Bond Valuation and Interest Rate Dynamics

Introduction to Bonds and Valuation

• Role of Bonds in Finance: Bonds serve as a critical source of financing for corporations. The return that investors receive from these bonds is a key factor in determining a firm’s cost of capital.

• Valuation Principles: The valuation of financial securities, such as bonds and stocks, relies on the principle that the value of an asset is equal to the present value (PV) of its expected future cash flows.

• General Course Context: This topic follows the study of the Time Value of Money (TVM), interest rates, and investment decision rules. It acts as a foundation for capital budgeting and the broader understanding of risk and return in capital markets.

Bond Terminology and the Indenture

• Bond Definition: A bond is a fixed-income security. It represents a debt contract wherein the issuer promises to make a stream of known future cash payments to the holder until a specific maturity date.

• Bond Indenture: This is the legal contract that specifies all terms related to the bond issue, including:

  • Repayment type.

  • Identification of the issuer.

  • Maturity date.

  • Collateral or security backing the debt.

• Maturity Date: The final date on which the principal must be repaid. Payments continue until this specific date is reached.

• Term: The duration remaining until the repayment date.

• Principal (Face Value / Par Value): The maturity value or principal amount of the bond. It is the notional amount used to calculate interest (coupon) payments.

• Coupon Rate: The interest rate stated on the bond, expressed as an Annual Percentage Rate (APR). This rate is fixed at issuance.

• Coupon Payment ($CPN$): The regular interest payments promised to the bondholder. The formula to calculate the coupon payment is:

  • $CPN = \frac{\text{Coupon Rate} \times \text{Face Value}}{\text{Number of Coupon Payments per Year}}$

Valuation Principles for Bonds

• The Valuation Process: To determine the value of a bond, two primary steps are required:

  1. Identify Cash Flows: Determine the size and timing of all future cash flows (coupons and face value).

  2. Discount Cash Flows: Apply a discount rate that reflects the default risk of the bond.

• Default Risk: While government bonds are often treated as risk-free, corporate bonds involve varying levels of credit risk that influence the required discount rate.

Zero-Coupon Bonds

• Definition: A zero-coupon bond (also known as a pure discount bond) does not make regular coupon payments.

• Pricing: These bonds always sell at a discount (a price lower than the face value). The investor's return comes from the difference between the purchase price and the face value received at maturity.

• Treasury Bills (T-bills): These are zero-coupon bonds issued by the U.S. government with maturities of up to one year.

• Yield to Maturity (YTM) of Zero-Coupon Bonds: The YTM is the discount rate that sets the present value of the promised terminal payment equal to the current market price.

  • Formula for Price ($P$): $P = \frac{FV}{(1 + YTM_n)^n}$

  • Formula for YTM: $YTM_n = \left( \frac{\text{Face Value}}{\text{Price}} \right)^{1/n} - 1$

• Example: One-Year Zero-Coupon Bond:

  • Face Value ($FV$): $100,000$

  • Initial Price ($P$): $96,618.36$

  • Calculation: $1 + r_1 = \frac{100,000}{96,618.36} = 1.035$

  • Result: $r_1 = 3.5\%$

  • In this case, the YTM for the one-year bond is $3.5\%$.

• YTM as Internal Rate of Return (IRR): Solving for the YTM of a zero-coupon bond is functionally identical to calculating the IRR of the investment.

Risk-Free Interest Rates and the Yield Curve

• Law of One Price: In a competitive market, all risk-free investments maturing at the same time must earn the same rate of return. Consequently, the YTM of a risk-free (default-free) zero-coupon bond serves as a proxy for the risk-free interest rate.

• Spot Interest Rate: This is another term used to describe the default-free, zero-coupon yield for a specific maturity "on the spot."

• The Yield Curve (Term Structure of Interest Rates): A graphical representation plotting the yields of risk-free zero-coupon bonds against their respective maturity dates.

• Shapes of the Yield Curve:

  • Normal (Rising): Yields increase as maturity lengthens.

  • Flat: Yields are consistent across different maturities.

  • Inverted (Falling): Short-term yields are higher than long-term yields.

• U.S. Treasury Case (April 7, 2024): Historical data shows fluctuations in the yield curve, with yields for U.S. government bonds ranging approximately between $4\%$ and $6\%$ across various maturities (from 1 month to 30 years).

Coupon Bonds

• Cash Flow Structure: Investors in coupon bonds receive two types of cash flows:

  1. Regular coupon interest payments (an annuity).

  2. The face value (par value) at maturity (a lump sum).

• U.S. Treasury Securities: Typical maturities for Treasury coupon-bearing notes and bonds include 2, 3, 5, 7, 10, 20, and 30 years.

• Bond Price Quotes: In financial markets, bond prices and yields are often used interchangeably. Bond traders typically quote the yield rather than the dollar price.

• Standard Practice: The coupon rate is usually set to match the prevailing market yields at the time of issuance, so the bond initially sells at par.

• General Coupon Bond Valuation Formula:

  • $P = CPN \times \frac{1}{y} \left( 1 - \frac{1}{(1 + y)^N} \right) + \frac{FV}{(1 + y)^N}$

  • Where $y$ is the YTM (discount rate per period) and $N$ is the number of periods.

Computing Yield to Maturity for Coupon Bonds

• Definition: The YTM for a coupon bond is the single discount rate that equates the present value of all remaining coupons and the face value to the bond's current market price.

• Assumptions for YTM:

  • The investor holds the bond until maturity.

  • All interim cash flows (coupons) are reinvested at the YTM rate.

  • There is no default on any payments.

• Numerical Example (Example 6.4):

  • Given: 5-year bond, $1,000$ face value, $2.2\%$ coupon rate with semiannual payments, current price = $963.11$.

  • Steps:

    • $N = 10$ periods (5 years $\times$ 2).

    • $PMT = 11$ ($2.2\% / 2 \times 1,000$).

    • $PV = -963.11$.

    • $FV = 1,000$.

  • Solving for rate ($y$) using Excel RATE(10, 11, -963.11, 1000) yields $1.5\%$

  • Annual Rate (APR): $1.5\% \times 2 = 3.0\%$.

Computing the Price of a Coupon Bond

• Relationship to Interest Rates: Bond prices are the present value of future cash flows. If market interest rates (required yields) drop, bond prices rise.

• Numerical Example (Example 6.5):

  • Given: The same 5-year bond ($2.2\%$ coupon, semiannual). Investors now demand a $2.0\%$ APR yield.

  • Steps:

    • $N = 10$.

    • $y = 1.0\%$ ($2\% / 2$).

    • $PMT = 11$.

    • $FV = 1,000$.

  • Solving for $PV$ using Excel PV(0.01, 10, 11, 1000) yields a price of $1,009.47$.

• Clean vs. Dirty Price: Valuation typically assumes the "Clean Price," which ignores accrued interest between coupon dates ("Dirty Price").

Premiums, Discounts, and Par

• Bond Trading Classifications:

  1. Premium Bond: Trades "above par." This occurs when the Bond Price > Face Value, which happens when the Coupon Rate > Yield to Maturity.

  2. Par Bond: Trades "at par." This occurs when the Bond Price $=$ Face Value, which happens when the Coupon Rate $=$ Yield to Maturity.

  3. Discount Bond: Trades "below par." This occurs when the Bond Price < Face Value, which happens when the Coupon Rate < Yield to Maturity.

• Example (Example 6.6): Comparing three 30-year bonds with a $5\%$ YTM:

  • $10\%$ Coupon Bond: Price = $176.86$ (Premium).

  • $5\%$ Coupon Bond: Price = $100.00$ (Par).

  • $3\%$ Coupon Bond: Price = $69.26$ (Discount).

Dynamic Behavior of Bond Prices

• Inverse Relationship: There is an inverse relationship between interest rates ($r$) and bond prices ($P$). As rates rise, prices fall, and vice versa.

• Fixed Income vs. Fixed Return: Because market interest rate changes are unpredictable, bond prices fluctuate. While the "income" (coupons) may be fixed, the "return" is uncertain until maturity due to capital gains or losses.

• Why Prices Change:

  1. Changes in Market Rates: Future cash flows remain unchanged, but the discount rate used to value them changes.

  2. Passage of Time: Bond prices tend to converge toward their face value as they approach maturity. A discount bond's price will rise toward par, while a premium bond's price will fall toward par over time.

• Capital Gain (Loss) Yield: Calculated as the change in price over the original price:

  • $\text{Capital Gain/Loss Yield} = \frac{\text{New Price} - \text{Original Price}}{\text{Original Price}}$

Questions & Discussion

• Q: Suppose an investment pays $100 interest annually for 4 years, and then $1000 at the end of year 4. How much will you pay for this investment today if your required rate of return is 10%?

  • A: $1000. Calculation: The cash flows represent a bond where the coupon rate (10%) equals the required rate of return (10%), which means the bond sells at par.

• Q: Is there default risk on government bonds?

  • A: Government bonds are generally considered risk-free in a domestic context, though valuation principles allow for the introduction of default risk as a factor in the discount rate for other bonds.

• Q: Suppose a 4-year zero-coupon risk-free bond is trading at $83.06 per $100 face value. Find its yield to maturity.

  • A: Using the formula $YTM = (100 / 83.06)^{1/4} - 1$, the result is approximately $4.75\%$.

• Q: If the yield curve is downward sloping, longer maturity bonds generally have ________ shorter maturity bonds.

  • A: Lower yields than.

• Q: What is the expected rate of return if you buy it [a bond] at the market price?

  • A: Yield to Maturity (YTM).

• Q: A $1000 par value 5% bond is trading to yield 4%. Which of the following is likely to be the bond’s market price?

  • A: $1100. (Since the yield is lower than the coupon rate, the bond must sell at a premium/above par).

• Q: Case Study - 8% Semiannual US Treasury Bond sold at par. If the Fed suddenly raises the fund rate to 9%, what happens?

  • A: The YTM adjusts to the new market interest rate (9%). The price will fall below par to the present value of future cash flows discounted at 9%. Given the calculation provided in the transcript: $Price = 40 \times \frac{1 - (1 + 0.045)^{-10}}{0.045} + \frac{1000}{(1 + 0.045)^{10}} = 960.44$.", "title": "Bond Valuation and Interest Rate Dynamics"}