Time Value of Money (TVM)

1
V 1.3 Nov 12

• Time Value of Money (TVM)
• Learning Objectives (Ch 4 Parts 1 2)
• Explain the TVM concept
• Compute the FV, PV, ROR and N of a Single Cash
  Flow Situation
• Compute the FV, PV, ROR, PMT and N of an Annuity
• Compute the PV of a Series of Uneven Cash Flows
• Perform All of the Above With Other-than-annual
  Payments/Compounding
• Perform All of the Above with Fractional
  (non-integer) Time Periods
• Perform All of the Above in Cases Where
  Compounding Periods Per Year Aren't Equal To
  Payments Per Year
• Apply TVM Computations To Common Business and
  Personal Finance Decisions Such as Borrowing
  Money, Planning for Retirement, and Investing
• Explain and Use the Fundamental Valuation Concept
  Process
• Explain Effective Annual Rate
• Use Effective Annual Rate To Make a Borrowing or
  Investing Decision
• Learn How To Use a Financial Calculator

2
V 1.2 June 10

• Time Value of Money
• When money is some how invested (and not just
  placed under a mattress or in a safe), the amount
  of money grows
• Although money loses value over time due to
  inflation, the amount of money in an account that
  earns a positive ROR will be greater in the
  future than what it is today
• Thus the money in the account has different
  values at different points in time
• This is what the term Time Value of Money
  refers to
• The ROR should compensate for opportunity cost,
  inflation and risk
• the increasing amount of money over time should
  more than make up for the value lost due to
  inflation and opportunity costs
• Time Lines / Cash Flow Diagrams
• For the rest of this course, we will deal with
  cash flows that occur over some period of time
• It is ever so helpful to be able to depict these
  cash flows graphically
• We will use Time Lines (also called Cash Flow
  Diagrams) to do this because they are.
• a means to visually depict cash flows, both
  positive and negative (incoming and outgoing) so
  we get a clearer idea of whats happening
• a means to inventory what we know and what we
  dont know about a problem
• a tool to help us decide on what we need to find
  do in order to solve the problem
• Your basic Time Line (without cash flows) looks
  like this

0
2
3
1
The Future ?
Today
Note the time units can be what ever they need
to be days, weeks, months, years, etc.
3

• Future Value
• Future Value (FV) (a noun)
• The amount to which an investment grows when it
  earns a positive rate of return.
• Compounding (a verb)
• The process of going from todays values
  (present values) to values at some future time
  (future values).
• Applying the effects of TVM
• The process of determining the Future Value of a
  cash flow or a series of cash flows.
• Converting present/previous/prior values to
  future values
• The Interest Rate/Rate of Return is the
  conversion factor
• Example (Simple Case) You deposit 100 in a
  savings account that pays 6 per year (1
  compounding period per year). What amount of
  money would you have in this account after 1
  year? (What is the future value of 100 _at_ 6
  after 1 year?)
• In this example, there is only one period (1
  year) therefore n 1
• Interest that is paid-out over only one interest
  earning period is called simple interest
• Draw a cash flow diagram (a time line with cash
  flows added)

PV 100
FV ?
r 6
0
1
or this..
PV 100
r 6
0
1
FV ?
4

• Example (continued)
• The future value includes the principle and the
  interest
• How much is the interest? Answer Int.
  Principle x Int. Rate
• 100 x 0.06
  6
• How much is the future value? Answer 100 6
  106

• Basic formula for finding Future Value FV
  PV(1 r)n
• PV Present Value of the principle or investment
• r interest rate, Yield, ROR (the symbol k or i
  can also be used)
• n number of periods
• FV Future Value of the investment,
• includes the amount invested plus the
  return/profit
• for a loan, this includes the principle plus all
  accumulated interest
• The (1 r)n portion of the formula is called the
  Future Value Interest Factor (FVIFr,n) and can
  be found on a FVIF table
• (1 r)n is also called the compounding factor
• When r is used in compounding, it is referred
  to as the compound rate
• You can think of r as an exchange rate i.e. the
  rate at which we can exchange money today for
  money in the future
• Example Same as before but use the basic formula
  to find FV.
• Draw a cash flow diagram

PV 100
FV ?
r 6
0
1
FV PV(1 r)n 100(1 0.06)1
100(1.06) 106
5

• Solution Method 2 Use your TI BA II Plus
  financial calculator
• 1) Clear your calculator 2nd, QUIT
• 2) Clear the TVM memories 2nd, CLR TVM
• 3) Set payments per year to 1 2nd, P/Y, 1,
  ENTER, CE/C
• 4) Enter parameters
• Enter number of periods 1, N
• Enter interest rate 6, I/Y (the calculator
  assumes you mean it as 6)
• Enter PV 100, PV
• Find FV CPT, FV and voila! FV (-) 106.00 Why
  is FV negative?
• Important TVM Concept
• 100 today is equivalent to 106 one year from
  now if the current investment opportunity is 6
• and
• 106 one year from now (_at_ 6 ROR) is equivalent
  to 100 today
• Compound Interest
• What happens when money is left in a bank account
  for more than one interest paying time period
  (for more than 1 compounding period)?
• Example You deposit 100 in a savings account
  that pays 6 per year. What amount of money
  would you have in this account after 2 years?
  (What is the future value of 100 _at_ 6 after 2
  years?) (Abbreviation notation FV100,6,2yr)

PV 100
r 6
0
1
2
FV ?
6

• Compound Interest (continued)
• Example (continued)
• The account collects interest after one year (6
  as per previous example) which results in a
  balance of 106 (FV1 106)
• During the second year, interest is paid on the
  one year balance (i.e. the 106 earns 6
  interest)
• FV2 100(1 0.06)(1 0.06)
• FV1
• FV2 100(1 0.06)2 112.36
• When an investment is held for more than one
  interest paying period, the interest is
  compounded (interest is paid on previously
  earned interest as well as on the principle)
• Each interest paying period is called a
  compounding period
• In this context, r is referred to as the
  compound rate
• Solution Method 2 Use the financial functions on
  your calculator
• 1) Clear TVM Memory 2nd, CLEAR TVM
• 2) Set/ensure payments per year 1 2nd, P/Y,
  1, ENTER, CE/C
• 3) Enter parameters
• Enter number of periods 2, N

7
Example (continued) Heres an explanation of what
happened at each time period
Cash Flow Diagram as Normally Drawn
PV
0
1
2
FV
Cash Flow Diagram Showing Implied Interest
Payments
PV
FV


0
1
2
Interest Payments (Implied Not Drawn)
FV
Cash Flow Diagram Showing Period-to-Period
Earnings Balances
r 6
PV
0
1
2
Beginning Balance 100.00
100.00 106.00 Interest Earned
0.00 6.00
6.36 Ending Balance 100.00
106.00 112.36
FV
100.00 x (1 0.06)
106.00 x (1 0.06)
8
Example What is the FV of 5000 _at_ 6 after 5
years? (Find FV5000,6,5)
FV ?
r 6
0
1
2
3
4
5
PV 5,000
Formula Solution
Calculator Financial Function Solution
9
Future Value as a Function of Time and Rate of
Return
5 4 3 2 1 0
r 15
Future Value of 1
r 10
r 5
r 0
2 4 6
8 10
Number of Periods

• Key Points
• The greater the interest rate/rate of return, the
  bigger the future value
• The longer the investment is held, the bigger the
  future value
• Which factor (r or time) has the greatest
  influence on FV?

10

• Present Value
• Money that is expected to be received or paid in
  the future does not have the same value as
  todays money because of TVM
• In order to determined what future money is worth
  in terms of todays dollars, we have to reverse
  the effects of TVM this is referred to as
  discounting
• Present Value (PV) (a noun) the value today (the
  value in "today's dollars") of a future cash flow
  or series of cash flows.
• Discounting (a verb)
• The process of going from future values to
  present values. (The reverse of compounding.)
• Reversing the effect of TVM
• The process of finding the Present Value of a
  future cash flow or series of cash flows.
• Converting future dollars into todays dollars
• Basic formula for finding Present Value PV FV
  / (1 r)n
• The 1/(1 r)n portion of the formula is called
  the Present Value Interest Factor (PVIFr,n) and
  can be found on a PVIF table
• 1/(1 r)n is also called the discounting
  factor
• When r is used to discount, it is referred to
  as the discount rate

11
Present Value Example (Simple Case) What is
the Present Value of 200 discounted for 1 year _at_
4 per year? (Abbreviation notation PV200,4,1)
In other words, how much would we have to deposit
today into an account that pays 4 per year in
order to have 200 one year from now? Draw a
cash flow diagram
FV 200
r 4
0
1
PV ?
Solution Method 1 Use the Formula PV FV / (1
r)n 200 / (1 0.04)1 200
/(1.04)1 200 / 1.04 192.30

• Solution Method 2 Use your financial calculator
• 1) Clear TVM Memory 2nd, CLEAR TVM
• 2) Set/ensure payments per year 1 2nd, P/Y,
  1, ENTER, CE/C
• 3) Enter parameters
• Enter number of periods 1, N
• Enter interest rate 4, I/Y
• Enter FV 200, FV
• Find PV CPT, PV and voila! PV (-) 192.30

• Why did we use 4 to discount the future cash
  flow?
• Answer
• It is the specified compound rate
• It is the rate at which we expect our investment
  to grow to achieve a future value
• Since discounting is the opposite of compounding,
  we must discount at that exact same rate in order
  to find the present value

12
Example Whats the PV of 5000 discounted for 3
years _at_12? (Find PV5000,12,3)
Formula Solution
Calculator Financial Function Solution
Heres an explanation of what happened at each
time period
0
2
3
1
r 12
PV
Beginning Balance 3,558.90 3,558.90
3,985.97 4,464.29 Interest
Earned 0.00 427.07
478.32 535.71
Ending Balance 3,558.90 3,985.97
4,464.29 5,000.00
3,558.90 x (1 0.12)
3,985.97 x (1 0.12)
4,464.29 x (1 0.12)
FV
13

• Very Important Financial Concepts
• Re-word the example from p. 12 You are
  considering purchasing a security that promises
  to pay 5,000 three years from now. What is this
  security theoretically worth today (or.. what
  is its fair market value?) if your best
  investment opportunity yields 12?
• Answer 3,558.90
• Financial Valaution Principle The present value
  of any financial asset depends on usable,
  after-tax cash flow it is expected to produce in
  the future
• But, due to the Time Value of Money, we have to
  discount those expected future cash flows in
  order to convert them into todays dollars.
• Thus The overriding, all important,
  never-to-be-forgotten Financial Valuation
  Process is The theoretical value (fair market
  value or no-arbitrage price) of any financial
  asset is determined by discounting all future
  expected cash flows to the present (i.e. find the
  PV _at_ t 0 of all cash flows) and adding them up
• This is the present value of the asset.
• This is what the asset is theoretically worth
  today, without profits, fees or other transaction
  costs
• This is the fair market value/no-arbitrage price
  of the asset today, without profits, fees or
  other transaction costs

14
Use the Financial Valuation Process to Make a
Decision Example Consider two different
investment options -Option A For an initial
cost of 9,000 you receive 12,600 at the end of
four years. -Option B For an initial investment
of 20,000 you receive 26,000 at the end of four
years -Your best investment opportunity yields 6
p.a. ( per annum). Which is the best investment
option? Step 1 Find the present value of the
future cash flows of both options Option A N4,
I/Y6, FV12600, CPT PV PV 9,980.38 Option B
N4, I/Y6, FV26000, CPT PV PV
20,594.44 Step 2 Compute the net profit from
each investment (Discounted Future Cash Flows
Initial Costs) Option A 9,980.38 - 9,000
980.38 Option B 20,594.44 - 20,000
594.44 Answer Option A is the better
investment Net Profit (Discounted Future Cash
Flows Initial Costs) is also called Net Present
Value (NPV) NPV PV of Benefits PV of
Costs Read Section 3.5 in your text book Point
This is a fundamental way to make a financial
decision Important TVM Rule It is only possible
to compare or combine values at the same point in
time i.e. if you want to compare or combine
values, they must have the same time value of
money (they must be in terms of t 0 dollars or
t 5 dollars or t 10 dollars, etc.)
15

• Other Key Points
• Compounding means we're making numbers bigger
  we're growing it we're going to the right on the
  timeline
• FVs are bigger numbers than PVs check your
  answer
• Discounting means we're making the numbers
  smaller we're shrinking it we're going to the
  left on the time line
• PVs are smaller numbers than FVs check your
  answer

Compounding
0
1
Discounting
16
Solving for Interest Rate (or IRR) Example
Your broker proposes an investment scheme that
will pay you 1000 one year from now for an
initial cost of 900 today. What is the annual
return on this investment? Draw a cash flow
diagram
FV 1000
r ?
0
1
PV 900

• Solve for r using the PV (or FV) formula
• FVn PV(1 r)n
• 1000 900(1 r)1
• 1000/900 (1 r)1
• (1.1111)1 1 r
• r 1.1111 - 1
• r 0.1111 11.1111 per year
• Calculator Financial Function Solution
• 1) Clear your calculator 2nd, CLEAR TVM
• 2) Set/ensure payments per year 1 2nd, P/Y,
  1, ENTER, CE/C
• 3) Enter parameters
• Enter number of periods 1, N
• Enter PV 900, /-, PV
• Enter FV 1000, FV
• Find I/Y CPT, I/Y and voila! I/Y 11.1111
  per year
• Solution Method 3
• (New - Old) / Old (1000 - 900) / 900 x 100
  11.1111

Note One of the two cash inputs must be negative
17
Solving for Interest Rate (or IRR)
Example Your broker proposes an investment
scheme that will pay you 1000 two years from now
for an initial cost of 900 today. What is the
annual return on this investment?
FV 1000
r ?
0
2
1
PV 900
Solve for r using the PV or FV formula
FVn PV(1 r)n 1000 900(1
r)2 1000/900 (1 r)2 (1.1111)1/2 1
r r 1.054093- 1
r 0.05493 5.4093 per year
Calculator Financial Function Solution
Important Point Rates of Return are always
expressed on an annual basis. Why? Why cant
you use (New-Old) / Old?
(New - Old) / Old (1000 - 900) / 900 x 100
11.1111 This is the return over 2 years, not an
annual rate
18
Example Your broker proposes an investment
scheme that will pay you 1000 two years from now
for an initial cost of 900 today. What is the
total return on this investment?
(New - Old) / Old (1000 - 900) / 900 x 100
11.1111
Why dont you just multiply the annual rate of
return (5.4093) by 2 (to produce 10.8186) ?
Solving for the number of periods (n)

• Example How long will it take to double an
  investment of 1000 _at_ 6 annual interest?
• Solve for n using the PV (or FV) formula
• FVn PV(1 r)n
• 2000 1000(1 0.06)n
• 2000/1000 (1.06)n
• LN(2) LN(1.06)n
• n LN(2)/LN(1.06) 0.6931 /
  0.05827 11.90 years
• Calculator Financial Function Solution
• 1) Clear your calculator 2nd, CLEAR TVM
• 2) Set/ensure payments per year 1 2nd, P/Y,
  1, ENTER, CE/C
• 3) Enter parameters
• Enter I/Y 6, I/Y
• Enter PV 1000, /-, PV

FV 2000
r 6
0
n ?
PV 1000
Note One of the two cash inputs must be negative
19

• Annuities
• Definition a series of equal payments made at
  fixed intervals for a specified number of periods
• Examples

Trust Fund
Pmt
Pmt
Pmt
Endowment (PV)
Pmt
Pmt
Pmt
Endowment (PV)
Loan
Principle (PV)
Pmt
Pmt
Pmt
Principle (PV)
Pmt
Pmt
Pmt
Note The payments occur at the end of each
period. These are examples of an Annuity in
Arrears, also called and Ordinary Annuity
20

• We can find the PV and FV of annuities
• Theres an added parameter, the payment (PMT)
• annuities involve periodic payments in addition
  to implied compound interest payments
• These additional payments occur throughout the
  life of the annuity at specified periods
• all of these additional payments are the same

Example (Future Value) If you deposited 300 a
year (at the end of the year) into a savings
account that pays 5 APR, what would the account
balance be after 3 years? (Abbreviation notation
FVA300,5,3)
300
300
300
r 5
3
0
2
1
FV ?
CF2(1 r)1
CF1(1 r)2
Formula Solution Process Compound all of the
cash flows to t 3 and add them up FV CF1(1
r)2 CF2(1 r)1 CF3 300(1 0.05)2
300(1 0.05)1 300 300(1.05)2
300(1.05)1 300 300(1.1025) 300(1.05)
300 330.7500 315.0000 300
945.75
21
Example (continued) ) If you deposited 300 a
year (at the end of the year) into a savings
account that pays 5 APR, what would the account
balance be after 3 years? (Abbreviation notation
FVA300,5,3) Note there are 5 parameters but
you only care about 4 of them (1 of them is
irrelevant) so if you know 3, you can find the
4th Which parameter is irrelevant in this
example?
300
300
300
r 5
3
0
2
1
FV ?

• Calculator Financial Function Solution
• 1) Clear your calculator 2nd, CLEAR TVM
• 2) Set/ensure payments per year 1
• 3) Enter parameters
• Enter N 3, N
• Enter I/YR 5, I/Y
• Enter Pmt 300, PMT
• Find FV CPT, FV and voila! FV (-) 945.75

22
Heres an explanation of what happened at each
time period
Cash Flow Diagram as Normally Drawn
300
300
300
3
0
2
1
FV ?
Cash Flow Diagram Showing Implied Interest
Payments
Specified Additional Payments
300
300
300
Interest Payments (Implied Not Drawn)
3
0
2
1
FV ?
Cash Flow Diagram Showing Period-to-Period
Earnings Balances
2
3
1
0
i 5
Beginning Balance 0.00
0.00 300.00
615.00 Interest Earned 0.00
0.00
15.00 30.75 Promised
Payment 0.00 300.00
300.00
300.00 Ending Balance 0.00
300.00 615.00
945.75
300 x (1 0.05)
615 x (1 0.05)
FV
23
Example (Future Value) If you deposited 500 a
year (at the end of the year) into a savings
account that pays 9 APR, what would the account
balance be after 2 years? (Abbreviation notation
FVA500,9,2)
Formula Solution
Calculator Financial Function Solution
24
Retirement Example (Future Value) If you
deposited 6000 a year (at the end of the year)
into a savings account that pays 9 p.a., How
much would you have saved up for your retirement
at age 65?
You start saving when you are 25
You start saving when you are 45
You start saving when you are 55
25
Example (Present Value) You recently won a court
settlement that promises to pay 5,000 a year (at
the end of the year) for 3 years. What is the
equivalent present value of this award? Your
opportunity cost of capital is 5.0000 per year.
(Abbreviation notation (PVA5000,5,3) Assume
that the number of payments per year and the
number of compounding periods per year are the
same.
5000
5000
5000
0
2
3
1
r 5
PV ?
Formula Solution Process Discount all of the
cash flows back to t 0 and add them up PV
CF1/(1 r)1 CF2/(1 r)2 CF3/(1 r)3
5000/(1 0.05)1 5000/(1 0.05)2 5000/(1
0.05)3 5000/(1.05)1 5000/(1.05)2
5000/(1.05)3 5000/1.05 5000/1.1025
5000/1.1576 4,761.9048 4,535.1474
4,319.2813 13,616.24
26

• Annuities (continued)
• Calculator Financial Function Solution
• 1) Clear your calculator 2nd, CLEAR TVM
• 2) Set/ensure payments per year 1 2nd, P/Y,
  1, ENTER, CE/C
• 3) Enter parameters
• Enter N 3, N
• Enter I/Y 5, I/Y
• Enter PMT 5000, PMT
• Find PVCPT, PV and voila! PV (-) 13,616.24
• Why did we use 5 to discount the future cash
  flows?
• Answer It is the Opportunity Cost of Capital
• ? This is the rate we expect our invest to grow
  (compound)
• The opposite of compounding is discounting
• We must use the same rate for the discount rate
  to find PV
• Important Assumption The number of payments per
  year corresponds to the number of compounding
  periods per year. Unless otherwise specified,
  this is usually the case. Later we will discuss
  what to do when this is not the case.

27

• Annuities (continued)
• Annuity Due
• The previous examples were ordinary annuities
  or annuity in arrears all payments occurred at
  the end of the period (deferred payments)
• An annuity in which payments occur at the
  beginning of a period is call an annuity due
  Examples?
• Example Your company is considering leasing a
  warehouse for 3 years _at_ 3,000 per year, paid at
  the beginning of each year. What is the PV of
  the lease if the appropriate opportunity cost is
  6?
• Draw a cash flow diagram

PV ?
i 6
3000
3000
3000
Formula Solution Process Discount all of the
cash flows back to t 0 and add them up PV CF1
CF2/(1 r)1 CF3/(1 r)2 3000
3000/(1 0.06)1 3000/(1 0.06)2 3000
3000/(1.06)1 3000/(1.06)2 3000
3000/1.06 3000/1.1236 3000
2,830.1887 2,669.9893 8,500.18
28

• Annuities (continued)
• Annuity Due (continued)
• Example (continued)
• Calculator Financial Function Solution
• 1) Clear your calculator 2nd, CLEAR TVM
• 2) Set/ensure payments per year 1
• 3) Set payment timing to beginning of year 2nd,
  BGN, 2nd, SET, CE/C (Note BGN should appear
  in your calculator display)
• 4) Enter parameters
• Enter N 3, N
• Enter I/Y 6, I/Y
• Enter Pmt 3000, PMT
• Find PV CPT, PV and voila! PV 8,500.18

29
Annuities (continued) Example What is the FV of
the lease in the previous example? Draw a cash
flow diagram
Formula Solution Process Compound all of the
cash flows to t 3 and add them up FV CF1(1
r)3 CF2(1 r)2 CF3 (1 r)1 3000(1
0.06)3 3000(1 0.06)2 3000(1 0.06)1
3000(1.06)3 3000(1.06)2 3000(1.06)1
3000(1.1910) 3000(1.1236) 3000(1.06)
3573.0000 3,370.8000 3,180.000
10,123.85
Calculator Financial Function Solution
30
Annuities (continued) Finding the Payment of an
Annuity (Ordinary or Due) Example You plan to
start a savings fund to pay for your childs
college education. You estimate that you will
need 150k, 15 years from now. Your financial
advisor/broker says he can earn your money 7.5.
You plan to make an annual contribution to this
fund at the end of each year. How large will
your contributions be? Draw a cash flow diagram
FV 150k
r 7.5
1
2
3
4
5
15
14
13
0
PMT ?
What kind of annuity is this?

• Calculator Financial Function Solution
• 1) Clear your calculator 2nd, CLEAR TVM
• 2) Set/ensure payments per year 1
• 3) Set payment timing to end of year (BGN
  should NOT appear in the display)
• 3) Enter parameters
• Enter I/Y 7.5, I/Y
• Enter FV 150000, FV
• Enter N 15, N
• Find PMT CPT, PMT and voila! PMT
  (-)5,743.08

31

• Annuities (continued)
• Finding the Interest Rate of an Annuity (Ordinary
  or Due)
• Numerical Solution
• Uses the equations we already covered
• Requires trial error, iterative solution
  techniques
• Youre on your own!
• Using Your Calculator
• Example You are considering either buying a
  computer or leasing one _at_ 725 per year (paid at
  the beginning of the year) for 3 years. The
  current value of the computer is 2,000. You
  dont have that much cash but you can get a loan
  for 8. Which would cost less, buying or
  leasing? Assume the computer is worth 0 at the
  end of 3 years (0 salvage value). (Hint Find
  what return the leasing company will earn and
  compare it to the cost of debt)
• Draw a cash flow diagram
• Leasing companys perspective

725
725
725
r ?
PV 2000
What kind of annuity is this?
32

• Annuities (continued)
• Finding the Interest Rate of an Annuity (Ordinary
  or Due)
• Example (continued)
• Calculator Financial Function Solution
• 1) Clear your calculator 2nd, CLEAR TVM
• 2) Set/ensure payments per year 1
• 3) Set payment timing to beginning of year (BGN
  should appear in the display)
• 3) Enter parameters
• Enter N 3, N
• Enter PV 2000, /-, PV
• Enter PMT 725, PMT
• Find I/Y CPT, I/Y and voila! I/Y 9.0206
• Answer Since you can borrow money _at_ 8,
  purchasing the computer is the better option
• Example (continued) What will your payments be
  if you borrow money to buy the computer? (Note
  loans are usually ordinary annuities)

Note One of the two cash inputs must be negative
33
Example (continued)

• Calculator Financial Function Solution
• 1) Clear your calculator 2nd, CLEAR TVM
• 2) Set/ensure payments per year 1
• 3) Set payment timing to end of year (BGN
  should not appear in the display)
• 3) Enter parameters
• Enter N 3, N
• Enter PV 2000, /-, PV
• Enter I/Y 8, I/Y
• Find PMT CPT, PMT and voila! PMT 776.07

But the annual payment for borrowing is greater
than that for leasing. How can borrowing (in
this case) be better than leasing? Response
Convert the lease from an annuity due to an
ordinary annuity and solve for the annual payment
using 9.0206 APR then compare to the loan
payments
34

• Annuities (continued)
• Finding the Period of an Annuity (Ordinary or
  Due)
• Numerical Solution
• With the equations we already covered, you can
  solve for n algebraically
• Youre on your own!
• Using Your Calculator

Example How long will it take for your savings
account to accumulate 1m if it pays 4 interest
per year (at the end of the year) and you deposit
10k per year at the end of each year?
FV 1m
r 4
1
2
3
4
5
n ?
n - 1
n - 2
0
PMT (-)10,000
What kind of annuity is this?

• Calculator Financial Function Solution
• 1) Clear your calculator 2nd, CLEAR TVM
• 2) Set/ensure payments per year 1
• 3) Set payment timing to end of year (Note BGN
  should NOT appear in the display)
• 3) Enter parameters
• Enter I/Y 4, I/Y
• Enter FV 1000000, FV
• Enter PMT 10000, /-, PMT
• Press N, and voila! N 41.04 years

Note One of the two cash inputs must be negative
35

• The Opportunity Cost of Capital (from Ch 5)
• The best available expected return offered in the
  market on an investment of comparable risk and
  length (term)
• The return the investor forgoes on an alternative
  investment of equivalent risk and term when the
  investor takes on the alternative investment
• Point one always uses his/her Opportunity Cost
  of Capital as the discount/compound rate when
  solving TVM problems, if the rate is not already
  given