Interest Rates and Bond Prices

Darren Webb Development Team Montego Data Limited
September 2003

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Contents

Introduction
Conclusion

Introduction

  • The time value of money
    A. The interest rate connects the value of dollars today with the value of dollars in the future
    B. Principle – The original amount of funds lent

  • Determining Future Value and Present Value of money
    A. Future Value – The future worth of a sum of money today
    B. Present Value – The current worth of a sum of money on a future date
    C. Compounding – Determining the future value of a current amount of money
    1. Financially speaking, compounding is the increase in the future value of funds resulting from interest on interest
    2. The compound rate will always be greater than the annual interest rate
    3. Compound factor = (1+i)n

      i. i = annual interest rate
      ii. n = number of years
    4. FV = PV*(1+i)n
    5. FV = 500 * (1+.06) 2
    6. FV = 500 * 1.1236
    7. FV = $561.80
    8. The future value of current dollar loan is positively related to: i. The interest rate
      ii. How far in the future the payment will be made
    D. Discounting – Determining the present value of a future amount of money
    1. Discount factor = 1/(1+i)n

    i. i = annual interest rate
    ii. n = number of years
    2. PV = FV*1/(1+i)n
    3. PV = 500 * 1/(1+.06) 2

    4. PV = 500 * .8900
    5. PV = $445.00
    6. The present value of a future stream of future values is:
    PV = FV1/(1+i) + FV2/(1+i)2 + FV3/(1+i)3 +…+ FVn/(1+i)n

    7. The present value of a future dollar payment is inversely related to:
    i. The interest rate
    ii. How far in the future the payment will be received

  • Determining the price of bonds
    A. Par value – The face value printed on a bond; the amount the bond originally sold for
    B. Annuity – A series of equal dollar payments for a specified number of years
    C. Coupon payment – Periodic dollar payments made to bond holders
    1. Coupon payment = principal * coupon rate
    D. The price of a bond with annual annuities is found by:
    P = C1/(1+i) + C2/(1+i)2 +C3/(1+i)3 +…+ Cn/(1+i)n +Fn/(1+i)n

    1. P = price of the bond
    2. C = coupon payment on the bond in year 1, 2, 3, etc.
    3. F = the face value of the bond
    4. i = interest rate
    5. n = the number of years to maturity
    E. The price of a bond with semiannual annuities is found by:
    P = C1/(1+i) + C2/(1+i)2 + C3/(1+i)3 +…+ Cn/(1+i)n +Fn/(1+i)n
    1. P = price of the bond
    2. C = coupon payment on the bond for semiannual payment 1, 2, 3, etc.
    3. F = the face value of the bond
    4. i = (interest rate)/2
    5. n = the number of semiannual periods to maturity

  • Effects of interest rate changes on bond prices
    A. Yield to maturity – The return on a bond held to maturity, which includes both the interest return and capital gains/losses
    B. Capital gains/losses – The difference between the par value and price paid for a bond
    C. Discount from par – When a bond sells below its face value
    1. Occurs because interest rates have increased since the bond was originally issued
    D. Premium above par – When a bond sells above its face value
    1. Occurs because interest rates have decreased since the bond was originally issued
    E. When the market interest rate rises, the price of existing bonds decreases
    F. When the market interest rate falls, the price of existing bonds increases
    G. What is the price of a $1,000 two-year bond with semiannual coupon payments of $40 if the interest rate is 8%
    1. P = 40/(1+.04) + 40/(1+.04)2 + 40/(1+.04)3 + 40/(1+.04)4 + 1000/(1+.04)4

    2. P = 40(.9615) + 40(.9246) + 40(.8890) + 40(.8548) + 1000(.8548)
    3. P = 38.46 + 36.99 + 35.56 + 34.19 + 854.80
    4. The price of the bond is $1000
    H. Suppose interest rates rise to 12%, what is the bond price
    1. P = 40/(1+.06) + 40/(1+.06)2 + 40/(1+.06)3 + 40/(1+.06)4 + 1000/(1+.06)4

    2. P = 40(.9434) + 40(.8900) + 40(.8396) + 40(.7921) + 1000(.7921)
    3. P = 37.74 + 35.60 + 33.58 + 31.68 + 792.10
    4. The price of the bond is $930.70
    5. Capital loss = $69.30
    I. Suppose interest rates fall to 4%, what is the bond price
    1. P = 40/(1+.02) + 40/(1+.02)2 + 40/(1+.02)3 + 40/(1+.02)4 + 1000/(1+.02)4

    2. P = 40(.9804) + 40(.9612) + 40(.9423) + 40(.9239) + 1000(.9239)
    3. P = 39.22 + 38.45 + 37.69 + 36.96 + 923.90
    4. The price of the bond is $1076.22
    5. Capital gain = $76.22

  • Fluctuations in interest rates and managing a bond portfolio
    A. If you think interest rates will rise, sell bonds to avoid capital losses on currently held bonds
    B. If you think interest rates will fall, buy bonds to capture capital gains
    C. Remember, the price of a bond is the discounted value of the future stream of income over the life of the bond

  • The determinants of interest rates
    A. Remember interest rates are determined by the supply of and demand for loanable funds
    1. Demand for loanable funds – The demand for borrowed funds by households, businesses, government, or foreign DSU’s
    2. Supply of loanable funds – The supply of borrowed funds originating from households, businesses, government, and foreign SSU’s or the Fed through its provision of reserves
    B. Changes in the demand for loanable funds
    1. Increases in GDP increase the demand for loanable funds
    2. Increase in demand raises the interest rate and increases the quantity of loanable funds
    i. Bond prices will fall
    C. Changes in the supply of funds
    1. Increase in supply decreases the interest rates and increases the quantity of loanable funds
    i. Bond prices will rise
    D. We can summarize the relationship as:
    i = f (Y+, M-)


  • Inflation and interest rates
    A. Nominal interest rate (money interest rate) – The market interest rate
    1. It includes both the real return and the expected rate of inflation
    B. Real interest rate – Return on an asset corrected for changes in the purchasing power of money
    1. It’s the nominal rate minus expected rate of inflation
    C. Money illusion – When spending units react to nominal changes caused by changes in prices, although real variables such as interest rates have not changed
    D. i = r + pe

    E. Effect of an increase in inflation expectations:
    1. Suppose nominal rate = 6% and inflation = 4%
    2. If inflation expectations rise to 8%, real falls to –2%
    3. Supply falls, demand increases, until real is restored at 2%, therefore raising the nominal to 10%
    4. Inflation has a positive effect on interest rates
    i = f (Y+, M-, pe+)


  • The cyclical movement of interest rates
    A. Interest rates tend to move pro-cyclically
    1. Move with the business cycle, rising during expansions and falling during recessions

    • Conclusion

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