Today's Lecture

1
Today's Lecture

• NPV
• Present Value
• Multi-Period
• Useful formulae
• Discount Rates

TIP If you do not understand something, a
sk me!
2
Another NPV Example

• What if the choice were trading 10 today for 20
  tomorrow.
  
• What are the market prices in this case
• Dollars today are always worth their face value,
  that is, their price is 1
  
• The price of dollars tomorrow is given by the
  interest rate

3
Present Value

• If the interest rate is 10, what is 20 tomorrow
  worth today?
  
• The amount of money you could borrow against this
  payment.
  
• Denote this amount of money as x

4
Another Example (contd)

• NPV18.18-108.18
• You should take this opportunity.
• This decision does not depend on how you
  personally trade off money today vrs money
  tomorrow. It is just like copper and aluminum.

5
Review

• Competitive Markets
• Both buy and sell at the same price
• Law of One Price
• Two securities with exactly the same cash flows
  must have exactly the same price
  
• NPV Rule

6
Definition NET PRESENT VALUE

• The additional value today of an investment
  opportunity.
  
• Net-Present-Value Rule (or the no-brainer rule)
• Take on any investment with a positive NPV.
• Reject any investment that has a negative NPV.
• Why do peoples' preferences not affect this rule?

7
Example 1

• I have an offer to sell my bike for 500. My
  brother also wants to buy the bike. He will pay
  me 545, but he can only make the payment in a
  year. If current interest rates are 10, which
  is the better deal?

8
Present Value Formula
9
NET Present Value Formula
10
Example 2

• Your buddy would like to start a business. He
  needs 10,000. If you lend him the money and the
  business is successful, he will give you 12,000
  in a year.
• If interest rates are 10 what should you do?
• What is the present value of 12,000?
• What is the net present value of this investment?
  
  
• Is 10 the correct rate to use?
• What would your answer be if rates were 25?

11
Multiperiods

• The key to not making a mistake is the TIMELINE

12
Example 3 - The Multi-Period Case

• Assume that the average college tuition costs
  20,000 dollars per annum (paid at the end of the
  year). For a freshman just starting college,
  what is the present value of the cost of a four
  year degree when the interest rate is 10?

13
Multiperiod PV Formula
14
Net Present Value

• Just the Present Value minus the cost of the
  investment
  
• Formula

15
Where are we?

• We understand the basic idea but ...
• it is a pain to add up these series --- is there
  an easier way?
  
• YES!

16
Perpetuity

• A certain (constant) cashflow forever (e.g. a
  consol bond).
  
• What is the present value of a perpetuity with
  cashflow C forever?

17
Perpetuity Timeline
18
Trick

• How much money would you have to put in the bank
  to get a constant cashflow stream forever?
  
• Formula

19
Example

• You made your fortune in the dot-com boom (and
  got out in time!). As part of your legacy, you
  want to endow an annual MBA graduation party at
  your alma matter. You want it to be a memorable,
  so you budget 30,000 per year for the party. If
  the university earns 8 per year on its
  investments, and if the first party is in one
  years time, how much will you need to donate to
  endow the party?

20
Solution

• PV C / r 30,000 / .08 375,000 today.

21
Example Contd

• Suppose instead the first party was scheduled to
  be held 2 years from today (the current entering
  class). How would this change the amount of the
  donation required?

22
Solution

• PV 375,000 / 1.08 347,222 today.

23
Growing Perpetuity

• A stream of cashflows that grows at a constant
  rate forever.
  
• What is the present value of growing perpetuity?
  
  
• where g is the growth rate

24
Timeline

• A growing perpetuity with first payment of 100
  that grows at a rate of 3 has the following
  timeline
  

25
The General Case

• In general, a growing perpetuity with first
  payment C and growth rate g will have the
  following series of cash flows
  

26
Trick

• Write the growing perpetuity as a perpetuity and
  apply the previous formula
  
• Definition
• Now write the PV formula as

27
Trick (continued)

• so the PV formulae from the previous slide
  becomes

28
Trick (continued)

• Now apply the perpetuity formula
• Substitute back for (1R)

29
Example 5

• Assuming the discount rate is 7 per annum, how
  much would you pay to receive 50, growing at 5,
  annually, forever?

30
Annuity

• Pays a constant payment for a fixed number of
  years (periods).
  
• What is the present value of an N period
  annuity?

31
Timeline
32
Trick

• Write an annuity as the difference between two
  perpetuities.
  
• An N period annuity is equal to a perpetuity
  minus another perpetuity whose first cashflow
  arrives in period N1.

33
Trick (contd)
34
Example

• You are the lucky winner of the 30 Million State
  Lottery. You can take your prize money either
  as
  
• 30 payments of 1M per year (starting today),
• 15M paid today. If the interest rate is 8,
  which option should you take?

35
Timeline
36
Growing Annuity

• Pays a constantly growing cashflow for a fixed
  number of years (N)

37
Timeline

• What is the present value?

38
Trick

• same as before!
• Express as the difference between two growing
  perpetuities.
  
• Derivation
• Let N be the number of period to maturity, C be
  the first payment, and g be the growth rate
  
• What is the first payment of the perpetuity that
  needs to to be subtracted off?

39
Subtract the two perpetuities
40
Example

• Assume that a college education means an
  additional 10,000/year in starting salary, and
  that this difference grows at 3 per annum.
  Assume a 7 annual discount rate and a 40 year
  working life.
• On graduation day, what is the value of the
  degree?
  
• Assuming that college costs about 20,000/annum
  (due in advance), what is the NPV of the
  investment opportunity?

41
Working Backwards

• Sometimes you know the PV, but you do not know
  the payment
  
• Example
• You are considering opening a business that
  requires an initial investment of 100,000. Your
  bank manager has agreed to lend you this money.
  The terms of the loan are that you will make
  equal annual payments for the next 10 years and
  will pay an interest rate of 8. What is your
  annual payment?

42
Solution
43
Internal Rate of Return

• Sometime you know the monthly payment and the PV
  and you would like to know what interest rate
  sets them equal
  
• You can also think of this as the return of the
  investment

44
Example

• Assume you wanted to purchase a BMW that cost
  40,000. The dealer is willing to let you have
  the car with zero down payment, so long as you
  are willing to pay off the car with 4 annual
  payments of 15,000. What interest rate is the
  dealer charging for this loan?

45
Timeline
46
IRR

• The internal rate of return (IRR) is the interest
  rate that sets the present value of an investment
  opportunity equal to zero.

47
The Discount Rate

• The discount rate is the correct rate to use to
  move a particular cash flow in time.
  
• We have not really addressed where this comes
  from.
  
• A deep question that we will not fully answer in
  this course.

48
Example

• Your cousin would like to buy your Acura.
  Unfortunately, he is just a student and has very
  little money. Instead of paying for the car, he
  offers to pay you 100/month forever. If the
  annual interest rate is 10, how much is he
  offering to pay for the car?
• What monthly interest rate would you demand on
  your deposit at the bank so that you would be
  indifferent between that and being paid 10
  annually?

49
Aside Annual vrs Monthly Compounding

• Say you deposit 1
• If you chose the annual interest deposit in one
  year you will have
  
• If you chose the monthly interest deposit in one
  year you will have
  
• So..

50
Example - Solution
51
General Idea

• Given a discount rate of rx per x-years, the
  equivalent discount rate ry per y-years is given
  by compounding (1rx) for y/x periods
  
• 1 ry (1 rx)y/x.

52
Interest Rate Quotes

• When a bank quotes an interest rate for a
  particular loan, it is usually not correct to use
  this quote directly as the discount rate
  
• The discount rate often has to be computed from
  the quoted rate based on the conventions of the
  quote.

53
Annual Percentage Rate (APR)

• This is the amount of interest you would earn in
  one year assuming that you rollover the loan but
  do not reinvest any interest payment paid during
  the year, that is, the loan is not compounded.

54
Example

• If a 3 month bond has a 8 APR, how much
  interest will I earn over the life of the bond?
  
• Since the APR quote does not include interest on
  interest and since a 3 month bond can be
  reinvested 4 times during the year, the bond will
  earn 2 interest over its life.

55
Effective Annual Rate (EAR)

• This is the amount of interest you would earn in
  one year assuming that you rollover the loan and
  reinvest all interest payments as often as is
  allowed by the terms of the loan, that is, the
  loan is compounded as often as possible during
  the year.

56
Example

• If a 3 month bond has a 8 EAR, how much interest
  will I earn over the life of the bond?
  
• Since the EAR quote does include interest on
  interest and since a 3 month bond can be
  reinvested 4 times during the year,

57
Example 6

• Using current rates, assume that the most you can
  afford in monthly mortgage payments is 1000. If
  you plan to use a 30 year fixed rate mortgage
  with an interest rate of 7.75 and a 1
  origination fee, how large a mortgage can you
  afford?

58
Next Lecture

• Topics
• Bonds
• Reading
• Chapter 5 Appendix, Sec. 5.4-5.9
• BD Chapters 6 and 6X