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Chapter 4. Present and Future Value. Future Value Present Value Applications IRR Coupon bonds Real vs. nominal interest rates. Present & Future Value. time value of money $100 today vs. $100 in 1 year not indifferent! money earns interest over time, and we prefer consuming today.

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chapter 4 present and future value
Chapter 4. Present and Future Value
  • Future Value
  • Present Value
  • Applications
    • IRR
    • Coupon bonds
  • Real vs. nominal interest rates
present future value
Present & Future Value
  • time value of money
  • $100 today vs. $100 in 1 year
    • not indifferent!
    • money earns interest over time,
    • and we prefer consuming today
example future value fv
example: future value (FV)
  • $100 today
  • interest rate 5% annually
  • at end of 1 year:

100 + (100 x .05)

= 100(1.05) = $105

  • at end of 2 years:

100 + (1.05)2 = $110.25

future value
future value
  • of $100 in n years if annual interest rate is i:

= $100(1 + i)n

  • with FV, we compound cash flow today to the future
rule of 72
Rule of 72
  • how long for $100 to double to $200?
  • approx. 72/i
  • at 5%, $100 will double in
    • 72/5 = 14.4
    • $100(1+i)14.4 = $201.9
present value pv
present value (PV)
  • work backwards
  • if get $100 in n years,

what is that worth today?

$100

PV

=

(1+ i)n

example
example
  • receive $100 in 3 years
  • i = 5%
  • what is PV?

$100

PV

=

=

$86.36

(1+ .05)3

slide9
With PV, we discount future cash flows
    • Payment we wait for are worth LESS
about i
About i
  • i = interest rate
  • = discount rate
  • = yield
  • annual basis
slide11
n

PV

PV

i

pv fv and i
PV, FV and i
  • given PV, FV, calculate I

example:

  • CD
  • initial investment $1000
  • end of 5 years $1400
  • what is i?
slide14
is it 40%?
  • is 40%/5 = 8%?
  • No….
  • i solves

i = 6.96%

applications
Applications
  • Internal rate of return (IRR)
  • Coupon Bond
application 1 irr
Application 1: IRR
  • Interest rate
    • Where PV of cash flows = cost
  • Used to evaluate investments
    • Compare IRR to cost of capital
example1
Example
  • Computer course
    • $1800 cost
    • Bonus over the next 5 years of $500/yr.
  • We want to know i where

PV bonus = $1800

solve the following
Solve for i?

Trial & error

Spreadsheet

Online calc.

Answer?

12.05%

Solve the following:
example2
Example
  • Bonus: 700, 600, 500, 400, 300
  • Solve

i = 14.16%

example3
Example
  • Bonus: 300, 400, 500, 600, 700
  • Solve

i = 10.44%

example annuity vs lump sum
Example: annuity vs. lump sum
  • choice:
    • $10,000 today
    • $4,000/yr. for 3 years
  • which one?
  • implied discount rate?
application 2 coupon bond
Application 2: Coupon Bond
  • purchase price, P
  • promised of a series of payments until maturity
    • face value at maturity, F

(principal, par value)

    • coupon payments (6 months)
slide24
size of coupon payment
    • annual coupon rate
    • face value
    • 6 mo. pmt. = (coupon rate x F)/2
what determines the price
what determines the price?
  • size, timing & certainty of promised payments
  • assume certainty

P =

PV of payments

example coupon bond
example: coupon bond
  • 2 year Tnote, F = $10,000
  • coupon rate 6%
  • price of $9750
  • what are interest payments?

(.06)($10,000)(.5) = $300

    • every 6 mos.
what are the payments
what are the payments?
  • 6 mos. $300
  • 1 year $300
  • 1.5 yrs. $300 …..
  • 2 yrs. $300 + $10,000
  • a total of 4 semi-annual pmts.
slide29
YTM solves the equation
  • i/2 is 6-month discount rate
  • i is yield to maturity
slide30
how to solve for i?
    • trial-and-error
    • bond table*
    • financial calculator
    • spreadsheet
slide31
price between $9816 & $9726
  • YTM is between 7% and 7.5%

(7.37%)

p f and ytm
P, F and YTM
  • P = F then YTM = coupon rate
  • P < F then YTM > coupon rate
    • bond sells at a discount
  • P > F then YTM < coupon rate
    • bond sells at a premium
slide33
P and YTM move in opposite directions
  • interest rates and value of debt securities move in opposite directions
    • if rates rise, bond prices fall
    • if rates fall, bond prices rise
slide35
YTM rises from 6 to 8%
    • bond prices fall
    • but 10-year bond price falls the most
  • Prices are more volatile for longer maturities
    • long-term bonds have greater interest rate risk
slide36
Why?
    • long-term bonds “lock in” a coupon rate for a longer time
    • if interest rates rise

-- stuck with a below-market coupon rate

    • if interest rates fall

-- receiving an above-market coupon rate

real vs nominal interest rates
Real vs. Nominal Interest Rates
  • thusfar we have calculated nominal interest rates
    • ignores effects of rising inflation
    • inflation affects purchasing power of future payments
example4
example
  • $100,000 mortgage
  • 6% fixed, 30 years
  • $600 monthly pmt.
  • at 2% annual inflation, by 2037
    • $600 would buy about half as much as it does today $600/(1.02)30 = $331
real interest rate i r
real interest rate, ir

nominal interest rate = i

expected inflation rate = πe

approximately:

i = ir + πe

  • The Fisher equation

or ir = i – πe

[exactly: (1+i) = (1+ir)(1+ πe )]

slide41
real interest rates measure true cost of borrowing
  • why?
    • as inflation rises, real value of loan payments falls,
    • so real cost of borrowing falls
inflation and i
inflation and i
  • if inflation is high…
  • lenders demand higher nominal rate, especially for long term loans
  • long-term i depends A LOT on inflation expectations
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