Bond Present Value and the Geometric Annuity Formula

Summary

The document explains how to simplify the present value of a bond’s fixed annual coupons using the finite geometric series formula. Discounting each coupon produces a sequence with first term and common ratio equal to the one-period discount factor. Summing those terms gives the familiar annuity expression: the coupon amount divided by the yield, multiplied by one minus the discount factor at maturity.

The answer highlights an algebraic error in the professor’s displayed step: the denominator in the annuity factor should be the rate r, not a power of (1+r). The bond’s face value is then discounted separately to maturity. This is a standard fixed-rate cash-flow calculation under a constant annual discount rate and annual coupon payments. The question’s bond has specified example inputs, but the answer does not calculate a numerical present value or discuss compounding conventions, changing rates, credit risk, or other features that would affect valuation in practice.

Key ideas

• A level coupon stream can be valued by summing a finite geometric series of discount factors.
• The annuity factor is the coupon amount divided by the discount rate, multiplied by one minus the maturity discount factor.
• The displayed coupon formula has an incorrect denominator; it should use r.
• The bond’s face value is discounted separately to its maturity date.
• The expression assumes a constant annual rate and annual coupon payments.

Tags

# Calculating present value of a bond (understanding a step)


# Calculating present value of a bond (understanding a step)












Our professor calculated the present value of a bond with $T=10$ years, $FV=10,000$€, $C=700$€ p.a. and an expected rate of return $r$. He wrote $$\begin{align}PV&=C\cdot\sum_{n=1}^{10}\frac{1}{(1+r)^n}+\frac{FV}{(1+r)^n}\\&=700\cdot\color{red}{\frac{1-\frac{1}{(1+r)^n}}{(1+r)^n}}+\frac{FV}{(1+r)^n}\end{align}$$ I don't understand the red step (I guess he transformed the sum).

In our lectures we have to different formulas for calculating the present value (with cash flows $C_n$): $$PV=\sum_{n=1}^T\frac{C_n}{(1+r)^n} \tag{1}$$ and annuity (value of $C$ received each year for $T$ years): $$PV = \frac{C}{r}\left(1-\frac{1}{(1+r)^T}\right).\tag{2}$$ I understand that if in $(1)$ $C_n=C\ \forall n=1,...,T:$ $$PV=C\cdot\sum_{n=1}^T\left(\frac{1}{(1+r)}\right)^n $$ Maybe it's connected to the geometric sum? Thanks for every help!

## Answer by user35980 (score 1)

https://quant.stackexchange.com/a/78090

Not sure what the issue is here, but $$S_n=\frac{a(x^n-1)}{x-1}$$ for the sum of the first $n$ terms of a series with first term $a$ and common ratio $x$. In your case $a=x=\frac{1}{1+r}$ so $$\sum_{n=1}^{10} \frac{1}{(1+r)^n}=\frac{1}{r}\left( 1-\frac{1}{(1+r)^{10}}\right)$$ which as @nbbo2 has pointed out is your red term with the denominator corrected to be $r$ instead of $(1+r)^n$.

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