Time value of money

1
Time value of money

• Some important concepts

2
Todays agenda

• Review of what we have learned in the last
  lecture
• Continue to discuss the concept of the time value
  of money
• present value (PV)
• discount rate (r)
• net present value (NPV)
• Learn how to draw cash flows of projects
• Learn how to calculate the present value of
  annuities
• Learn how to calculate the present value of
  perpetuities
• Inflation, real interest rates and nominal
  interest rates, and their relationship

3
What have we learned in the last lecture

• The motivation for the study of the financial
  market
• The seven functions of a financial market
• The cost of capital
• The present value concept
• The NPV rule
• The difference between capital budgeting and the
  investment in the financial market (simply called
  investment)

4
Example 1

• John got his MBA from SFSU. When he was
  interviewed by a big firm, the interviewer asked
  him the following question
• A project costs 10 m and produces future cash
  flows, as shown in the next slide, where cash
  flows depend on the state of the economy.
• In a boom economy payoffs will be high
• over the next three years, there is a 20 chance
  of a boom
• In a normal economy payoffs will be medium
• over the next three years, there is a 50 chance
  of normal
• In a recession payoffs will be low
• over the next 3 years, there is a 30 chance of a
  recession
• In all three states, the discount rate is 8 over
  all time horizons.
• Tell me whether to take the project or not

5
Cash flows diagram in each state

• Boom economy
• Normal economy
• Recession

3 m
8 m
3 m
-10 m
2 m
7 m
1.5 m
-10 m
0.9 m
1 m
6 m
-10 m
6
Example 1 (continues)

• The interviewer then asked John
• Before you tell me the final decision, how do you
  calculate the NPV?
• Should you calculate the NPV at each economy or
  take the average first and then calculate NPV
• Can your conclusion be generalized to any
  situations?

7
Calculate the NPV at each economy

• In the boom economy, the NPV is
• -10 8/1.08 3/1.082 3/1.0832.36
• In the average economy, the NPV is
• -10 7/1.08 2/1.082 1.5/1.083-0.613
• In the bust economy, the NPV is
• -10 6/1.08 1/1.082 0.9/1.083 -2.87
• The expected NPV is
• 0.22.360.5(-.613)0.3(-2.87)-0.696

8
Calculate the expected cash flows at each time

• At period 1, the expected cash flow is
• C10.280.570.366.9
• At period 2, the expected cash flow is
• C20.230.520.311.9
• At period 3, the expected cash flows is
• C30.230.51.50.30.91.62
• The NPV is
• NPV-106.9/1.081.9/1.0821.62/1.083
• -0.696

9
Perpetuities

• We are going to look at the PV of a perpetuity
  starting one year from now.
• Definition if a project makes a level, periodic
  payment into perpetuity, it is called a
  perpetuity.
• Lets suppose your friend promises to pay you 1
  every year, starting in one year. His future
  family will continue to pay you and your future
  family forever. The discount rate is assumed to
  be constant at 8.5. How much is this promise
  worth?

C
C
C
C
C
C
PV ???
Yr1
Yr2
Yr3
Yr4
Yr5
Timeinfinity
10
Perpetuities (continue)

• Calculating the PV of the perpetuity could be
  hard

11
Perpetuities (continue)

• To calculate the PV of perpetuities, we can have
  some math exercise as follows

12
Perpetuities (continue)

• Calculating the PV of the perpetuity could also
  be easy if you ask George

13
Calculate the PV of the perpetuity

• Consider the perpetuity of one dollar every
  period your friend promises to pay you. The
  interest rate or discount rate is 8.5.
• Then PV 1/0.08511.765, not a big gift.

14
Perpetuity (continue)

• What is the PV of a perpetuity of paying C every
  year, starting from year t 1, with a constant
  discount rate of r ?

C
C
C
C
C
C
t1
t2
t3
t4
T5
Timetinf
Yr0
15
Perpetuity (continue)

• What is the PV of a perpetuity of paying C every
  year, starting from year t 1, with a constant
  discount rate of r ?

16
Perpetuity (alternative method)

• What is the PV of a perpetuity that pays C every
  year, starting in year t1, at constant discount
  rate r?
• Alternative method we can think of PV of a
  perpetuity starting year t1. The normal formula
  gives us the value AS OF year t. We then need
  to discount this value to account for periods 1
  to t
• That is

17
Annuities

• Well, a project might not pay you forever.
  Instead, consider a project that promises to pay
  you C every year, for the next T years. This
  is called an annuity.
• Can you think of examples of annuities in the
  real world?

C
C
C
C
C
C
PV ???
Yr1
Yr2
Yr3
Yr4
Yr5
TimeT
18
Value the annuity

• Think of it as the difference between two
  perpetuities
• add the value of a perpetuity starting in yr 1
• subtract the value of perpetuity starting in yr
  T1

19
Example for annuities

• you win the million dollar lottery! but wait, you
  will actually get paid 50,000 per year for the
  next 20 years if the discount rate is a constant
  7 and the first payment will be in one year, how
  much have you actually won (in PV-terms) ?

20
My solution

• Using the formula for the annuity

21
Example
You agree to lease a car for 4 years at 300
per month. You are not required to pay any money
up front or at the end of your agreement. If
your opportunity cost of capital is 0.5 per
month, what is the cost of the lease?
22
Solution
23
Lottery example

• Paper reports Todays JACKPOT 20mm !!
• paid in 20 annual equal installments.
• payment are tax-free.
• odds of winning the lottery is 13mm1
• Should you invest 1 for a ticket?
• assume the risk-adjusted discount rate is 8

24
My solution

• Should you invest ?
• Step1 calculate the PV
• Step 2 get the expectation of the PV
• Pass up this this wonderful opportunity

25
Mortgage-style loans

• Suppose you take a 20,000 3-yr car loan with
  mortgage style payments
• annual payments
• interest rate is 7.5
• Mortgage style loans have two main features
• They require the borrower to make the same
  payment every period (in this case, every year)
• The are fully amortizing (the loan is completely
  paid off by the end of the last period)

26
Mortgage-style loans

• The best way to deal with mortgage-style loans is
  to make a loan amortization schedule
• The schedule tells both the borrower and lender
  exactly
• what the loan balance is each period (in this
  case - year)
• how much interest is due each year ? ( 7.5 )
• what the total payment is each period (year)
• Can you use what you have learned to figure out
  this schedule?

27
My solution
Ending balance
Total payment
Interest payment
Principle payment
year
Beginning balance
0
20,000
1,500
6,191
7,691
13,809
1
7,154
13,809
1,036
6,655
7,691
2
7,154
7,691
0
7,154
537
3
28
Future value

• The formula for converting the present value to
  future value
• present value at time zero
• future value in year i
• discount rate during the i years

29
Manhattan Island Sale
Peter Minuit bought Manhattan Island for 24 in
1629. Was this a good deal? Suppose the interest
rate is 8.
30
Manhattan Island Sale
Peter Minuit bought Manhattan Island for 24 in
1629. Was this a good deal?
To answer, determine 24 is worth in the year
2003, compounded at 8.
FYI - The value of Manhattan Island land is well
below this figure.
31
Inflation

• What is inflation?
• What is the real interest rate?
• What is the nominal interest rate?

32
Inflation rule

• Be consistent in how you handle inflation!!
• Use nominal interest rates to discount nominal
  cash flows.
• Use real interest rates to discount real cash
  flows.
• You will get the same results, whether you use
  nominal or real figures

33
Example

• You own a lease that will cost you 8,000 next
  year, increasing at 3 a year (the forecasted
  inflation rate) for 3 additional years (4 years
  total). If discount rates are 10 what is the
  present value cost of the lease?

34
Inflation

• Example - nominal figures

35
Inflation

• Example - real figures