Introduction
Bond valuation is a technique for determining the fair price of a bond. As a fixed-income security, the value of a bond is determined by the present value of its expected future cash flows, which include periodic interest payments and the principal amount to be received at maturity.
Pricing Bonds Using the Present Value of Future Cash Flows
The value of a bond is calculated by the present value of its expected future payments. The formula to calculate the price of a bond is:
Bond Price = ∑ (C / (1 + r)^n) + F / (1 + r)^T
Where:
- C = Coupon payment
- r = Discount rate or yield to maturity
- n = Number of periods until the payment
- F = Face value of the bond
- T = Total number of periods until maturity
Calculating Yield to Maturity (YTM)
Yield to Maturity (YTM) is the total return anticipated on a bond if the bond is held until it matures. YTM is expressed as an annual rate and is a complex calculation that can be found using a trial-and-error method, or more accurately, with a financial calculator. Here’s a simplified version of the calculation:
YTM ≈ (C + (F – P) / T) / ((F + P) / 2)
Where:
- C = Annual coupon payment
- F = Face value of the bond
- P = Current price of the bond
- T = Number of years to maturity
Bond Duration and Convexity
1. Bond Duration
Duration measures the sensitivity of a bond’s price to a change in interest rates and is expressed in years. Macaulay duration is a common measure of bond duration and can be calculated as follows:
Macaulay Duration = ∑ (t × C / (1 + r)^t) / Bond Price + (T × F / (1 + r)^T) / Bond Price
2. Bond Convexity
Convexity is a measure of the curvature in the relationship between bond prices and bond yields. Convexity shows how much a bond’s duration changes when the yield to maturity changes. The formula for calculating convexity is:
Convexity = ∑ (t × (t + 1) × C / (1 + r)^(t + 2)) / Bond Price + (T × (T + 1) × F / (1 + r)^(T + 2)) / Bond Price
Conclusion
Bond valuation is a critical process for investors in the debt markets. Understanding the underlying principles such as pricing, yield to maturity, duration, and convexity helps investors make informed decisions and better manage their investment risks.
