# Fair price of a coupon paying bond

Consider a coupon paying bond with a maturity of $$3$$ years, that pays coupon annually. Let $$c$$ be the coupon rate (percentage) and let $$F$$ be the face value. This means that the holder of the bond receives $$cF$$ at the end of year $$1$$, $$cF$$ at the end of year $$2$$ and $$(cF+F)$$ at the end of year $$3$$. Furthermore, let $$r$$ denote the interest rate.

Let $$B_0$$ denote the price of the bond at the beginning of year $$1$$ and let $$B_t$$, $$t= 1, 2, 3$$, denote the bond price at the end of the respective year. It is quite common to price the bond by taking the sum of the future discounted cash flows:

$$B_0 = \frac{ cF }{ 1 + r } + \frac{ cF }{ { ( 1 + r ) }^2 } + \frac{ cF + F}{ { ( 1 + r ) }^3 }.$$

The formula can be obtained by reasoning in the following way:

The cash flow $$cF$$ at time $$1$$ should be worth $$cF / (1 + r)$$ at time $$0$$. The cash flow $$cF$$ at time $$2$$ should be worth $$cF / { (1 + r) }^2$$ at time $$0$$. The cash flow $$cF+F$$ at time $$3$$ should be worth $$(cF + F) / { (1 + r) }^3$$ at time $$0$$.The sum of these three terms results in the above-mentioned formula.

However, there seems to be a problem in constructing a formal arbitrage-free market, in which the price of this bond at times $$t = 1, 2, 3$$ will be given by the sum of the future discounted cash flows. What I mean is the following:

Let $$D_t = (1+r)^t$$, $$t = 0, \ldots, 3$$, be the value of a bank account which pays $$(1+r)$$ for an investment of $$1$$ dollar in $$1$$ year. This will be our numeraire. Now consider a formal market consisting of $$B$$ and $$D$$.

At time $$0$$:

\begin{align} &B_0 = \frac{ cF }{ 1 + r } + \frac{ cF }{ { ( 1 + r ) }^2 } + \frac{ cF + F }{ { ( 1 + r ) }^3 } \\ &D_0 = 1 \end{align}

At time $$1$$:

\begin{align} &B_1 = \frac{ cF }{ 1 + r } + \frac{ cF + F }{ { ( 1 + r ) }^2 } \\ &D_1 = 1+r \end{align}

At time $$2$$:

\begin{align} &B_2 = \frac{ cF + F }{ 1 + r } \\ &D_2 = { ( 1 + r ) }^2 \end{align}

At time $$3$$:

\begin{align} &B_3 = F \\ &D_3 = { ( 1 + r ) }^3 \end{align}

Now, for no-arbitrage, we need for the price process $$\frac{B_t}{D_t}$$ to be a martingale, or, to put it simply, it should hold that

$$\frac{B_{t+1}}{D_{t+1}} = \frac{B_t}{D_t}, \quad t = 0, 1, 2.$$

However, this equation is clearly violated for all $$t = 0, 1, 2$$. I think that the problem comes from the fact that this formal market does not account for the possibility of the reinvestment of the coupon payments. As a result, the investor, who holds one unit of $$B$$ in the above-mentioned market and follows a self-financing strategy, seems to obtain only $$F$$ at the end of year $$3$$. However, he could have actually obtained $$cF { ( 1 + r ) }^2 + cF (1+r) + (cF +F)$$ by reinvesting the cash flows in the numeraire.

It seems that for the price process of the bond we should rather consider $$B_t = B_0 { ( 1 + r ) }^t$$, $$t = 0, 1, 2, 3$$. In this case the martingale property holds, and the bond price naturally reflects the time value of money. However, in such a formal market, even though arbitrage would be excluded, the price of the bond would reflect an economic reality in which the person who buys the bond from the original investor at times $$t = 1, 2, 3$$ is also entitled to obtain all the prior coupon payments with the time value of money associated with them.

I am wondering if someone could suggest a formal market in which the price process of the bond will be consistent with the conventional bond pricing formula and exclude arbitrage at the same time.

The process you are using for $$B_{t}$$ is the price of the bond rather than the value to you of the portfolio in which you buy and hold a bond and reinvest coupons.
If we set $$V_{t}$$ to be the value of this strategy, we would have as you mention
\begin{align} &V_0 = \frac{ cF }{ 1 + r } + \frac{ cF }{ { ( 1 + r ) }^2 } + \frac{ cF + F }{ { ( 1 + r ) }^3 } \\ &V_1 = cF + \frac{ cF }{ ( 1 + r ) } + \frac{ cF + F }{ { ( 1 + r ) }^2 } \\ &V_2 = cF ( 1 + r ) + cF + \frac{ cF + F }{ ( 1 + r ) } \\ \end{align}
• It is clear that the wealth of the investor holding the bond should be given by $B_0 { ( 1 + r ) }^t$, $t=0, 1, 2, 3$. My question concerns constructing a formal market in which the bond price will be given as the sum of the discounted future cash flows. In such a market the no-arbitrage property is ensured by the prices of the assets and is independent of any particular strategy. – Holden Apr 26 '19 at 21:49