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**Understanding Interest Rates**

Chapter 4

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**Present Value: Discounting the Future**

Present Value (PV) - the value today of a payment promised in the future A dollar paid to you one year from now is less valuable than a dollar paid to you today.

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**Discounting the Future (PV) First need to Understand Future Value (FV)**

Let i = 0.10 (10 percent interest rate) In one year: $100 +(.10 x $100) = $100 x (1+.10) = $110 In two years: $110+(.10 x $110) = $110 x (1+.10) = $121 OR $100 x (1+.10)2 = $121

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**Future value In three years: $121+(.10 x $121)= $121x(1+.10) = $133**

OR $100 x (1+.10)3 = $133 In n years: FV = $100 x (1+ .10)n

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Future Value Future value in n years of an investment of PV today at interest rate i (i is measured as a decimal, 10% = .10) FVn = PV x (1+i)n Calculate one plus the interest rate (measured as a decimal), raised to the nth power, multiply by the amount invested (present value PV).

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Future Value Note: When computing Future Value, both n and i must be measured in same time units—if i is annual, then n must be in years. The future value of $100 in 18 months at 5% annual interest rate is: FV = 100 *(1+.05)1.5

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Future Value The future value of $100 in one month at a 5% annual interest rate is: FV = $100 *(1+.05)1/12 = $ (1+.05)1/12 converts the annual interest rates to a monthly rate. (1+.05)1/12 = , which converted to percentage is % or 0.41%(rounded) Note: 0.05/12 =

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**Basis Point Faction of a percentage point is called basis point.**

A basis point is one-one hundredth of a percentage point One basis point (bp) = 0.01 percent. On the previous slide: 0.41% is 41 basis points.

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**Discounting the Future PV Reverses the FV Calculation**

FVn = PV x (1+i)n PV = FVn /(1+i)n In General:

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Present Value

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**Present Value Example 1:**

Present Value of $100 received in 5 years discounted at an interest rate of 8%. PV = $100 / (1.08)5 = $68.05 Example2: PV of $20,000 received 20 years from now discounted at 8% is: PV = $20,000 / (1+0.08)20 = $20,000/ = $4,291 Discounted at 9%: PV = $20,000 / (1+0.09)20 = $20,000/ = $3,568

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**Present Value Example3:**

PV of $20,000 received 19 years from now discounted at 8% is: PV = $20,000 / (1+0.08)19 = $20,000/ = $4,634 In general, present value is higher: 1. The higher the future value of the payment (CF). 2. The shorter the time period until payment (n). 3. The lower the interest rate. (i)

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**Mishkin Discusses Four Types of Credit Market Instruments**

Simple Loan Fixed Payment Loan (I will just mention this) Coupon Bond - Special case: consol bond Discount Bond

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**We will focus on Bonds and look at three types of bonds**

Coupon Bonds: which make periodic interest payments and repay the principal at maturity. U.S. Treasury Bonds and most corporate bonds are coupon bonds. Discount or Zero-coupon bonds: which promise a single future payment, such as a U.S. Treasury Bill. Consols: which make periodic interest payments forever, never repaying the principal that was borrowed. (There aren’t many examples of these.)

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Yield to Maturity -YTM The interest rate that equates the present value (PV) of cash flow payments (CF) received from a debt instrument with its value today Given values for PV, CF and n, solve for i.

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**Yield to Maturity - One Year Simple Loan**

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**Fixed Payment Loan - YTM**

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**Bond price is a present value calculation. YTM is the interest rate. **

Bonds: Our Objectives Bond price is a present value calculation. YTM is the interest rate. Supply and Demand determine the price of bonds. - We will also discuss loanable funds and money supply/money demand. Explain why bonds are risky.

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Coupon Bond Price A simple contract-

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**Present Value of Coupon Payments**

Price of a Coupon Bond Present Value of Coupon Payments Present Value of Principal Payment Present Value of Coupon Bond (PB) = Present Value of yearly coupon payments (C) + Present Value of the Face Value (FV), where: i = interest rate and n = time to maturity

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**Example: Price of a n-year Coupon Bond**

Coupon Payment (C) =$100, Face value (FV) = $1,000, and n = time to maturity Given values for i and n, we can determine the bond price PB Definition: Coupon Payment / Face Value = Coupon Rate

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**Price of a 10-year Coupon Bond**

If n = 10, i = 0.10, C = $100 and FV = $1000. What’s the Coupon Rate? Excel Example

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**Price of a 10-year Coupon Bond**

If n = 10, i = 0.12, C = $100 and FV = $1000. What’s the Coupon Rate?

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**YTM - Coupon Bond Suppose n = 10, PB = $950, C = $100 and FV = $1000.**

What’s the Coupon Rate? What’s the YTM? YTM = or 10.85%

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C = the coupon payment FV = Face Value P = Price n = years to maturity

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**Using the Approximation Formula**

Previous example: n = 10, PB = $950, C = $100 and FV = $1000.

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**For our 10-year, $1000 FV bond with a $100 coupon payment selling at $950:**

Coupon rate = 10% YTM = 10.85% App. YTM = 10.77% Current Yield = 10.52%

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Current Yield Two Characteristics of Current Yield Is a better approximation of yield to maturity, nearer price is to par (face value) and longer is maturity of bond 2. Change in current yield always signals change in same direction as yield to maturity

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**When the 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 are negatively related The yield to maturity is greater than the coupon rate when the bond price is below its face value

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**Consol – Special Case Coupon Bond**

Infinite maturity No face value. Fixed coupon payment of C forever. Pconsol = C/(1+i) + C/(1+i)2 + C/(1+i)3 + … + C/(1+i)t As t goes to infinity this collapses to: PConsol = C / i => i = C / P $2,000 = $100/.05

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**Discount or Zero Coupon Bond**

Definition: A discount bond is sold at some price P, and pays a larger amount (FV) after t years. There is no periodic interest payment. Let P = price of the bond, i= interest rate, n = years to maturity, and FV = Face Value (the value at maturity):

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**Zero Coupon Bonds - Price**

Examples: Assume i=4% Price of a One-Year Treasury Bill with FV = $1,000: Price of a Six-Month Treasury Bill with FV = $1,000: Price of a 20-Year zero coupon bond at 8% and FV = $20,000:

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YTM - Zero Coupon Bonds

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**Zero Coupon Bonds - YTM i = 1.0657 -1=> i =6.57%**

For a discount bond with FV = $15,000 and P = $4,200, and n = 20, the interest rate (or yield to maturity) would be: i = => i =6.57% Note: This is the formula for compound annual rate of growth

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**Zero Coupon Bonds - YTM i = 1.06368 – 1 = .06368 or 6.368%**

For a discount bond with FV = $10,000 and P = $6,491, and n = 7, the interest rate (or yield to maturity) would be: i = – 1 = or 6.368%

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**From a Coupon Bond to Zero Coupon Bonds (called Strips)**

Create n+1 discount bonds

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**Yield on a Discount Basis**

SKIP

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**Distinction Between Interest Rates and Return**

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Holding Period Return the return from holding a bond and selling it before maturity. the holding period return can differ from the yield to maturity.

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**Holding Period Return Example:**

You purchase a 10-year coupon bond, with a 10% coupon rate, at face value ($1000) and sell one year later. Current Yield Capital Gain

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**One year holding period return =**

If the interest rate one year later is the same at 10%: One year holding period return = or 10.0%

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Holding Period Return If the interest rate one year later is lower, say at 8%: One year holding period return = or 22.5% Where did the $1125 come from?

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**Holding Period Returns**

If the interest rate in one year is higher at 12%: One year holding period return = or -.70% Where did the $893 come from. You need to know how to calculate the $1125 and $893

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**Key Conclusions From Table 2**

The return equals the yield to maturity (YTM) only if the holding period equals the time to maturity A rise in interest rates is associated with a fall in bond prices, resulting in a capital loss if the holding period is less than the time to maturity The more distant the maturity, the greater the percentage price change associated with an interest-rate change

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**Key Conclusions From Table 2**

The more distant a bond’s maturity, the lower the rate of return that occurs as a result of an increase in the interest rate Even if a bond has a substantial initial interest rate, its return can be negative if interest rates rise

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Interest-Rate Risk Change in bond price due to change in interest rate Prices and returns for long-term bonds are more volatile than those for shorter-term bonds There is no interest-rate risk for a bond whose time to maturity matches the holding period

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**Reinvestment (interest rate) Risk**

If investor’s holding period exceeds the term to maturity proceeds from sale of bond are reinvested at new interest rate the investor is exposed to reinvestment risk The investor benefits from rising interest rates, and suffers from falling interest rates

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**Real and Nominal Interest Rates**

Nominal interest rate (i) makes no allowance for inflation Real interest rate (r) is adjusted for changes in price level so it more accurately reflects the cost of borrowing Ex ante real interest rate is adjusted for expected changes in the price level (πe) Ex post real interest rate is adjusted for actual changes in the price level (π)

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**Real and Nominal Interest Rates**

Fisher Equation: i = r + πe From this we get - rex ante = i - πe rex post = i - π

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**Inflation and Nominal Interest Rates**

Mankiw

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**Inflation and Nominal Interest Rates**

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**Real and Nominal Interest Rates**

Real Interest Rate: Interest rate that is adjusted for expected changes in the price level r = i -πe if i = 5% and πe = 3% then: r = 5% - 3% = 2% if i = 8% and πe = 10% then r = 8% - 10% = -2%

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**A measure in inflationary expectations**

i = r + πe πe = i - r

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