\ \hline 1 & 3.00\% & 3.00\% \\ \hline 2 & 4.00\% & 4.02\% \\ \hline 3 & 5.00\% & 5.07\% \end{array}$$

To calculate the arbitrage-free price, each cash flow is discounted using the same maturity spot rate as the date of the cash flow.

$$\text{Arbitrage-free price } (P)=\frac{6}{1.03}+\frac{6}{{1.0402}^2}+\frac{106}{{1.057}^3}=102.76$$

Reading 29: The Arbitrage-Free Valuation Framework

LOS 29(b) Calculate the arbitrage-free value of an option-free, fixed-rate coupon bond.

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