Chapter 4 present and future value
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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

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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


Chapter 4 present and future value

  • 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


Chapter 4 present and future value

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?


Chapter 4 present and future value

  • 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?


Chapter 4 present and future value

i = 9.7%


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)


Chapter 4 present and future value

  • 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


Chapter 4 present and future value

  • i where P = PV(pmts.) is known as the yield to maturity (YTM)


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.


Chapter 4 present and future value

  • YTM solves the equation

  • i/2 is 6-month discount rate

  • i is yield to maturity


Chapter 4 present and future value

  • how to solve for i?

    • trial-and-error

    • bond table*

    • financial calculator

    • spreadsheet


Chapter 4 present and future value

  • 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


Chapter 4 present and future value

  • 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


Maturity bond price volatility

Maturity & bond price volatility


Chapter 4 present and future value

  • 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


Chapter 4 present and future value

  • 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


Chapter 4 present and future value

  • so interest charged by a lender reflects the loss due to inflation over the life of the loan


Real interest rate i r

real interest rate, ir

nominal interest rate = i

expected inflation rate = πe

approximately:

i = ir + πe

  • The Fisher equation

    orir = i – πe

    [exactly: (1+i) = (1+ir)(1+ πe )]


Chapter 4 present and future value

  • 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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