# Bond Valuation

This is an introductory page in Fixed Income. If you are unfamiliar with any of the terms, you can refer to the Fixed Income Glossary.

The basic principle of bond valuation, is that the bond's value should be equal to the present value of all of its expected (future) cash flows.

We will work through the simple case of a zero-coupon bond, and then build it up by adding the complications like having a coupon and having different interest rates.

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## Pricing a zero coupon bond

A zero coupon bond is one that simply pays the principal on maturity. It has a single cash payment of $P at time T.

Hence, if the interest rate is \(i \% \), then the present value of this bond is simply

\[ PV = \frac{ $P } { (1 + i \% ) ^ T }. \]

## Pricing a fixed coupon bond

If the bond were to pay an additional coupon of \( c \% \) during each time period, then we can find the present value of the bond by summing up the present value of each of these time payments. Hence, the present value of this bond is

\[ PV = \frac{ c\% $P } { ( 1 + i \% ) ^ 1 } + \frac{ c\% $P } { ( 1 + i \% ) ^ 2 } + \frac{ c\% $P } { ( 1 + i \% ) ^ 3 } + \ldots + \frac{ c\% $P } { ( 1 + i \% ) ^ {T-1} } + \frac{ c\% $P + $P} { ( 1 + i \% ) ^ T }. \]

Using the present value of an annuity, we can simplify it to

\[ PV = \frac{ c \% $P } { i \% } \left ( 1 - \frac{ 1} { (1+ i \% ) ^ T} \right) + \frac{ $P } { (1 + i \% ) ^ T }. \]

In the expression above, we have written the present value as the sum of the coupon payments, and of the final lump sum payments. The process by which an investor buys a bond and separates it into the interest payment and principal payment, is known as **stripping** the bond.

Note that in the special case when \( c = i \), the expression simplifies to \( PV = $P \).

## Pricing a fixed coupon bond with variable interest rate

In the above version, we assumed that the interest rate in each time period is a constant \( i \% \). This is generally not the case, and we instead have to look up the Yield curve in order to determine the appropriate interest rate to use for each time period.

Say that the yield rate at time \( t \) is \( y _ t \% \). Then, the present value of the bond is:

\[ \small PV = \frac{ c\% $P } { ( 1 + y_1 \% ) ^ 1 } + \frac{ c\% $P } { ( 1 + y_2 \% ) ^ 2 } + \frac{ c\% $P } { ( 1 + y_3 \% ) ^ 3 } + \ldots + \frac{ c\% $P } { ( 1 + y_{T-1} \% ) ^ {T-1} } + \frac{ c\% $P + $P} { ( 1 + y_T \% ) ^ T }. \]

There is often no simple way to find this sum, and we typically enter the values into a spreadsheet to determine the theoretical price of the bond.

## Pricing a variable coupon bond with variable interest rate

Finally, in this version, we let the coupon rate vary. Say that the coupon rate at time \( t \) is \( c _ t \% \). Then, the present value of the bond is

\[ \small PV = \frac{ c_1\% $P } { ( 1 + y_1 \% ) ^ 1 } + \frac{ c_2 \% $P } { ( 1 + y_2 \% ) ^ 2 } + \frac{ c_3\% $P } { ( 1 + y_3 \% ) ^ 3 } + \ldots + \frac{ c_{T-1}\% $P } { ( 1 + y_{T-1} \% ) ^ {T-1} } + \frac{ c_T \%$ P + $P} { ( 1 + y_T \% ) ^ T }. \]

Similarly, there is no simple formula for this sum, even for the most common case where the coupon rate is tied to the yield rate.

## Bond Pricing

See Bond Prices for more details.

In the above, we made the assumption that each of these payments will be made. As it is possible for a bond issuer to default on their payments, we should also also account for that probability, and update the payments in each of the time periods \(t\).

In practice, the price of a bond would correlate with the price of other similar liquid instruments or benchmarks. This is known as relative valuation, where the credit rating of the bond will also be taken into consideration.