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### Interest Rates

Chapter 4

Valuing Debt

- In 1945, U.S. Treasury bills offered a return of 0.4%. At their 1981 peak, they offered a return of over 17%.
- Why does the same security offer radically different yields at different times?

Valuing Debt

- In January 2000, the U.S. Treasury could borrow for 1 year at an interest rate of 6.2%, but it had to pay a rate of about 7% for 20-year loans.
- Why do bonds maturing at different dates offer different rates of interest?

Valuing Debt

- In January 2000, the U.S. government could issue long-term bonds at a rate of about 7%.
- You could not have borrowed at that rate. Why not?

5 Major Sources of Rate Differences in Bonds

- Term to maturity
- Default risk
- To default on a bond is to fail to pay the interest when interest is due or to fail to pay the principal at maturity
- Bond ratings

- Tax treatment
- The value of the tax factors to the investor depends on the investor’s marginal income tax rate
- After tax yield=Before tax yield (1-T)

- Marketability
- time required to effect the sale
- spread between the current market price and realized price at the time of the sale.

- Special features
- call option.
- put option.
- convertible option.

Characteristics of Bonds

- Details contained in the indenture.
- Administered through a trustee.
- Secured versus unsecured.
- Mortgage/debenture

- Senior or junior or subordinated.
- Call features.
- Bond rating.

Types of Bonds

- Coupon/Zero-coupon bonds
- Municipal bonds
- Revenue/General Obligation

- Junk bonds
- Consols
- Eurobonds/Foreign bonds

Sources of Bond Information

- http://www.moodys.com
- http://www.bondsonline.com/
- http://www.wsj.com
- http://bonds.yahoo.com

Bond Features

- When a corporation (or government) wants to borrow money, it often sells a bond.
- An investor gives the corporation money for the bond.
- The corporation promises to give the investor:
- Regular coupon payments every period until the bond matures.
- The face value of the bond when it matures.

The Bond Pricing Formula

C1 C2 C3 Cn + Fn

P = + + + … +

(1 + r)1 (1 + r)2 (1 + r)3 (1 + r)T

- The price of the bond today is the present value of all future cash flows (coupon payments and principal).

Bond Features

- Consider a bond with three years to maturity, a coupon rate of 8%, and a $1000 face value. If the current market rate is 10%, what is the price of the bond.
$80 $80 $1080

P = + +

(1.10)1 (1.10)2 (1.10)3

P=$80(0.9091)+$80(0.8264)+$1080(0.7513)

P=$950.24

Bond Rates and Yields

- The coupon rate is the annual dollar coupon expressed as a percentage of the face value.
Coupon rate = $80/$1000 = 8.0%

- The current yield is the annual coupon divided by the price:
Current yield = $80/$950.24 = 8.42%

- The yield to maturity is the rate that makes the price of the bond just equal to the present value of its future cash flows.
YTM = 10%

Example

- Bond A has 4 years remaining to maturity. Interest is paid annually; the bond has a $1,000 par value; and the coupon interest rate is 9%.
- What is the current yield and yield to maturity at a current market price of $829?
- What is the current yield and yield to maturity at a current market price of $1,104?

Par, Premium and Discount Bonds

- If a bond’s coupon rate is equal to the market rate of interest (the bond’s yield), the bond will always sell at par.
- Bonds selling at below par are called discount bonds.
- Bonds selling above par are called premium bonds.

Pure Discount Bond

- Zero-coupon bonds pay no coupon payment but promise a single payment at maturity.
- Value of a pure discount bond:
P = F / (1 + r)T

Perpetual Bonds

- A consol pays coupons forever. It never matures.
- P=C/r

Bond Pricing Theorem I

- Bond prices and market interest rates move in opposite directions.

Bond Pricing Theorem II

- When coupon rate = YTM, price = par value.
- When coupon rate > YTM, price > par value (premium bond)
- When coupon rate < YTM, price < par value (discount bond)

Bond Pricing Theorem III

- A bond with longer maturity has higher relative (%) price change than one with shorter maturity when interest rate (YTM) changes. All other features are identical.

Bond Pricing Theorem IV

- A lower coupon bond has a higher relative (%) price change than a higher coupon bond when interest rate (YTM) changes. All other features are identical.

High Coupon Bond

Discount Rate

Low Coupon Bond

Coupon Rate and Bond Price VolatilityConsider two otherwise identical bonds.

The low-coupon bond will have much more volatility with respect to changes in the discount rate

Interest Rate Risk

- Price Risk.
- Reinvestment Risk.
- Price Risk versus Reinvestment Risk.
- Duration.

Interest Rate Risk Example

- Suppose you buy three securities:
- A one-year bill with a face value of $10,000.
- A 5-year strip with a face value of $10,000.
- A 30-year strip with a face value of $10,000.

- The market interest rate is 6%, so their prices are:
- T-bill: P=$10,000/1.06=$9,434
- 5-year strip: P=$10,000/(1.06)5=$7,473
- 30-year strip: P=$10,000/(1.l06)30=$1,741

Example, continued

- After a year, you need your money and you must liquidate your portfolio. Suppose market rates have risen from 6% to 8%.
- T-bill: P=$10,000
- 5-year strip: P=$10,000/(1.08)4=$7,350
- 30-year strip: P=$10,000/(1.08)29=$1,073

Reinvestment-Rate Risk

- Suppose that you do not need your money after one year but want to leave it invested until you retire in 30 years.
- On the 30-year strip, the holding-period yield equals the market yield at the time you bought the bond.
- How much you make on the other two investments will depend on how you reinvest the money when the bond matures.

Reinvestment-Rate Risk

- Reinvestment-rate risk
- the risk associated with reinvestment at uncertain interest rates.

- Considerations:
- What is your time horizon?
- Do you want to play it safe?
- Do you think interest rates will rise or fall?

Lessons

- If you hold a bill or strip to maturity, the holding-period yield will equal the market yield at the time that you bought it.
- If you sell a bill or strip before maturity, the holding-period yield depends on the market yield at the time of the sale.
- The higher the market yield at the time of the sale, the lower the market price.
- The greater the bill’s or the strip’s remaining time to maturity, the greater the sensitivity of its market price to market yield.

Return Versus Yield to Maturity

- Rate of return measures the cash flows received during a period relative to the amount invested at the beginning
- For a bond held for one year, the return is computed as follows:

Nominal versus Real Rates

- The real rate of interest is the fundamental long-run interest rate in the economy. It is called the “real” rate of interest because it is determined by the real output of the economy.
- It is estimated to be on average about 3 percent. It varies between 2 and 4 percent.

- The nominal rate of interest is the observed rate of interest.
- Nominal rate ~ Real rate + Inflation

Calculating Interest Rates

- Nominal Versus Real Interest Rates
- Nominal Interest Rates—Money amount of interest received
- Real Interest Rates—Purchasing power of interest received
- Real interest rate is the nominal interest adjusted for inflation

Real interest rate = Nominal rate – Inflation rate

- Where:
- “ex-ante” is based on the expected rate of inflation
- “ex-post” is based on the actual or realized rate of inflation

Supply and Demand Determine the Interest Rate

- Interest rate is price of credit or borrowing money
- Market for Credit or Loanable Funds
- Supply of Funds—Upward sloping, lenders are willing to extend more credit at higher interest rates
- Demand for Funds—Downward sloping, borrowers are willing to borrow less at higher interest rates
- Equilibrium—Intersection of supply and demand, no tendency to change

Why Does the Interest Rate Fluctuate

- U.S. Treasury bond yields change day to day
- Movement along a single curve—Changes in the interest rate results in a movement along a single demand or supply curve
- Shifts of a Curve—Change in determinants of supply or demand (other than interest rate) causes the respective curve to shift
- Changes in Equilibrium—Shift of either the supply or demand curve will reflect a change in the equilibrium interest rate

Borrowing (Demand)

- Business firms
- finance inventory or buy capital equipment

- Households
- buy cars, consumer goods, or homes

- State and local government
- provide infrastructure or public services

- Federal government
- finance Federal Budget Deficit

- INCREASES IN BORROWING
- SHIFT DEMAND TO RIGHT AND RAISE INTEREST RATES

Lending or Credit (Supply)

- Financial institutions or individuals lend to market
- Government authorities may restrict lending by banks
- Ability of individuals to lend depends on their savings—less savings results in lower amount of lending
- DECREASES IN LENDING
- SHIFT SUPPLY TO LEFT AND RAISE INTEREST RATE

The Importance of Expectations

- Effect of a change in expectations of increasing inflation
- Demand—Borrowers increase demand since they will be repaying in depreciated dollars and desire to purchase before the prices increase
- Supply—Lenders decrease supply since they will be repaid with money of diminished purchasing power

- SHIFTS OF THE DEMAND AND SUPPLY CURVE WILL CAUSE THE INTEREST RATE TO INCREASE

The Importance of Expectations

- Self-fulfilling Prophesies
- If individuals and institutions expect inflation and interest rates to increase, they will alter behavior that causes the higher rates that were anticipated

Cyclical and Long-term Trends in Interest Rates

- Level of interest rates tends to rise during cyclical expansion and fall during recessions.
- During economic expansion:
- Firms and households increase borrowing—demand curve right
- FED usually tightens credit during expansion—supply curve left

Cyclical and Long-term Trends in Interest Rates

- Level of interest rates on upward long-term trend between 1950 and 1981
- Large federal budget deficit forced US Treasury to increase borrowing—pushing up interest rates
- Expectations of increasing inflation

- Since 1981 rates have trended downward
- Federal deficits continued to increase in 1980’s
- Expectations of lower inflation has been major reason for fall of interest rates.

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