Bond Valuation Lecture 7: Present Value Of Bond Cash Flows

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Bond valuation is fundamentally rooted in the principles of the time value of money, which posits that a dollar today is worth more than a dollar received in the future. The current price of a bond is calculated as the present value of its future cash flows—namely, interest payments and the principal repayment—discounted at the prevailing market interest rate. The market rate is influenced by factors such as the bond’s term to maturity, creditworthiness, and tax considerations. The simplest form of bond, for valuation purposes, is a zero-coupon bond, which makes no interest payments until maturity, at which point it pays the par value.

For example, a 20-year zero-coupon bond with a face value of $1,000, discounted at an annual yield of 6%, might have a price around $306.56. Over the 20-year horizon, the accrued interest income for the investor is the difference between the purchase price and the face value, which equals approximately $693.44, reflecting the compounded discount rate. Zero-coupon bonds sell at deep discounts because they do not provide periodic coupon payments, and their valuation hinges solely on the present value of the face amount at maturity.

In contrast, bonds with periodic interest payments, such as a 20-year bond with a 7% coupon paid semiannually, involve valuing an annuity stream of interest payments plus the present value of the face value. When interest payments are semiannual, the annual market rate must be converted into a semiannual rate, which then determines the discount factors for calculations. The bond’s price is composed of the present value of the coupon payments—an annuity—and the present value of the face value paid at maturity.

Using the present value formulas, the bond’s price can be computed as: P = (PMT × (1 – (1 + i)^-N) / i) + (FV / (1 + i)^N), where PMT is the semiannual coupon payment, N is the total number of semiannual periods, and i is the semiannual discount rate. The sum of these components yields the bond’s current market value. Empirical examples illustrate how a bond’s valuation changes with market interest rates, coupon rates, and time to maturity.

For instance, consider a 15-year bond with a face value of $1,000, a 5.5% coupon rate, paid semiannually, and a current market rate of 6.5%. Applying the valuation formula reveals the bond’s intrinsic value as approximately $905.09, which is below its market price of $940. Consequently, the bond appears overvalued, and an investor might decide against purchasing it. These calculations are typically performed using financial calculators or spreadsheet functions due to their complexity.

Bond Yields and Their Significance

Bond yields are critical metrics that reflect the return an investor can expect, given the price paid and the future cash flows. The most straightforward yield measure is the current yield, calculated as the annual coupon payment divided by the bond’s current market price. While easy to compute, the current yield only considers income from coupons and ignores capital gains or losses due to price fluctuations.

The more comprehensive measure is the yield to maturity (YTM), which accounts for all future coupon payments and the repayment of face value, discounted at the YTM rate itself. It represents the total annualized return an investor receives if the bond is held until maturity. Computing YTM involves solving the present value equation for the discount rate, often through iterative methods or financial calculators, because it cannot generally be isolated algebraically.

For example, a bond priced at $962.81 with four years remaining and a 3.5% coupon rate yields a YTM of approximately 4.53%, while its current yield is about 3.64%. The YTM provides a better estimate of expected returns, especially when bonds are purchased at premiums or discounts, which influence capital gains or losses.

Callable bonds, which can be redeemed before maturity at a specified call price, require an adjusted yield calculation called the yield to call (YTC). The YTC assumes the bond will be called at the earliest possible date, influencing its valuation and risk profile. If the likelihood of call increases as interest rates decline, investors may prefer to analyze YTC to assess potential returns accurately.

Municipal Bonds and Tax Considerations

Municipal bonds (munis) generally offer lower yields than corporate bonds or Treasuries because of their tax-exempt status, especially at the federal level. The tax advantage makes them particularly attractive to high-income investors, who can compute their taxable equivalent yields by adjusting the nominal yield for their marginal tax rate. For instance, a muni bond with a 4% yield is equivalent to a taxable bond with a higher yield, calculated as: taxable equivalent yield = municipal yield / (1 – tax rate).

Analyzing after-tax yields is essential for investors aiming to maximize after-tax income. For example, with a 35% marginal tax rate, a municipal bond yielding 4% would be equivalent to a taxable bond yielding about 6.15%. Therefore, high-tax-bracket investors might prefer munis if their taxable equivalent yield exceeds that of corporate bonds.

Comparing Yields and Investment Decisions

The relationship between different bond yields explains many market behaviors. When a bond is traded at par, the coupon rate, current yield, and YTM tend to align. However, bonds trading at a premium or discount exhibit inverse relationships among these yields. Specifically, for premium bonds, YTM and current yield are lower than the coupon rate, while for discount bonds, they are higher.

Investors must also heed bond features such as callability. When interest rates fall, issuing companies may redeem callable bonds early, prompting investors to evaluate the yield to call instead of YTM for a more accurate assessment. Such features add complexity to bond valuation and risk analysis.

Conclusion

Bond valuation merges concepts of present value, interest rate risk, credit risk, and tax considerations. Proper understanding of various yields—current yield, yield to maturity, yield to call, and taxable equivalent yield—equips investors to make informed decisions aligned with their investment objectives and tax situations. As market conditions change, the relative attractiveness of bonds shifts, demanding sophisticated valuation and comparison techniques to optimize investment returns.

References

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