Future Value, Present Value and Interest Rates

1
Chapter 4

• Future Value, Present Value and Interest Rates

2
The Major Questions

• How can we compare payments at different dates?
• What is a bond?
• What is the relationship between interest rates
  and inflation?

3
Valuing Monetary Payments Now and in the Future

• Fundamental to studying financial instruments is
  the ability to value payments made at different
  times.
• Tools Future value and Present Value

4
Future ValueDefinition

• The value on a future date of an investment made
  today.
• If you invest 100 today at 5 percent interest
  per year, in one year you will have 105.

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Future ValueOne Year

• Future Value Present Value Interest
• FV PV PV x i
• 105 100 100 x (0.05)

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Future ValueOne Year

• FV PV PV x i
• PV x (1i)
• Future Value in one year Present Value x (one
  plus interest rate)

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Future ValueTwo Years

• 100100(0.05)100(0.05) 5(0.05) 110.25
• Present Value of the Initial Investment
  Interest on the initial investment in the 1st Yr
  Interest on the initial investment in the 2nd
  Yr Interest on the Interest from the 1st Yr in
  the 2nd Yr Future Value in Two Years

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Future Value General Formula

• Future value of an investment of PV in n years
  at interest rate i
• FVn PV x (1i)n
• (Remember The interest rate is measured is a
  decimal so if 5, i .05)

9
Minuit and Manhattan Island

• On May 24, 1626 Peter Minuit purchased Manhattan
  Island from the Canarse native Americans for
  about 24 worth of trinkets, beads and knives.
• The Purchase took place at what is now Inwood
  Hill Park in upper Manhattan.

10
The power of compounding

• If the tribe had taken cash instead and invested
  it to earn 6 per year, how much would the tribe
  have today, 382 years later?
• FV 24 x (10.06)382
• 111,442,737,812
• More than 100 billion!

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Future Value Caution

• Time (n) interest rate (i) must be in same
  time units
• If i is at annual rate, then n must be in years.
• Future Value of 100 in 18 months at 5 annual
  interest rate is
• FV 100 x (1.05)1.5

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Another Example Payday Loan

• Payday Loan targets high-risk borrowers.
• Provides short-term loan for 7 to 30 days.
• 80 of all payday loans across the country are
  less than 300.
• A borrower writes a postdated check for 300 and
  receives cash from Payday Loan.
• Payday Loan holds on to the check until the
  following payday, before depositing it in its own
  account.

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Another Example Payday Loan

• Application for payday loans is simple (taking
  less than 1 hour)
• A home address
• A valid checking account
• A drivers license and Social Security number
• A couple of pay stubs showing wage, pay dates.
• A minimum earning of 1000/month.
• Payday Loan charges a fee of 15 - 30 for each
  100 advances.

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Another Example Payday Loan

• Payday loans have grown from about 8 billion in
  1999 to between 40-50 billion in 2004.
• In 2004, Payday loans generated an estimated 6
  billion in finance charges.
• In Kansas, the number of payday loan outlets has
  increased ten-fold in the past 10 years.
• Why is Payday Loan so successful?

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Another Example Payday Loan

• A the minimum 15 fee for a 100 loan for 2
  weeks.
• What is the annual interest rate?
• The bi-weekly interest rate 15
• Compounding for 26 weeks, the future value of a
  100 loan is
• 100(10.15)26 3786
• The annual interest rate is
• (3786 -100)/100 3686 !

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Rule of 72

• Invest 100 at 5 annual interest
• How long will it take for you to have 200?
• The Rule of 72
• Divide the annual interest rate into 72
• So 72/514.4 years.
• 100 x1.0514.4 202

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Present Value Definition

• Present Value (PV) is the value today (in the
  present) of a payment that is promised to be made
  in the future.

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Present ValueOne Year

• Solve the Future Value Formula for PV
• FV PV x (1i)
• so
• Present Value Future Value divided by
  one plus interest rate

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Present ValueOne Year Example

• 100 received in one year, i5
• Note FV PVx(1i) 95.24x(1.05) 100

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Present ValueGeneral Formula

• Present Value of payment received n years in
  the future

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Present ValueExample

• Present Value of 100 received in 2½ yrs at
  interest rate of 8.
• Note FV PVx(1i)n 82.50x (1.08)2.5 100

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A letter to New York Times, June 25, 1990

• I was startled and dismayed by an earlier Times
  editorial supporting Government borrowing as the
  appropriate way to deal with the bailout of
  bankrupt savings and loan institutions. Borrowing
  may be politically expedient it is, however,
  wrong, from both an economic and a moral point of
  view. The straightforward, and least damaging way
  to deal with this fiasco, is to pay off the 130
  billion loss with a temporary three- to four-year
  surcharge on income taxes.
• The economics are simple
• Borrowing will turn a 130 billion loss into a
  500 billion drain over 20 to 30 years. It
  will add 10 billion to 15 billion annually in
  interest costs to the Federal budget deficit,
  when interest costs constitute, after defense,
  the largest Federal expenditure

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Present ValueImportant Properties

• Present Value is higher
• The higher future value of the payment. (FV
  bigger)
• The shorter time period until payment. (n
  smaller)
• The lower the interest rate. (i smaller)

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Examples

• There are two Federal government bonds the first
  one has a payment of 1500 in 2014, the second
  one has a payment of 1000 in 2019. Which bond
  has a higher value today?
• There are two Federal government bonds the first
  one has a payment of 1000 in 2014, the second
  one has a payment of 1500 in 2019. Which bond
  has a higher value today?
• If the interest rate rises, which bond will have
  a bigger price decline?

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Present Valueof 100 Payment

• As the interest rate rises from 1 to 5, a
  payment due
• 5 year falls by 16.80
• 10 years falls by 29.14
• Changes in interest rates have much bigger impact
  on the present values of more distant future
  payments.

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Numerical Example 1early retirement

• Can you retire when youre 40?
• Assume
• Live to 85
• Interest rate 4
• Want to have 100,000 per year
• You will need

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Numerical Example 2 pay your debts

• You borrow 10,000 at 6 from your cousin for 20
  years.
• In 20 years you will owe him
• 10,000 X (16)20 32,071.36
• He proposes that you can either pay him
  32,071.36 in 2028, or pay him (32071.36/20)
  1603.568 every year for the next 20 years.
• Which option should you choose?

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Example 2 pay your debts

• The present value of 32,071.36 in 20 years is
  10,000 given 6 interest rate.
• 32.071.36/(16)20 10,000
• The present value of the 2nd payments is
• 1603.568/(16)1603.568/(16)21603.568/(16)
  20 18,392.80

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Example 2 continued

• What should be the correct annual payment?
• The present value of future payments should be
  equal to the value of todays loan.
• Let C be the annual payment for the 20 years,
  given interest rate i 6, we have
• We can find C to be 871.85.
• This is an example of fixed payment loan.

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Bond Basics

• Bond A promise to make a series of payments on
  specific future date
• Bond Price Present Value of payments

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Coupon Bond
1000 Face Value50-yr, 3½ coupon bond issued on
May 1, 1945.
Coupons
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Coupon Bond

• A type of loan
• Monetary payments during the life of the loan
• Loan repaid at maturity
• Coupon Rate ic the annual payments
  borrower pays
• Maturity Date n when the annual payments stop
  and the principal is paid
• Principal F the face value of the bond
• Treasury Bills have 0 coupon rate.

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Coupon BondValuing the Principal

Present value of Bond Principal Face value
divided by one plus the interest rate raised to n
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Coupon BondValuing the Coupon Payments

• Value of Coupon Payments Present value of the
  sequence
• Note that C ic x F
• The coupon rate ic can be different from i,
  which is the market interest rate when the bond
  is traded

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Price of Coupon BondPrincipal Coupons

• Price of Coupon Bond (PCB)
• Present value of Coupon Payments (PCP) Present
  Value of the Principal (PBP)

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Bond PricingImportant Property

• The price of a bond (PCB) and the interest rate
  (i) are inversely related
• i? ? PCB ?
• Interest rate risk is the main risk for bond
  investors.
• Another risk is the risk of default.

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U.S. Bond Index

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Interest Rates and Bond Prices

• Suppose a bond has 4 years till maturity. It pays
  1000 at maturity with annual coupons of 50.
• If interest rates are 5 then its price is
• 50/1.05 50/(1.05)4 1,000/(1.05)4 1000
• If interest rates rise to 6 then the price falls
  to
• 50/1.06 50/(1.06)4 1,000/(1.06)4 965.4
• The capital loss is
• (965.34-1000)/1000 -3.466

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Interest Rates and Bond Prices

• Suppose a bond has 20 years till maturity. It
  pays 1000 at maturity with annual coupons of
  50.
• If interest rate i 5 then its price is
• 50/1.05 50/(1.05)20 1,000/(1.05)20
  1000
• If interest rate i rises to 6 then the price
  falls to
• 50/1.06 50/(1.06)20 1,000/(1.06)20
  885.30
• The capital loss is
• (885.30-1000)/1000 -11.47

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Real and Nominal Interest Rates

• Borrowers care about the resources required to
  repay.
• Lenders care about the purchasing power of the
  payments they received.
• Neither cares solely about the number of dollars,
  they care about what the dollars buy.

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Real and Nominal Interest Rates

• Nominal Interest Rates (i)
• Interest Rates expressed in current dollar
  terms.
• Real Interest Rates (r)
• Nominal Interest Rate adjusted for inflation.

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Real and Nominal Interest Rates

• Nominal interest rate Real Interest Rate
  Expected Inflation
• i r ?e
• (This is called the Fisher Equation)

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Nominal Interest Rate, Inflation Rate and Real
Interest Rate
Nominal Interest Rate Real Interest Rate
Expected Inflation
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Real and Nominal Interest Rates

• Countries with high nominal interest rates have
  high inflation
• ?? ? i?