How to Derive The Bond Pricing Formula

When you calculate the price of a bond, you are determining the maximum price you would want to pay for the bond, based on how its coupon rate compares to the average rate most investors are currently receiving in the bond market.

To calculate a bond’s price, we can use the basic present value (PV) formula to sum all present values of all future cashflows:

$$Bond\ Price = \frac{future\ cashflow\ 1}{(1 + i)^1} + \frac{future\ cashflow\ 2}{(1 + i)^2} + \frac{future\ cashflow\ 3}{(1 + i)^3} + \dots + \frac{future\ cashflow\ n}{(1 + i)^{n}} + \frac{M}{(1 + i)^{n}}$$

where $$M$$ is the bond’s value at maturity (par value). When the periodic payments are all the same, we can rewrite it as follows:

$$Bond\ Price = \frac{C}{(1 + i)^1} + \frac{C}{(1 + i)^2} + \frac{C}{(1 + i)^3} + \dots + \frac{C}{(1 + i)^{n}} + \frac{M}{(1 + i)^{n}}$$

This bond-pricing formula can be tedious to calculate because you have to add the present value of each future coupon payment. We can use the present value of ordinary annuity formula:

$$Bond\ Price = C \frac{(1 - (\frac{1}{(1+i)^n})}{i} + \frac{M}{(1 + i)^n}$$

Proof

Consider the following sum of future cash flows discounted to present value:

$$s=\frac{C}{(1 + i)^1} + \frac{C}{(1 + i)^2} + \frac{C}{(1 + i)^3} + \frac{C}{(1 + i)^{4}} + \frac{C}{(1 + i)^{5}}$$

Since the periodic payments are all the same, the formula is considered a geometric series:

$$s=\frac{C}{(1 + i)^1} + \frac{C}{(1 + i)^2} + \frac{C}{(1 + i)^3} + \frac{C}{(1 + i)^{4}} + \frac{C}{(1 + i)^{5}}=a + ar + ar^2 + ar^3 + ar^4$$ with $a=\frac{C}{(1 + i)}$ and $r=\frac{1}{(1 + i)}$

I have already proved the deriving the perpetuity formula how to derive the perpetuity formula. We know that the sum of the first $$k$$ terms of a geometric sequence is:

$$s=a + ar + ar^2 + ar^3 + \dots + ar^{k-1}= a\frac{(1 - r^k)}{(1 - r)}$$

For our case

$$s=\frac{C}{(1 + i)^1} + \frac{C}{(1 + i)^2} + \frac{C}{(1 + i)^3} + \frac{C}{(1 + i)^{4}} + \frac{C}{(1 + i)^{5}}=a + ar + ar^2 + ar^3 + ar^4=a\frac{(1 - r^5)}{(1 - r)}$$

with $$a=\frac{C}{(1 + i)}$$ and $$r=\frac{1}{(1 + i)}$$.

$$s=\frac{C}{(1 + i)^1} + \frac{C}{(1 + i)^2} + \frac{C}{(1 + i)^3} + \frac{C}{(1 + i)^{4}} + \frac{C}{(1 + i)^{5}}=a\frac{(1 - r^5)}{(1 - r)}=\frac{C}{(1 + i)}\frac{(1 - (\frac{1}{(1 + i)})^5)}{(1 - (\frac{1}{(1 + i)}))}=\frac{C}{(1+i)}(\frac{\frac{(1+i)^5 - 1}{(1+i)^5}}{\frac{(1+i) - 1}{(1+i)}})=\frac{C}{(1+i)}\frac{(1+i)((1+i)^5 -1)}{(1+i)^5((1+i)-1)}=\frac{C(1+i)^5(1 - \frac{1}{(1+i)^5})}{(1+i)^5((1+i)-1)}=\frac{C(1 - \frac{1}{(1+i)^5})}{(1+i)-1}=C\frac{(1 - \frac{1}{(1+i)^5})}{i}$$

Since we are calculating a bond price we need to also add the discounted principle, $$\frac{M}{(1 + i)^5}$$ that we get after $$n$$ periods so the final formula is:

$$Bond\ Price = C \frac{(1 - (\frac{1}{(1+i)^5})}{i} + \frac{M}{(1 + i)^5}$$
Written on August 15, 2021