Price = Σ[C/(1+r)^t] + F/(1+r)^n
Bond price equals present value of coupon payments plus present value of face value, discounted at YTM.
Master the fundamental principle of fixed income valuation: a bond's price is the sum of the present value of all its future payments.
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Bond Price: Definition and Core Components
The **Bond Price** is the current market value of a bond. This price represents the total amount an investor must pay today to purchase the right to receive the bond's scheduled future cash flows. A bond is essentially a fixed-rate loan that the investor makes to the issuer.
The Two Cash Flow Streams
A standard bond generates two distinct types of cash flows for the investor:
- Coupon Payments: A series of fixed, periodic interest payments (an annuity) paid from the issue date until maturity.
- Face Value Repayment: A single lump sum payment (the principal, or par value) paid on the maturity date.
The bond price calculation involves finding the **Present Value (PV)** of each of these future cash flows and summing them up.
The Bond Price Calculation Formula
The bond price is equal to the Present Value of the annuity stream (coupons) plus the Present Value of the lump sum (face value).
The Calculation Identity
The total price is separated into its two component PVs:
Bond Price = PV (Coupons) + PV (Face Value)
The Detailed Formula
This combined formula discounts all future cash flows using the **Yield to Maturity (YTM)** ($r$), which acts as the discount rate:
Bond Price = Sum [ C / (1+r)^t ] + F / (1+r)^T
Where $C$ is the periodic coupon payment, $F$ is the face value, $r$ is the YTM, $t$ is the payment period, and $T$ is the total number of periods remaining.
Present Value of Coupons and Face Value
Each component of the formula requires careful application of the Present Value concept.
PV of the Coupon Stream (Annuity)
The coupon payments form an annuity. Their total PV is calculated by discounting each individual coupon payment back to the present. The periodic coupon payment ($C$) is calculated by dividing the annual coupon rate by the payment frequency (usually semiannually, or twice a year).
PV of the Face Value (Lump Sum)
The face value ($F$) is received only once, at maturity. Its PV is calculated by discounting the full lump sum back over the entire remaining time period ($T$). Since this value is received last, it is subject to the greatest discounting effect.
The Inverse Relationship Between Price and Yield
The discount rate used to price the bond is the **Yield to Maturity (YTM)**. The YTM is the single greatest driver of the bond price, and their relationship is always inverse.
Interest Rate Movement
When market interest rates (and thus the required YTM) rise, the calculated present value of the bond's fixed cash flows falls, causing the **Bond Price to Decrease**. Conversely, when market rates fall, the present value of the cash flows increases, causing the **Bond Price to Rise**.
Duration and Volatility
The sensitivity of the bond price to changes in YTM is measured by the bond's **Duration**. Bonds with longer maturities and lower coupon rates have higher duration, meaning their prices will fluctuate more dramatically for a given change in interest rates.
Pricing Scenarios: Par, Premium, and Discount
The relationship between the bond's **Coupon Rate** and the market's **Yield to Maturity (YTM)** determines whether the bond will trade at par, a premium, or a discount.
1. Par Bond
Occurs when the Coupon Rate **equals** the market YTM. The bond price equals its face value (e.g., 1,000 dollars).
2. Premium Bond
Occurs when the Coupon Rate is **greater than** the market YTM. The bond price is higher than its face value. New investors are willing to pay a premium because the bond's fixed interest payments are higher than prevailing market rates.
3. Discount Bond
Occurs when the Coupon Rate is **less than** the market YTM. The bond price is lower than its face value. The price is discounted because the bond's fixed interest payments are lower than new bonds being issued at the higher prevailing market rate.
Conclusion
The **Bond Price** is the calculated sum of the Present Value of its two cash flow components: the coupon annuity stream and the face value lump sum, discounted at the Yield to Maturity (YTM).
The fundamental rule of fixed income is the **inverse relationship between price and yield**. Understanding this relationship is crucial for investors to determine if a bond should trade at par, a premium (when the Coupon Rate exceeds YTM), or a discount (when the Coupon Rate is less than YTM).
Bond price is the present value of all future cash flows (coupons + face value) discounted at the yield to maturity.
Premium bonds trade above par when coupon exceeds YTM; discount bonds trade below par when YTM exceeds coupon.
Use this calculator to determine fair value and assess if a bond offers attractive returns relative to market rates.