Bonds, Interest Rates, and Debt Security Valuation

Fundamentals of Bonds and Interest Rates

• Role in the Economy:

  • Interest rates play a crucial role in driving overall economic performance and business activity in the United States.

  • Bonds represent a critical source of debt financing for corporations and government entities.

• Dual Measurement of Interest Rates:

  • An interest rate measures two distinct financial dimensions simultaneously:

    1. Cost to the borrower: The price incurred for obtaining capital upfront.

    2. Reward (return) to the lender: The compensation received for supplying capital and deferring consumption.

• Mechanics of Corporate Bond Issuance:

  • By issuing and selling bonds, corporations borrow capital immediately while legally committing to repay the principal and interest in the future.

  • Bond purchasers act as lenders who provide upfront funds in exchange for a contractual, predictable stream of future income.

• Core Attributes of Debt Securities:

  • Fixed Payments: Contractual coupon payments and/or face value payouts.

  • Expiration Date: A specified maturity date at which the debt contract concludes.

  • Variable Price: The market trading price fluctuates dynamically based on interest rate movements and overall market demand.

Taxonomy of Debt Securities

• Discount Bond (Zero-Coupon Bond):

  • Definition: A debt security that makes a single payment equal to its face value ($\text{FV}$) at maturity, without paying intermediate periodic interest.

  • Maturity Horizon: Typically expires in $1$ year or less.

  • Pricing: Sold at a discount relative to its face value ($P < \text{FV}$). The investor's yield is derived entirely from the difference between the discounted purchase price and the face value paid at maturity.

  • Examples: Treasury bills (T-bills), commercial paper, and large negotiable certificates of deposit (CDs).

  • Numerical Example: A $1$-year Treasury bill with a face value of $\$1,000$ selling at a current price of $\$950$. The bond pays $\$1,000$ in one year, with a current market cost of $\$950$.

  • Visual Certificate Components:

    1. Issuer: Identifies the issuing authority (e.g., The United States of America / Treasury Department).

    2. Maturity Date: Explicitly states the expiration date (e.g., a $6$-month discount bond maturing April 9, 1970).

    3. Face Value: Indicates the principal amount paid upon presentation at maturity (e.g., $\$100,000$).

    4. Price Omission: Market transaction price is never printed on the certificate.

  • Certificate Historical Details:

    • Denomination: One Hundred Thousand Dollars ($\$100,000$).

    • Serial Number: $1017055\text{A}$.

    • Statutory Authority: Second Liberty Bond Act, as amended; Treasury Department Circular No. 418, Revised.

    • Issue Date: October 9, 1969; Due Date: April 9, 1970.

    • Signatory: Secretary of the Treasury David M. Kennedy; Portrait: Albert Gallatin; Form Code: D 1966-2.

• Coupon Bond:

  • Definition: A debt security that pays fixed annual or semi-annual coupon interest payments ($C$) throughout its term, followed by the full return of the face value ($\text{FV}$) upon its maturity date.

  • Key Parameters: Face value ($\text{FV}$), coupon payment amount ($C$) or coupon rate ($\text{CR} = \frac{C}{\text{FV}}), maturity date, and issuing entity.\n * **Examples:** U.S. Treasury bonds (maturities over 10$years), U.S. Treasury notes (maturities between$1$and$10 years), and corporate bonds.\n * **Numerical Example:** A 2$-year coupon bond with a face value of$\$1,000$and a$5\%$coupon rate selling for$\$1,020.\n * Annual coupon payment calculation: C = 5\% \times \$1,000 = \$50.\n * Cash flow schedule: Makes two annual payments of \$50$each, plus the principal repayment of$\$1,000 at the end of Year 2.\n * Current price: \$1,020.\n * **Visual Certificate Components:**\n 1. **Issuer:** Identifies the issuing government or corporate entity.\n 2. **Maturity Date:** Specifies when principal repayment occurs (e.g., an 8-year bond maturing February 15, 1987).\n 3. **Face Value:** Indicates the nominal principal (e.g., \$1,000).\n 4. **Coupon Rate:** States the fixed nominal interest rate (e.g., 9\% Treasury Note).\n 5. **Coupons:** Detachable physical coupons for collecting semi-annual interest (e.g., \$45.00 paid twice per year on August 15 and February 15).\n 6. **Price Omission:** Market trading price is never printed on the bond paper.\n * **Certificate Historical Details:**\n * Denomination: One Thousand Dollars (\$1,000).\n * Serial Number: 10656$; CUSIP:$912827\text{ JK }9.\n * Series: Series B-1987; Circular No.: 2-79.\n * Issue Date: February 15, 1979; Due Date: February 15, 1987.\n * Signatory: Secretary of the Treasury W.H. Blumenthal; Portrait: Abraham Lincoln.\n\n\n\n* **Consol Bond (Perpetuity):**\n * **Definition:** A perpetual debt instrument that makes fixed coupon payments (C) forever in perpetuity. It carries no face value and no final maturity date.\n * **Pricing Formula:**\n        P = \frac{C}{i}\n * **Yield Formula:**\n        i = \frac{C}{P}\n * **Numerical Example:**\n * Calculate the price of a consol bond that pays an ongoing annual coupon payment of C = \$5$forever at a interest rate (yield) of$i = 8\% = 0.08:\n            P = \frac{5}{0.08} = 62.5\n * The fair value/price of this consol bond is \$62.50.\n\n\n# Pricing Mechanics and Present Value\n\n* **General Valuation Principle:**\n * The market price (P) of any bond is equal to the sum of the present values of all its expected future cash flows discounted back to the present at the market interest rate (yield).\n\n* **Pricing a One-Year Discount Bond:**\n * **Formula:**\n        P = \frac{\text{FV}}{1 + i}\n * **Worked Calculation:**\n * Price of a 1$-year discount bond with a face value of$\text{FV} = \$100$at a market interest rate (yield) of$i = 5\% = 0.05:\n            P = \frac{\100}{1 + 0.05} = \frac{\100}{1.05} = \$95.24\n\n* **Pricing a Multi-Year Coupon Bond:**\n * **General Formula:**\n        P = \sum_{t=1}^{n} \frac{C}{(1 + i)^t} + \frac{\text{FV}}{(1 + i)^n}\n * **Worked Calculation:**\n * Price of a 3$-year coupon bond with a face value of$\text{FV} = \$1,000$, an annual coupon rate of$10\%$($C = \$100$), and a yield of$i = 2\% = 0.02:\n            P = \frac{\100}{1 + 0.02} + \frac{\100}{(1 + 0.02)^2} + \frac{\100}{(1 + 0.02)^3} + \frac{\1,000}{(1 + 0.02)^3}\n            P = \frac{\100}{1.02} + \frac{\100}{1.0404} + \frac{\100}{1.061208} + \frac{\1,000}{1.061208}\n            P = \$98.039 + \$96.117 + \$94.232 + \$942.322 = \$1,230.70\n\n\n# Yield to Maturity (YTM) and Current Yield\n\n* **Yield to Maturity (YTM) Defined:**\n * The exact interest rate (i$) that equates the total present value of all contractual future cash flows to the security's current market price ($P).\n\n* **Inverse Relationship:**\n * A bond's price and its yield to maturity are always inversely linked: as interest rates/yields increase, the bond's market price decreases, and vice versa.\n\n* **Yield to Maturity Equations:**\n * **One-Year Discount Bond:**\n        \text{YTM} = i = \frac{\text{FV} - P}{P} \times 100\%\n * Where \text{FV}$represents the fixed face value and$P represents the current variable price.\n * **One-Year Coupon Bond:**\n * Present value equation:\n            P = \frac{C}{1 + i} + \frac{\text{FV}}{1 + i}\n * Explicit formula for YTM (i):\n            i = \frac{C + \text{FV} - P}{P}\n * **Multi-Year (n > 2) Coupon Bond:**\n * Requires solving for i in the multi-period discount equation:\n            P = \frac{C}{1 + i} + \frac{C}{(1 + i)^2} + \dots + \frac{C}{(1 + i)^n} + \frac{\text{FV}}{(1 + i)^n}\n * To calculate YTM, four factors are integrated: fixed Face Value (\text{FV}$), fixed Coupon Payment ($C$), fixed Maturity ($n$), and variable Price ($P$). Solving analytically for$n > 2 is computationally complex.\n\n* **Empirical Relationship Between Bond Price and YTM:**\n * Data for a 10\%$coupon rate bond maturing in$10$years with a face value of$\text{FV} = \$1,000$($C = \$100 annually):\n\n\n\n * **Price vs. YTM Schedule:**\n * Price = \$1,200$$\rightarrow$Yield to Maturity =$7.13\%\n * Price = \$1,100$$\rightarrow$Yield to Maturity =$8.48\%\n * Price = \$1,000$$\rightarrow$Yield to Maturity =$10.00\%\n * Price = \$900$$\rightarrow$Yield to Maturity =$11.75\%\n * Price = \$800$$\rightarrow$Yield to Maturity =$13.81\%\n\n* **Core Coupon Bond Pricing Theorems:**\n 1. When a coupon bond is priced exactly at its face value (P = \text{FV}$), its yield to maturity equals its coupon rate ($\text{YTM} = \text{CR} = 10.00\%).\n 2. When a coupon bond is priced below its face value (P < \text{FV}$), its yield to maturity is strictly greater than its coupon rate ($\text{YTM} > \text{CR}).\n 3. When a coupon bond is priced above its face value (P > \text{FV}$), its yield to maturity is strictly less than its coupon rate ($\text{YTM} < \text{CR}).\n\n* **Current Yield (CY):**\n * **Definition:** The annual coupon payment divided by the market price of the bond.\n * **Formula:**\n        \text{CY} = \frac{C}{P}\n * **Coupon Rate (CR) Formula:**\n        \text{CR} = \frac{C}{\text{FV}}\n * **Universal Ordering Relationships (For Any Coupon Bond Maturity):**\n * If P < \text{FV}$(Discount Bond):$\text{YTM} > \text{CY} > \text{CR}\n * If P > \text{FV}$(Premium Bond):$\text{YTM} < \text{CY} < \text{CR}\n * If P = \text{FV}$(Par Bond):$\text{YTM} = \text{CY} = \text{CR}\n\n\n# Holding Period Rates of Return and Interest-Rate Risk\n\n* **YTM vs. Holding Period Return:**\n * Yield to maturity measures the return earned **only** if the security is held continuously until its final maturity date.\n * Liquidating a bond before maturity generates a holding period rate of return (R) that often diverges from YTM due to capital gains or losses resulting from price changes.\n\n* **One-Year Holding Period Return Formula:**\n * Total return (R) over a single year holding period consists of current income plus price change relative to initial price:\n        R = \frac{C}{P_0} + \frac{P_1 - P_0}{P_0}\n        R = \text{Current Yield} + \text{Rate of Capital Gain/Loss}\n * Where P_0$is the purchase price at the beginning of the period and$P_1 is the selling price at the end of the period.\n\n* **Capital Loss Hazard:**\n * If market interest rates rise significantly, driving down the resale price P_1$, the rate of capital loss ($\frac{P_1 - P_0}{P_0}$) can exceed the current yield ($\frac{C}{P_0}$), producing a negative overall rate of return ($R < 0).\n\n* **Interest-Rate Risk:**\n * **Definition:** The exposure to fluctuations in a bond's market price caused by changes in prevailing interest rates.\n * **Maturity Dependence:** The magnitude of interest-rate risk increases with maturity; longer-term bonds experience greater price swings for any given change in market interest rates than shorter-term bonds.\n\n\n# Real Interest Rates and Inflation Dynamics\n\n* **Erosion of Purchasing Power:**\n * Because most coupon payments and principal obligations are contractually fixed in nominal dollars, inflation erodes the real value of these cash flows over time.\n * Investors focus on real interest rates (purchasing power return) rather than nominal interest rates when evaluating investment opportunities.\n\n* **Fisher Equation (Real Interest Rate Calculation):**\n * **Formula:**\n        r = i - \pi^e\n * Where r represents the real interest rate\n * Where i represents the nominal interest rate\n * Where \pi^e$represents the expected inflation rate</p></li></ul></li><li><p><strong>Inflation Expectations and Wealth Redistribution:</strong></p><ul><li><p>Determining the real rate requires investors to formulate explicit predictions regarding future inflation ($\pi^e$$).

  • If actual inflation exceeds expected inflation during the life of the loan, bond buyers (lenders) suffer an unexpected loss of real purchasing power, benefiting borrowers who repay obligations in depreciated currency.