###### Extensions of VaR

Conditional Value at Risk (CVaR) Rockafellar and Uryasev introduced conditional value-at-risk (CVaR) in... **Read More**

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

Under the * traditional valuation approach*, a

Under the * arbitrage-free valuation approach*, each cash flow is discounted at its

Perhaps an example will help illustrate. Consider a three-year U.S. Treasury note with a 10% coupon rate, paid semiannually. Considering the cash flows per $100 of par value, we would have six payments of $3 (one payment after every six months) and a final principal payment of $100.

- The traditional valuation approach would discount each of these cash flows using the same discount rate without considering their timing nor the shape of the yield curve.
- The arbitrage-free valuation approach would view each cash flow as a zero-coupon instrument maturing on the date the cash flow is received. The three-year Treasury note would be viewed as a package of seven zero-coupon instruments maturing over a three-year period at six-month intervals. This would deny market participants an opportunity to realize an arbitrage profit by “stripping” the bond and selling the individual cash flows at a higher aggregate value than it would cost to purchase the Treasury in the market.

An arbitrage-free value is the present value of expected future values using Treasury spot rates for option-free bonds. Arbitrage-free valuation usually involves three main steps:

Estimate the future cash flows.**Step 1**:Determine the appropriate discount rates that should**Step 2**:

be used to discount each cash flow.Calculate the present value of the expected future cash flows by applying the appropriate discount rates determined in Step 2.**Step 3**:

Thus, the following formula is used:

$$ PV=\frac{PMT}{\left(1+S_1\right)^1}+\frac{PMT}{\left(1+S_2\right)^2}+\ldots+\frac{PMT+FV}{\left(1+S_n\right)^N} $$

Where:

\(PMT\) is the periodic coupon.

\(FV\) is the face value.

\(S_1\), \(S_2,\) and \(S_N\) are the spot rates for periods 1 to \(N\).

An 8% semi-annual coupon bond is priced at $800. It has a remaining term to maturity of 2 years. Given the following benchmark spot rates, the value of the bond, if its face value is $1,000, is *closest* to:

$$ \begin{array}{c|c} \textbf{Year} & \textbf{Spot Rates} \\ \hline 0.5 & 16\% \\ \hline 1 & 17\% \\ \hline 1.5 & 16\% \\ \hline 2 & 15\% \end{array} $$

**Solution**

$$ \text{Present value} =\frac{40}{1.08}+\frac{40}{{1.085}^2}+\frac{40}{{1.08}^3}+\frac{1,040}{{1.075}^4}=$881.52 $$

Note that $40 is the semi-annual coupon.

## 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:

- $102.69.
- $102.76.
- $102.94.

Solution

The correct answer is B.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.*