The Meaning of Interest Rates - Comprehensive Study Guide

Measuring Interest Rates and the Concept of Present Value

  • Foundational Concept: To understand the study of money, banking, and financial markets, one must precisely define the term "interest rate." The yield to maturity (YTM) is identified as the most accurate measure of the interest rate.

  • The Principle of Present Value: A dollar paid to an individual one year from today is less valuable than a dollar paid today because a dollar held today can be deposited to earn interest. Specifically, if a dollar is deposited today at an interest rate ii, it will become 1×(1+i)1 \times (1 + i) in one year.

  • Future Value Calculations:

    • Assuming an interest rate of i=0.10i = 0.10:

    • In one year: $100×(1+0.10)=$110\text{\$100} \times (1 + 0.10) = \text{\$110}

    • In two years: $110×(1+0.10)=$121\text{\$110} \times (1 + 0.10) = \text{\$121}, which is the same as $100×(1+0.10)2\text{\$100} \times (1 + 0.10)^2

    • In three years: $121×(1+0.10)=$133\text{\$121} \times (1 + 0.10) = \text{\$133}, which is the same as $100×(1+0.10)3\text{\$100} \times (1 + 0.10)^3

    • In n years: $100×(1+i)n\text{\$100} \times (1 + i)^n

  • Simple Present Value Formula: The value of a future cash flow today is determined by the formula:     PV=CF(1+i)nPV = \frac{CF}{(1 + i)^n}     Where:

    • PVPV = today’s (present) value

    • CFCF = future cash flow (payment)

    • ii = the interest rate

    • nn = number of periods (years)

  • Time-Line Comparisons: It is impossible to directly compare payments scheduled for different points in a timeline without first converting them to their present values.

  • Practical Example: The Lottery Jackpot: Consider a $20 million New York State Lottery jackpot that pays $1 million per year for 20 years.

    • Is this really worth $20 million today? No.

    • In a present value sense, the jackpot is worth significantly less because the dollars received in the future must be discounted.

    • For example, if the interest rate is 10%10\%, the present value of the final $1 million payment received 20 years from now would only be approximately 1,000,000(1+0.10)20\frac{1,000,000}{(1 + 0.10)^{20}}.

Types of Credit Market Instruments

  • Four Primary Instruments:

    1. Simple Loan: The borrower receives a principal amount and repays the principal plus interest at a specified maturity date.

    2. Fixed-Payment Loan: The borrower makes a fixed periodic payment (comprising both principal and interest) throughout the life of the loan.

    3. Coupon Bond: A bond that pays the owner a fixed interest payment (coupon) every year until the maturity date, at which time a specified final amount (face value or par value) is repaid.

    4. Discount Bond: A bond (such as a U.S. Treasury bill) that is purchased at a price below its face value. The face value is repaid at the maturity date, and the bond does not make any periodic interest payments.

Yield to Maturity (YTM)

  • Definition: The yield to maturity is the interest rate that equates the present value of all future cash flow payments received from a debt instrument with its value today.

  • Yield to Maturity on a Simple Loan:

    • If PV=amount borrowed=$100PV = \text{amount borrowed} = \text{\$100}, CF=cash flow in one year=$110CF = \text{cash flow in one year} = \text{\$110}, and n=1n = 1:

    • $100=$110(1+i)1\text{\$100} = \frac{\text{\$110}}{(1 + i)^1}

    • (1+i)×$100=$110(1 + i) \times \text{\$100} = \text{\$110}

    • (1+i)=$110$100(1 + i) = \frac{\text{\$110}}{\text{\$100}}

    • i=1.10−1=0.10=10%i = 1.10 - 1 = 0.10 = 10\%

    • Conclusion: For simple loans, the simple interest rate is exactly equal to the yield to maturity.

  • Fixed-Payment Loan Formula: To find the YTM, solve for ii in the following equation:     LV=FP1+i+FP(1+i)2+FP(1+i)3+...+FP(1+i)nLV = \frac{FP}{1 + i} + \frac{FP}{(1 + i)^2} + \frac{FP}{(1 + i)^3} + \text{...} + \frac{FP}{(1 + i)^n}     Where:

    • LVLV = loan value

    • FPFP = fixed yearly payment

    • nn = number of years until maturity

  • Coupon Bond Formula: The price of a coupon bond (PP) is the present value of all coupon payments plus the present value of the face value:     P=C1+i+C(1+i)2+C(1+i)3+...+C(1+i)n+F(1+i)nP = \frac{C}{1 + i} + \frac{C}{(1 + i)^2} + \frac{C}{(1 + i)^3} + \text{...} + \frac{C}{(1 + i)^n} + \frac{F}{(1 + i)^n}     Where:

    • PP = price of the coupon bond

    • CC = yearly coupon payment

    • FF = face value of the bond

    • nn = years to maturity

  • Relationships in Coupon Bonds:

    • When a coupon bond is priced at its face value, the yield to maturity equals the coupon rate.

    • The price of a coupon bond and the yield to maturity have a negative relationship; as YTM rises, bond price falls.

    • The yield to maturity is greater than the coupon rate when the bond price is below its face value (discounted bond).

  • Sample Yields on a 10%-Coupon-Rate Bond (FV=$1,000FV = \text{\$1,000}, Maturity = 10 years):

    • Price = $1,200: YTM = 7.13%

    • Price = $1,100: YTM = 8.48%

    • Price = $1,000: YTM = 10.00%

    • Price = $900: YTM = 11.75%

    • Price = $800: YTM = 13.81%

  • Consol (Perpetuity): A bond with no maturity date and no principal repayment that pays fixed coupon payments forever.

    • Formula: Pc=CicP_c = \frac{C}{i_c}

    • Yield of a consol: ic=CPci_c = \frac{C}{P_c}

    • This equation provides the current yield, which serves as an easy-to-calculate approximation of the yield to maturity for coupon bonds.

  • Discount Bond Yield to Maturity:

    • For a one-year discount bond: i=F−PPi = \frac{F - P}{P}

    • Where FF is face value and PP is current price.

    • The YTM equals the increase in price over the year divided by the initial price.

    • Like coupon bonds, YTM and price are negatively related.

Distinction Between Interest Rates and Returns

  • Rate of Return (RET): The total payments received by the owner plus the change in the instrument's value, expressed as a fraction of the purchase price.     RET=C+Pt+1−PtPtRET = \frac{C + P_{t+1} - P_t}{P_t}     Where:

    • PtP_t = price of bond at time tt

    • Pt+1P_{t+1} = price of bond at time t+1t+1

    • CC = coupon payment

  • Decomposition of Return:

    • Current Yield (ici_c): ic=CPti_c = \frac{C}{P_t}

    • Rate of Capital Gain (gg): g=Pt+1−PtPtg = \frac{P_{t+1} - P_t}{P_t}

    • Therefore: RET=ic+gRET = i_c + g

  • Key Relationships Regarding Returns:

    1. The return equals the yield to maturity only if the holding period matches the time to maturity.

    2. A rise in interest rates leads to a fall in bond prices, resulting in a capital loss if the time to maturity is longer than the holding period.

    3. The longer the bond's maturity, the greater the percentage price change associated with a change in the interest rate.

    4. The longer the bond's maturity, the lower the rate of return when interest rates increase.

    5. Negative returns are possible even if a bond has a high initial interest rate, provided interest rates rise sufficiently.

  • Table: Returns on 10%-Coupon-Rate Bonds when Rates Rise from 10% to 20%:

    • 30 Years to Maturity: Initial Price: $1,000; Price Next Year: $503; Capital Gain: -49.7%; Return: -39.7%

    • 20 Years to Maturity: Initial Price: $1,000; Price Next Year: $516; Capital Gain: -48.4%; Return: -38.4%

    • 10 Years to Maturity: Initial Price: $1,000; Price Next Year: $597; Capital Gain: -40.3%; Return: -30.3%

    • 5 Years to Maturity: Initial Price: $1,000; Price Next Year: $741; Capital Gain: -25.9%; Return: -15.9%

    • 2 Years to Maturity: Initial Price: $1,000; Price Next Year: $917; Capital Gain: -8.3%; Return: +1.7%

    • 1 Year to Maturity: Initial Price: $1,000; Price Next Year: $1,000; Capital Gain: 0.0%; Return: +10.0%

Interest-Rate Risk and Volatility

  • Interest-Rate Risk: The riskiness of an asset's return that results from interest-rate changes.

  • Maturity Factors: Prices and returns for long-term bonds are significantly more volatile than for short-term bonds.

  • Bond Holding Rule: There is no interest-rate risk for any bond whose time to maturity exactly matches the investor's holding period.

  • Negative Interest Rates: Recent historical experiences in Japan and several European countries demonstrate that interest rates do not always have to be positive.

Real Versus Nominal Interest Rates

  • Nominal Interest Rate: The interest rate that makes no allowance for inflation.

  • Real Interest Rate: The interest rate adjusted for changes in the price level to more accurately reflect the true cost of borrowing.

    • Ex-ante Real Interest Rate: Adjusted for expected changes in the price level (used for decision making).

    • Ex-post Real Interest Rate: Adjusted for actual changes in price level (measures what actually happened).

  • Fisher Equation: Describes the relationship between nominal and real interest rates:     i=ir+πei = i_r + \frac{\pi^e}{}     (Note: Slide notation shows i = i_r + \text{\pi}^e)     Where:

    • ii = nominal interest rate

    • iri_r = real interest rate

    • \text{\pi}^e = expected inflation rate

  • Incentives: When the real interest rate is low, there are greater incentives to borrow and fewer incentives for lenders to lend. The real interest rate is a superior indicator of the true incentives in the credit market.

  • Historical Context (1953–2020): Real and nominal rates often diverge significantly, especially during periods of high inflation. Real rates are typically constructed by estimating expected inflation based on past interest rates, inflation, and time trends, and then subtracting that value from the nominal interest rate (Methodology by Frederic S. Mishkin, 1981).