Time Value of Money

1
Time Value of Money

• Many financial decisions require comparisons of
  cash payments at
• different dates
• Example 2 investments that require an initial
  investment of 100
• Timing Inv 1 Inv 2
• After 1 year 30 20
• After 2 years 30 20
• After 3 years 30 40
• After 4 years 30 60
• If you should choose one of them, which would you
  choose?

2
Compounding

• Future Value amount to which an investment will
  grow after earning interest
• Compounding the process of accumulating interest
  in an investment over time to earn more interest
• Compound interest Interest earned on both the
  initial principal and the reinvested interest
  from prior periods

3
Future Value

• FV of 100 in 2 years if k10
• Time principal Interest
• 0 100 0
• 1 100 10
• 2 110 11
• So 100 today ? 121 in 2 years

4
simple and compounded interest?

• What is the difference between simple and
• compounded interest?
• Compound interest assumes accumulated interest is
  reinvested (therefore, interest earns interest).
• Simple interest assumes interest is not
  reinvested. Interest is earned each period on the
  original principal only.

5
Present Value and Discounting

• Present Value value today of a future cash flow
• PV is simply the reverse of future value
• PV works backward through time, while future
  value goes forward through time
• Discounting finding present value of some future
  
• amount

6
Example

• 3 different ways to find future value of a single
  cash flow
• Find FV of 100 in 2 years _at_ 10
• FV2 100(110)2 formula
• 100 FVIF2,10 table
• 100 PV 10 i 2 n FV
  financial calculator
• in general FVn PV (1i)n
• PV, FV formulas are based on this equation
• 4 variables given any 3, you can calculate the
  4th

7
solving for n

• in how many years will 100 grow to 121 _at_ i 10
• Formula way
• 100(110)n 121
• (110)n 1.21
• n ln(110)ln 1.21?

Table way 100 FVIFn,10 121 FVIFn,10
1.21 Refer to FVIF Table. Look down the 10
column to find 1.21. Financial calculator
way 100 PV 10 i 121 FV n
8
Solving for i

• At what rate of return will 100 grow to 121 in
  2 years
• Formula way
• 100(1i)2 121
• (1i)2 1.21
• 1i (1.21)1/2 1.10
• i 0.10 10
• Table way
• 100 FVIF2,i 121
• FVIF2.i 1.21
• Refer to FVIF Table. Look across 2 period row to
  find 1.21.
• Financial calculator way
• 100 PV 2 n 121 FV i

9
Present and future value of multiple cash flows

• Calculate PV(FV) of each cash flow and add them
  up e.g. i10

• PV 100/(110) 300/(110)2 400/(110)3
  formula way
• PV 100 PVIF1,10 300 PVIF2,10 400
  PVIF3,10 table way
• 10 i 0 CFi 100 CFi 300 CFi
  400 CFi NPV financial calculator

10
Valuing Level Cash Flows Annuities and
Perpetuities

• We often deal with situations where cash flows
  are same
• throughout the problem. For example, a car loan,
  rent
• payment etc.
• An annuity is a level stream of cash flows for a
  fixed period of
• time. Cash flow must be the same in each period.
• Ordinary annuity Payments are at the end of
  period
• Annuity due Payments are at the beginning of
  period
• Unless stated otherwise, assume you deal with
  ordinary annuity

11
Future value of an annuity

• FVA3 A (1i)2 A (1i) A formula way
• A (1i)2 (1i) 1
• A FVIFA3,i table way
• A PMT r i 3 n FV
  financial calculator way
• again given any 3, we can solve for the 4th

12
Present value of an annuity

• PVA3 A/(1i)3 A/(1i)2 A/(1i)
  formula way
• A 1/(1i)3 1/(1i)2 1/(1i)
• A PVIFA3,i table way
• A PMT r i 3 n PV
  financial calculator way

13
deriving the PVIFA3,i formula

• use sum of infinite geometric series formula
• asa one can show that

14
deriving the PVIFA3,i formula
15
Perpetuities

• A special case of an annuity is when the cash
  flows continue forever.
• The most common application of perpetuities in
  finance is preferred stock
• Preferred stock offers a fixed cash dividend
  every period (usually every quarter) forever.
• The dividend never increases in value, so its
  similar to a bond with a fixed interest payment.
• Present value of a perpetuity

16
Comparing Interest Rates

• How do you compare interest rates?
• Rates can be quoted monthly, annually or
  something in between,
• and it quickly becomes confusing to try and
  determine the real
• interest rate.
• Stated Rate ( also called APR, Quoted Rate,
  Nominal Rate) rate
• before considering any compounding effects
• e.g. 10 APR quarterly compounding
• Periodic Rate APR/( of times compounding occurs
  in a year)
• It is the effective or real rate. It considers
  the compounding
• effects.

17
Effective Annual Rate

• Effective Annual Rate (EAR)
• Rate on an annual basis that reflects all
  compounding
• effects
• EAR (1APR/n)n 1
• You can compare different interest rate
  quotations
• by using EAR

18
Note in TVM problems

• Timing of cash flows tells you what the period is
• Find and use the periodic rate that is consistent
  with the period definition

19
Loan Amortization There are many different kinds
of loans available

• Pure discount loan
• With such a loan, the borrower receives money
  today and repays a single lump sum at some time
  in the future.
• Interest-only loans
• This kind of loan repayment plan calls for
  the borrower to pay interest each period and
  repay
• the entire principal at some point in the
  future.

20
different types of loans

• Amortized loans
• With a pure discount or interest-only loan,
  the principal is paid all in once. An alternative
  is an amortized loan where the lender may require
  the borrower to repay parts of the loan amount
  over time. The process of paying off a loan by
  making
• regular principal reductions is called
  amortizing the loan.
• Partially amortizing loan
• Similar to amortized loan except the borrower
  makes a single, much larger final payment called
  a balloon to pay off the loan.

21
Example

• You get a 10,000 car loan. It is a five year
  amortized loan with
• annual installments. 12 is the interest rate
  charged by the bank.
• Develop the amortization schedule.