Calculating The Current Price Of A Bond

The current price of a bond is calculated as the Present Value (PV) of all its expected future cash flows, which consist of the periodic coupon payments (interest) and the final repayment of the face value (principal) at maturity.

This calculation is known as bond valuation, and the discount rate used is the bond’s Yield to Maturity (YTM), which is the current market interest rate for bonds with similar risk and maturity.

The Bond Pricing Formula

The general formula for the current price ( $P$ ) of a bond with periodic coupon payments is:

$P = \sum_{t=1}^{N} \frac{C}{(1 + r)^t} + \frac{F}{(1 + r)^N}$

Where:

$P$ = Current Price of the Bond

$C$ = Periodic Coupon Payment (e.g., if annual coupon rate is $5\%$ and face value is $$1,000$ , and payments are semi-annual, $C = ($1,000 \times 0.05) / 2 = $25$ )

$F$ = Face Value (or Par Value) to be repaid at maturity (typically $$1,000$ or $$100$ )

$r$ = Yield to Maturity (YTM) per period (i.e., the annual YTM divided by the number of payments per year)

$N$ = Total Number of Periods until maturity (i.e., Years to Maturity multiplied by the number of payments per year)

$t$ = The period number

This formula essentially consists of two parts:

The Present Value of the Annuity (Coupon Payments): The sum of the present values of all future periodic coupon payments.
The Present Value of the Lump Sum (Face Value): The present value of the face value repayment at maturity.

Steps for Calculation

Here’s how to calculate the bond price, often assuming semi-annual payments, which is common for corporate and government bonds:

Determine Periodic Values:
- Periodic Coupon ( $C$ ): $C = (\text{Annual Coupon Rate} \times \text{Face Value}) / 2$
- Periodic YTM ( $r$ ): $r = \text{Annual YTM} / 2$
- Total Periods ( $N$ ): $N = \text{Years to Maturity} \times 2$
Calculate the Present Value of Coupon Payments: Use the present value of an annuity formula for the coupon stream:
$\text{PV}_{\text{Coupons}} = C \times \left[ \frac{1 - (1 + r)^{-N}}{r} \right]$
Calculate the Present Value of Face Value: Use the present value of a lump sum formula for the principal repayment:
$\text{PV}_{\text{Face Value}} = F \times (1 + r)^{-N}$
Sum the Present Values:
$\text{Bond Price} = \text{PV}_{\text{Coupons}} + \text{PV}_{\text{Face Value}}$

Bond Pricing Relationships

The relationship between the bond’s Coupon Rate and the prevailing Yield to Maturity ( $r$ ) determines if the bond trades at a premium, discount, or par:

Relationship	Bond Price vs. Face Value	Description	Example (Business)
Coupon Rate > YTM	Premium Price ( $P > F$ )	The bond’s fixed payments are more attractive than current market rates, so investors pay more than the face value.	In 2024, Nestlé S.A. bonds issued years ago with a $4\%$ coupon might trade at a premium if new, similar bonds are only yielding $2.5\%$ .
Coupon Rate = YTM	Par Price ( $P = F$ )	The bond’s payments are equal to the market’s required return.	A newly issued German Government Bond (Bund) with a $1.5\%$ coupon trading when the market YTM for similar bonds is also $1.5\%$ .
Coupon Rate < YTM	Discount Price ( $P < F$ )	The bond’s fixed payments are less attractive than current market rates, so investors require a discount on the price to achieve the higher market yield.	A bond issued by Toyota Motor Corporation with a $1.0\%$ coupon might trade at a discount if rising global interest rates have pushed the required market YTM for similar bonds to $3.0\%$ .