Present Value and Future Value

1
where to begin, Im playing for time
2
Present Value and Future Value

• Furnace Advertisement
• Furnace costs 2,000
• Energy Savings 200/year
• Claim The furnace will pay for itself in 10
  years

Is this true?
3
Present Value and Future Value

• Powerball Lottery
• Claim Current Prize is worth 20 million
• Payout is 800,000 per year for 25 years
• Claim The state will make money on the lottery,
  even if it only sells 20 million in tickets for
  the 20 million prize.

Is this true?
4
Present Value and Future Value

• 1626, Peter Minuit bought Manhattan from the
  Man-a-hat-a Indians for goods valued at 24
• The 12800 acres are now valued at 627
  million/acre or 8 trillion unimproved
• This was a heck a deal for the Dutch

Is this true?
5
Present Value and Future Value

• Problem
• 1 one year from now is not equal to 1 today
• Need a way to create equivalencies between
  dollars received at different times
• Mechanism (r rate of interest)
• Opportunity cost of spending 1 today
• (1 r)1 (1 r)
• at r 0.1 opportunity cost is 1.10 next period

6
Future Value

• Future Value the value in dollars at a future
  point in time of a sum of money today.
• Compounding successive application of interest
  payments to generate future values.
• Period 0 Period 1 Period 2
• 1 (1r)1 (1r)(1r)1
• (1r)21

7
Future Value

• Generally, 1 today is worth (1r)T T years from
  now
• At r 0.1
• Period 0 1
• Period 1 (1 .1) 1.10
• Period 2 (1 .1)2 1.21
• Period 3 (1 .1)3 1.33
• Period 40 (1 .1)40 45.26

8
Future Value

• Man-a-hat-a Indians
• How much is 24 in 1626 worth today if they
• just collected interest?
• 1 in 1626 is worth (1r)T now,
• T 2006-1626 380
• At r 0.1 24(1r)380 1,286,564 trillion
• At r 0.08 24(1r)380 120.6 trillion
• At r 0.07 24(1r)380 35.2 trillion
• At r 0.06 24(1r)380 99.2 billion
• At r 0.05 24(1r)380 2.7 billion

breakeven
9
Application

• Suppose you want to be a millionaire when you
  retire. How much should you start putting away
• FV 1 million
• A annual amount invested
• FV A((rT1 1)/r)
• See handout for derivation

10
Application

• Suppose you want to be a millionaire when you
  retire. How much should you start putting away
• FV A((rT1 1)/r)
• Current age 18 Millionaire by 40? 50? 60?

11
Present Value

• Present Value the value in current period
  dollars of a sum of money to be received at some
  point in the future.
• Discounting successive application of interest
  rates to generate present values.
• Period 0 Period 1
• 1 (1r)1

12
Present Value

• Present Value the value in current period
  dollars of a sum of money to be received at some
  point in the future.
• Discounting successive application of interest
  rates to generate present values.
• Period 0 Period 1
• 1/(1r) (1r)1
• 1/(1r) 1

13
Present Value

• Generally
• Period 0 Period 1 Period 2
• 1 (1r)1 (1r)21
• 1/(1r)2 1/(1r) 1

14
Present Value

• Generally
• Period 0 Period T
• 1 (1r)T1
• 1/(1r)T 1

15
Present Value

• Generally, 1 T periods in the future will be
  worth 1/(1r)T now
• At r 0.1 compute present value of 1 in Period
  X
• Period Present Value
• 1 1/(1 .1) 0.91
• 2 1/(1 .1)2 0.83
• 3 1/(1 .1)3 0.75
• 40 1/(1 .1)40 0.02

16
Furnace

• Generally, 200 T periods in the future will be
  worth 200/(1r)T now
• At r 0.1
• Year Present Value
• 1 200/(1 .1) 181.82
• 2 200/(1 .1)2 165.29
• 3 200/(1 .1)3 150.26
• 10 200/(1 .1)10 77.11

ADD UP THESE RETURNS
17
Furnace

• Generally, 200 T periods in the future will be
  worth 200/(1r)T now
• At r 0.1
• Year Present Value
• 1 200/(1 .1) 181.82
• 2 200/(1 .1)2 165.29
• 3 200/(1 .1)3 150.26
• 10 200/(1 .1)10 77.11
• Present Value 1,429
• At r 0.12, this furnace investment would never
  break even

18
POWERBALL

• At r 0.1 1 24 years from now is worth 10
• Present Value of 800,000 per year for 25 years
  starting this year
• PV (A/r)(1 r (1/(1r)24)) 8 million
• Derivation in handout

If the payment starts in the next period, the
formula is PV (A/r)(1 (1/(1r)25))
19
POWERBALL

• At r 0.1 1 24 years from now is worth 10
• Present Value of 800,000 per year for 25 years
  starting now
• PV (A/r)(1 r (1/(1r)24)) 8 million
• State net 20 million - 8 million
• 12 million
• Derivation in handout

20
Conclusions Regarding Present and Future Value

• General Formula
• PV Present Value
• FV Future Value
• FVT (1r)T PV0 (Compounding)
• PV0 FVT / (1r)T (Discounting)

21
Bonds

• Principal The value of the bond at maturity
• The face value on the bond
• Future value
• Individuals buy bonds at the present value of the
  principal
• Interest rate is implied by the relationship
  between current price and principal

22
Bonds

• Suppose bond matures in one period
• PBOND PV FV/(1r)
• Interest rate yield implied by
• (1r) FV/PV
• If bond matures T periods from now
• PBOND PV FV/(1r)T
• Interest rate yield implied by
• (1r)T FV/PV

23
Bonds

• Suppose FV 10,000
• PBOND 9500
• Maturity in one period
• Interest rate yield is
• (1r) FV/PV (10,000/9,500) 1.053
• r 0.053

24
How does bond market respond to rising inflation
expectations?
PV
Supply of FV 10,000 bonds
Initial equilibrium sets price implied r 0.053
9,500
D low inflation
Bonds issued
25
How does bond market respond to rising inflation
expectations?
PV
Supply of FV 10,000 bonds
9,500
D low inflation
Bonds issued
With higher inflation, 10,000 one year from now
will be worth less, so demand shifts left
26
How does bond market respond to rising inflation
expectations?
PV
Supply of FV 10,000 bonds
New lower equilibrium price implied r 0.081
9,500
D low inflation
9,250
D high inflation
Bonds issued
(1 r) (FV/PV) (10000/9250) 1.081
27
Can you outguess the market?

• Suppose you expect that price of bond will fall
  tomorrow because the Federal Reserve Board of
  Governors is going to raise the reserve rate
  (the interest rate charged to banks by the Fed).
  
• What will you do?

28
Fundamental value of stocks

• Stock share of ownership in the firm
• Stockholder has a share of the future earnings of
  the firm
• Stock price should be the present value of the
  stream of future earnings per share

29
Fundamental value of stocks

• Stock price should be the present value of the
  stream of future earnings per share (E)
• PV Pstock E/r
• Price Earnings (PE) ratio Pstock/E (1/r)
• Very high PE ratios imply having to pay a lot per
  of expected earnings

30
Price-Earnings Ratios as a Predictor of
Twenty-Year Returns A twenty-year modification
of the plot by wRobert Shiller (Figure 10.1 from
Shiller, Robert (2005). Irrational Exuberance (2d
ed.). Princeton University Press
31
www2.standardandpoors.com/spf/xls/index/SP500EPSES
T.XLS
32
How do we add savings and borrowing into the
consumer demand model?Let Yt income in year
tr interest ratePt price level in year tCt
consumption in year t
33
TWO PERIOD BUDGET CONSTRAINT Y0 Y1/(1r) P0C0
P1/(1r) C1 C0 Y0 Y1/(1r)/ P0 -
P1/P0(1r)C1
C0
Y0 Y1/(1r)/P0
Borrow to consume more in period 1
Slope - P1/P0(1r)
C0 Y0 /P0
Save to raise period 2 consumption
(1r)Y0 Y1/P1
C1 Y1 /P1
C1
34
TWO PERIOD BUDGET CONSTRAINT Y0 Y1/(1r) P0C0
P1/(1r) C1 C0 Y0 Y1/(1r)/ P0 -
P1/P0(1r)C1
C0
Y0 Y1/(1r)/P0
Slope - P1/P0(1r)
c0
U0
(1r)Y0 Y1/P1
c0
C1
35
As r increases, slope gets flatter Tendency to
consume less today, more tomorrow
C0
Slope - P1/P0(1r)
c0
U0
c1
c0
C1
c1
36
Debt payment as of disposable income
2005 1 bankruptcy per 50 households
Law change makes bankruptcy more difficult
American Bankruptcy Institute http//www.abiworld
.org/statcharts/CDebt.pdf
37
Total Household debt relative to GDP, 1916-2008
38
TWO PERIOD BUDGET CONSTRAINT WHAT IF CURRENT AND
FUTURE CONSUMPTION ARE SUBSTITUTES? SUPPOSE
P1 RISESThen Current consumption rises
OUT
C0
Y0 Y1/(1r)/P0
c1
Slope - P1/P0(1r) GETS STEEPER
c0
U0
(1r)Y0 Y1/P1
c0
c1
C1