### Face, Par and Market Value of a Bond

The face value, or par value of a vanilla bond can be determined by discounting its expected future cash flows according to some appropriate discount rate. A bond can have an alternative value, called its market value, the difference being that the market value is its value as observed today in the market. A bond whose current market value is less than its face or par value is called a discount bond because it is currently worth less than its fair value. Conversely if its market value is greater than its face or par value then it is called a premium bond.

### Present Value

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If one makes the assumption that interest rates, $i$ are constant then one may calculate a bond’s present value, or abbreviated to $PV$.

If money could never be invested and earn interest then there would be no time value of money effect would not be observed and the present value of financial instruments would simply coincide with its future value. The fact is that money can always be invested in something that we perceive as having future value and hence we must always compensate for this perceived increase in value over time. The way we do this is to apply a multiplicative factor to the present value to obtain the future value. This factor is usually a number that looks like $(1+i)$ where $i$ is termed an interest rate or the embodiment of the time value of money that forces any present value of money to increase over time such that the future value $FV$ is greater than the present value $PV$:

$FV = (1+i)\cdot PV$

What if we knew what the future value of money was and instead wanted to know the present value of that money? We simply re-arrange the above formula to obtain:

$PV = \frac{FV}{(1+i)}$

We call the fraction $\frac{FV}{(1+i)}$ the future value discounted at a rate of interest to the present value.

The formula as it stands only tells us the present value of a future value known exactly 1 time step ahead. If there are multiple time steps, like months or years, then we have to apply the formula for each one until we make it all the way back to the current time. If there are $n$ of these time-steps then we use the following amended formula:

$PV = \frac{FV}{(1+i)^n}$

The interest rate $i$, of course must be expressed in terms of the lengths of these time-steps. Thus, if the time-steps are months, then the interest rate must be expressed per month, etc.

Now, what if the future amount of money was in fact several cash flows occurring at each of the time steps? Can we accommodate for that? Can we still calculate the present value of multiple amounts of money occurring at every time-step, like a interest payment on a savings account occurring every month? Yes we can:

$PV = \sum_{t=1}^n \frac{FV_t}{(1+i)^t}$

From now on we will replace $FV$ with $C$ to represent a future cash flow.

### Present Value of a Bond

A bond is nothing but a sequence of future cashflows, these future cashflows are called coupons. We now know how to calculate the present value of each of these coupons:

$PV = \sum_{n=1}^N \frac{C}{(1+i)^n}$

A bond also has 1 special cashflow, called the re-payment of notional, $M$, that occurs at the maturity date of the bond and works to re-pay the investor with an equivalent amount that the investor initially paid for the bond. It too, has a present value:

$PV = \frac{M}{(1+i)^N}$

but it is only 1 cashflow and it occurs $N$ time steps in to the future.

Combining these two we can obtain a formula for the present value of a bond:

$PV = \left(\sum_{n=1}^N \frac{C}{(1+i)^n}\right) + \frac{M}{(1+i)^N}$

Or, since the coupon payments are constant we note that the part in parentheses is actually a geometric series with initial value $a = C$, multiplicative factor $(1+i)$ and $n$ terms. The Geometric Series formula is

$\sum_{k=0}^{n-1} ar^k = a\frac{1-r^n}{1-r}$

Hence we get an alternative formula for the present value of a bond:

$PV = \frac{C\left(1-(1+i)^n\right)}{1-(1+i))} + M(1+i)^{-N} = \frac{C\left(1-(1+i)^n\right)}{-i} + M(1+i)^{-N}$

where $N$ is the number of payments or time-steps in to the future.

### Next Article

Bond Valuation II – Zero-Coupon Bonds