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CHAPTER 2 Time Value of Money. Future value Present value Annuities Rates of return Amortization. Time lines. 0. 1. 2. 3. I%. CF 0. CF 1. CF 2. CF 3. Show the timing of cash flows.

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Chapter 2 time value of money

CHAPTER 2Time Value of Money

Future value

Present value

Annuities

Rates of return

Amortization


Time lines
Time lines

0

1

2

3

I%

CF0

CF1

CF2

CF3

  • Show the timing of cash flows.

  • Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.


Drawing time lines

$100 lump sum due in 2 years

0

1

2

I%

100

3 year $100 ordinary annuity

0

1

2

3

I%

100

100

100

Drawing time lines


Drawing time lines1

0

2

3

1

I%

-50

100

75

50

Drawing time lines

Uneven cash flow stream (mixed stream)


Compounding and discounting single sums

Today

Future

Compounding and Discounting Single Sums

We know that receiving $1 today is worth more than $1 in the future. This is due to opportunity costs.

The opportunity cost of receiving $1 in the future is the interestwe could have earned if we had received the $1 sooner.


Compounding vs discounting

Today

Future

?

Today

Future

?

Compounding VS Discounting

  • Translate $1 today into its equivalent in the future(compounding).

  • Translate $1 in the future into its equivalent today(discounting).



Compound interest and future value
Compound Interest and Future Value

  • 1) Future Value - single sumsIf you deposit $100 in an account earning 6%, how much would you have in the account after 1 year?

PV = -100 FV = 106

Mathematical Solution:

FV = PV (FVIF i, n )

FV = 100 (FVIF .06, 1 ) (use FVIF table, or)

FV = PV (1 + i)n

FV = 100 (1.06)1 = $106


Compound interest and future value1
Compound Interest and Future Value

  • 2) Future Value - single sums

  • If you deposit $100 in an account earning 6%, how much would you have in the account after 5 years?

Mathematical Solution:

FV = PV (FVIF i, n )

FV = 100 (FVIF .06, 5 ) (use FVIF table, or)

FV = PV (1 + i)n

FV = 100 (1.06)5 = $133.82


Compound interest and future value2
Compound Interest and Future Value

  • 3) Future Value - single sums

  • If you deposit $100 in an account earning 6% with quarterly compounding, how much would you have in the account after 5 years?

Mathematical Solution:

FV = PV (FVIF i, n )

FV = 100 (FVIF .015, 20 ) (can’t use FVIF table)

FV = PV (1 + i/m) m x n

FV = 100 (1.015)20 = $134.68


Compound interest and future value3
Compound Interest and Future Value

  • 4) Future Value - single sums

  • If you deposit $100 in an account earning 6% with monthly compounding, how much would you have in the account after 5 years?

Mathematical Solution:

FV = PV (FVIF i, n )

FV = 100 (FVIF .005, 60 ) (can’t use FVIF table)

FV = PV (1 + i/m) m x n

FV = 100 (1.005)60 = $134.89


Compound interest and future value4
Compound Interest and Future Value

  • 5) Future Value - continuous compounding

  • What is the FV of $1,000 earning 8% with continuous compounding, after 100 years?

Mathematical Solution:

FV = PV (e in)

FV = 1000 (e .08x100) = 1000 (e 8)

FV = $2,980,957.99


What is the future value fv of an initial 100 after 3 years if i yr 10

0

1

2

3

10%

100

FV = ?

What is the future value (FV) of an initial $100 after 3 years, if I/YR = 10%?

  • Finding the FV of a cash flow or series of cash flows is called compounding.

  • FV can be solved by using the step-by-step (formula/table), financial calculator, and spreadsheet methods (excel).


Solving for fv the step by step and formula methods
Solving for FV:The step-by-step and formula methods

  • After 1 year:

    • FV1 = PV (1 + I) = $100 (1.10) = $110.00

  • After 2 years:

    • FV2 = PV (1 + I)2 = $100 (1.10)2 =$121.00

  • After 3 years:

    • FV3 = PV (1 + I)3 = $100 (1.10)3 =$133.10

  • After N years (general case):

    • FVN = PV (1 + I)N


Solving for fv the calculator method
Solving for FV:The calculator method

  • Solves the general FV equation.

  • Requires 4 inputs into calculator, and will solve for the fifth. (Set to P/YR = 1 and END mode.)

3

10

-100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

133.10



Present value
Present Value

  • 1) Present Value - single sums

  • If you receive $100 one year from now, what is the PV of that $100 if your opportunity cost is 6%?

Mathematical Solution:

PV = FV (PVIF i, n )

PV = 100 (PVIF .06, 1 ) (use PVIF table, or)

PV = FV / (1 + i)n

PV = 100 / (1.06)1 = $94.34


Present value1
Present Value

  • 2) Present Value - single sums

  • If you receive $100 five years from now, what is the PV of that $100 if your opportunity cost is 6%?

Mathematical Solution:

PV = FV (PVIF i, n )

PV = 100 (PVIF .06, 5 ) (use PVIF table, or)

PV = FV / (1 + i)n

PV = 100 / (1.06)5 = $74.73


Present value2
Present Value

  • 3) Present Value - single sums

  • What is the PV of $1,000 to be received 15 years from now if your opportunity cost is 7%?

Mathematical Solution:

PV = FV (PVIF i, n )

PV = 100 (PVIF .07, 15 ) (use PVIF table, or)

PV = FV / (1 + i)n

PV = 100 / (1.07)15 = $362.45


Present value3
Present Value

  • 4) Present Value - single sums

  • If you sold land for $11,933 that you bought 5 years ago for $5,000, what is your annual rate of return?

Mathematical Solution:

PV = FV (PVIF i, n )

5,000 = 11,933 (PVIF ?, 5 )

PV = FV / (1 + i)n

5,000 = 11,933 / (1+ i)5

.419 = ((1/ (1+i)5)

2.3866 = (1+i)5

(2.3866)1/5 = (1+i) i = .19


Present value4
Present Value

  • 5) Present Value - single sums

  • Suppose you placed $100 in an account that pays 9.6% interest, compounded monthly. How long will it take for your account to grow to $500?

Mathematical Solution:

PV = FV / (1 + i)n

100 = 500 / (1+ .008)N

5 = (1.008)N

ln 5 = ln (1.008)N

ln 5 = N ln (1.008)

1.60944 = .007968 N N = 202 months


What is the present value pv of 100 due in 3 years if i yr 10
What is the present value (PV) of $100 due in 3 years, if I/YR = 10%?

  • Finding the PV of a cash flow or series of cash flows is called discounting (the reverse of compounding).

  • The PV shows the value of cash flows in terms of today’s purchasing power.

0

1

2

3

10%

PV = ?

100


Solving for pv the formula method
Solving for PV:The formula method

  • Solve the general FV equation for PV:

    • PV = FVN / (1 + I)N

    • PV = FV3 / (1 + I)3

      = $100 / (1.10)3

      = $75.13


Solving for pv the calculator method
Solving for PV:The calculator method

  • Solves the general FV equation for PV.

  • Exactly like solving for FV, except we have different input information and are solving for a different variable.

3

10

0

100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-75.13


Solving for i what interest rate would cause 100 to grow to 125 97 in 3 years
Solving for I:What interest rate would cause $100 to grow to $125.97 in 3 years?

  • Solves the general FV equation for I.

  • Hard to solve without a financial calculator or spreadsheet.

3

-100

0

125.97

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

8


Solving for n if sales grow at 20 per year how long before sales double
Solving for N:If sales grow at 20% per year, how long before sales double?

  • Solves the general FV equation for N.

  • Hard to solve without a financial calculator or spreadsheet.

20

-1

0

2

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

3.8


Hint for single sum problems
Hint for single sum problems

  • In every single sum present value and future value problem, there are four variables:

    FV, PV, i and n.

  • When doing problems, you will be given three variables and you will solve for the fourth variable.

  • Keeping this in mind makes solving time value problems much easier!


Annuities

0

1

2

3

4

Annuities

  • Annuity: a sequence of equal cash flows, occurring at the end of each period.


Examples of annuities
Examples of Annuities:

  • If you buy a bond, you will receive equal semi-annual coupon interest payments over the life of the bond.

  • If you borrow money to buy a house or a car, you will pay a stream of equal payments.


Annuities1
Annuities

  • 1) Future Value – annuity

  • If you invest $1,000 each year at 8%, how much would you have after 3 years?

Calculator Solution:

P/Y = 1 I = 8 N = 3

PMT = -1,000

FV = $3,246.40


Future value annuity if you invest 1 000 each year at 8 how much would you have after 3 years
Future Value - annuityIf you invest $1,000 each year at 8%, how much would you have after 3 years?

Mathematical Solution:

FV = PMT (FVIFA i, n)

FV = 1,000 (FVIFA .08, 3) (use FVIFA table, or)

FV = PMT (1 + i)n - 1

i

FV = 1,000 (1.08)3 - 1 = $3246.40

.08


Annuities2
Annuities

  • 1) Present Value - annuity

  • What is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

Calculator Solution:

P/Y = 1 I = 8 N = 3

PMT = -1,000

PV = $2,577.10


Present Value - annuityWhat is the PV of $1,000 at the end of each of the next 3 years, if the opportunity cost is 8%?

Mathematical Solution:

PV = PMT (PVIFA i, n)

PV = 1,000 (PVIFA .08, 3) (use PVIFA table, or)

1

PV = PMT 1 - (1 + i)n

i

1

PV = 1000 1 - (1.08 )3 = $2,577.10

.08


What is the difference between an ordinary annuity and an annuity due

Ordinary Annuity (end)

0

1

2

3

i%

PMT

PMT

PMT

Annuity Due (beginning)

0

1

2

3

i%

PMT

PMT

PMT

What is the difference between an ordinary annuity and an annuity due?


Solving for fv 3 year ordinary annuity of 100 at 10
Solving for FV:3-year ordinary annuity of $100 at 10%

  • $100 payments occur at the end of each period, but there is no PV.

3

10

0

-100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

331


Solving for pv 3 year ordinary annuity of 100 at 10
Solving for PV:3-year ordinary annuity of $100 at 10%

  • $100 payments still occur at the end of each period, but now there is no FV.

3

10

100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-248.69


Solving for fv 3 year annuity due of 100 at 10
Solving for FV:3-year annuity due of $100 at 10%

  • Now, $100 payments occur at the beginning of each period.

  • FVAdue= FVAord(1+I) = $331(1.10) = $364.10.

  • Alternatively, set calculator to “BEGIN” mode and solve for the FV of the annuity:

BEGIN

3

10

0

-100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

364.10


Solving for pv 3 year annuity due of 100 at 10
Solving for PV:3-year annuity due of $100 at 10%

  • Again, $100 payments occur at the beginning of each period.

  • PVAdue= PVAord(1+I) = $248.69(1.10) = $273.55.

  • Alternatively, set calculator to “BEGIN” mode and solve for the PV of the annuity:

BEGIN

3

10

100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-273.55


What is the present value of a 5 year 100 ordinary annuity at 10
What is the present value of a 5-year $100 ordinary annuity at 10%?

  • Be sure your financial calculator is set back to END mode and solve for PV:

    • N = 5, I/YR = 10, PMT = 100, FV = 0.

    • PV = $379.08


What if it were a 10 year annuity a 25 year annuity a perpetuity
What if it were a 10-year annuity? A 25-year annuity? A perpetuity?

  • 10-year annuity

    • N = 10, I/YR = 10, PMT = 100, FV = 0; solve for PV = $614.46.

  • 25-year annuity

    • N = 25, I/YR = 10, PMT = 100, FV = 0; solve for PV = $907.70.

  • Perpetuity

    • PV = PMT / I = $100/0.1 = $1,000.


Other cash flow patterns
Other Cash Flow Patterns perpetuity?

  • Perpetuities

  • Suppose you will receive a fixed payment every period (month, year, etc.) forever. This is an example of a perpetuity.

  • You can think of a perpetuity as an annuity that goes on forever.


Present value of a perpetuity
Present Value of a Perpetuity perpetuity?

  • When we find the PV of an annuity, we think of the following relationship:

PV = PMT (PVIFA i, n )


1 perpetuity?

n

1 -

(1 + i)

i

Mathematically,

(PVIFA i, n ) =

We said that a perpetuity is an annuity where n = infinity. What happens to this formula when n gets very, very large?


1 perpetuity?

n

1 -

(1 + i)

i

1

i

When n gets very large,

this becomes zero.

So we’re left with PVIFA =


PMT perpetuity?

PV =

i

Present Value of a Perpetuity

  • So, the PV of a perpetuity is very simple to find:


PMT $10,000 perpetuity?

i .08

= $125,000

PV = =

What should you be willing to pay in order to receive $10,000 annually forever, if you require 8% per year on the investment?


The power of compound interest
The Power of Compound Interest perpetuity?

A 20-year-old student wants to save $3 a day for her retirement. Every day she places $3 in a drawer. At the end of the year, she invests the accumulated savings ($1,095) in a brokerage account with an expected annual return of 12%.

How much money will she have when she is 65 years old?


Solving for fv if she begins saving today how much will she have when she is 65

45 perpetuity?

12

0

-1095

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

1,487,262

Solving for FV:If she begins saving today, how much will she have when she is 65?

  • If she sticks to her plan, she will have $1,487,261.89 when she is 65.

  • N = 45, I/YR = 12, PMT = -1095, PV = 0; solve for FV = $1,487,262.


Solving for fv if you don t start saving until you are 40 years old how much will you have at 65
Solving for FV: perpetuity?If you don’t start saving until you are 40 years old, how much will you have at 65?

  • If a 40-year-old investor begins saving today, and sticks to the plan, he or she will have $146,000.59 at age 65. This is $1.3 million less than if starting at age 20.

  • Lesson: It pays to start saving early.

25

12

0

-1095

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

146,001


Solving for pmt how much must the 40 year old deposit annually to catch the 20 year old
Solving for PMT: perpetuity?How much must the 40-year old deposit annually to catch the 20-year old?

  • To find the required annual contribution, enter the number of years until retirement and the final goal of $1,487,261.89, and solve for PMT.

25

12

0

1,487,262

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-11,154.42


What is the pv of this uneven cash flow stream

4 perpetuity?

0

1

2

3

10%

100

300

300

-50

90.91

247.93

225.39

-34.15

530.08 = PV

What is the PV of this uneven cash flow stream?


Solving for pv uneven cash flow stream
Solving for PV: perpetuity?Uneven cash flow stream

  • Input cash flows in the calculator’s “CFLO” register:

    • CF0 = 0

    • CF1 = 100

    • CF2 = 300

    • CF3 = 300

    • CF4 = -50

  • Enter I/YR = 10, press NPV button to get NPV = $530.087. (Here NPV = PV.)


Uneven cash flows

-10,000 2,000 4,000 6,000 7,000 perpetuity?

0 1 2 3 4

Uneven Cash Flows

  • Sorry! There’s no quickie for this one. We have to discount each cash flow back separately.


-10,000 2,000 4,000 6,000 7,000 perpetuity?

0 1 2 3 4

period CF PV (CF)

0 -10,000 -10,000.00

1 2,000 1,818.18

2 4,000 3,305.79

3 6,000 4,507.89

4 7,000 4,781.09

PV of Cash Flow Stream: $ 4,412.95


0 perpetuity?

1

2

3

10%

100

133.10

0

1

2

3

4

5

6

0

1

2

3

5%

100

134.01

Will the FV of a lump sum be larger or smaller if compounded more often, holding the stated I% constant?

  • LARGER, as the more frequently compounding occurs, interest is earned on interest more often.

Annually (n=1): FV3 = $100(1.10)3 = $133.10

Semiannually (nxm=1x2=2): FV6 = $100(1.05)6 = $134.01


Classifications of interest rates
Classifications of interest rates perpetuity?

  • 1) Nominal rate (INOM) – also called the quoted or state rate. An annual rate that ignores compounding effects.

    • INOM is stated in contracts. Periods must also be given, e.g. 8% Quarterly or 8% Daily interest.

  • 2) Periodic rate (IPER) – amount of interest charged each period, e.g. monthly or quarterly.

    • IPER = INOM / M, where M is the number of compounding periods per year. M = 4 for quarterly and M = 12 for monthly compounding.


Classifications of interest rates1
Classifications of interest rates perpetuity?

  • 3) Effective (or equivalent) annual rate (EAR = EFF%) – the annual rate of interest actually being earned, accounting for compounding.

    • EFF% for 10% semiannual investment

      EFF% = ( 1 + INOM / M )M - 1

      = ( 1 + 0.10 / 2 )2 – 1 = 10.25%

    • Should be indifferent between receiving 10.25% annual interest and receiving 10% interest, compounded semiannually.


Annual percentage yield apy
Annual Percentage Yield (APY) perpetuity?

Which is the better loan:

  • 8% compounded annually, or

  • 7.85% compounded quarterly?

  • We can’t compare these nominal (quoted) interest rates, because they don’t include the same number of compounding periods per year!

    We need to calculate the APY.


Annual percentage yield apy1

APY = perpetuity?( 1 + ) m - 1

quoted rate

m

APY = ( 1 + ) 4 - 1

APY = .0808, or 8.08%

.0785

4

Annual Percentage Yield (APY)

  • Find the APY for the quarterly loan:

  • The quarterly loan is more expensive than the 8% loan with annual compounding!


Why is it important to consider effective rates of return
Why is it important to consider effective rates of return? perpetuity?

  • Investments with different compounding intervals provide different effective returns.

  • To compare investments with different compounding intervals, you must look at their effective returns (EFF% or EAR).

  • See how the effective return varies between investments with the same nominal rate, but different compounding intervals.

    EARANNUAL 10.00%

    EARQUARTERLY 10.38%

    EARMONTHLY 10.47%

    EARDAILY (365) 10.52%


When is each rate used
When is each rate used? perpetuity?

  • INOM written into contracts, quoted by banks and brokers. Not used in calculations or shown on time lines.

  • IPER Used in calculations and shown on time lines. If M = 1, INOM = IPER = EAR.

  • EAR Used to compare returns on investments with different payments per year. Used in calculations when annuity payments don’t match compounding periods.


What is the fv of 100 after 3 years under 10 semiannual compounding quarterly compounding
What is the perpetuity?FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?


Can the effective rate ever be equal to the nominal rate
Can the effective rate ever be equal to the nominal rate? perpetuity?

  • Yes, but only if annual compounding is used, i.e., if M = 1.

  • If M > 1, EFF% will always be greater than the nominal rate.


What s the fv of a 3 year 100 annuity if the quoted interest rate is 10 compounded semiannually
What’s the perpetuity?FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually?

  • Payments occur annually, but compounding occurs every 6 months.

  • Cannot use normal annuity valuation techniques.

0

1

2

3

4

5

6

5%

100

100

100


Method 1 compound each cash flow

110.25 perpetuity?

121.55

FV3 = 331.80

Method 1:Compound each cash flow

FV3 = $100(1.05)4 + $100(1.05)2 + $100

FV3 = $331.80

0

1

2

3

4

5

6

5%

100

100

100


Method 2 financial calculator
Method 2: perpetuity?Financial calculator

  • Find the EAR and treat as an annuity.

  • EAR = ( 1 + 0.10 / 2 )2 – 1 = 10.25%.

3

10.25

0

-100

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

331.80


Find the pv of this 3 year ordinary annuity
Find the perpetuity?PV of this 3-year ordinary annuity.

  • Could solve by discounting each cash flow, or …

  • Use the EAR and treat as an annuity to solve for PV.

3

10.25

100

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

-247.59


Loan amortization
Loan amortization perpetuity?

  • Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, etc.

  • Financial calculators and spreadsheets are great for setting up amortization tables.

  • EXAMPLE: Construct an amortization schedule for a $1,000, 10% annual rate loan with 3 equal payments.


Step 1 find the required annual payment
Step 1: perpetuity?Find the required annual payment

  • All input information is already given, just remember that the FV = 0 because the reason for amortizing the loan and making payments is to retire the loan.

3

10

-1000

0

INPUTS

N

I/YR

PV

PMT

FV

OUTPUT

402.11


Step 2 find the interest paid in year 1
Step 2: perpetuity?Find the interest paid in Year 1

  • The borrower will owe interest upon the initial balance at the end of the first year. Interest to be paid in the first year can be found by multiplying the beginning balance by the interest rate.

    INTt = Beg balt (I)

    INT1 = $1,000 (0.10) = $100


Step 3 find the principal repaid in year 1
Step 3: perpetuity?Find the principal repaid in Year 1

  • If a payment of $402.11 was made at the end of the first year and $100 was paid toward interest, the remaining value must represent the amount of principal repaid.

    PRIN = PMT – INT

    = $402.11 - $100 = $302.11


Step 4 find the ending balance after year 1
Step 4: perpetuity?Find the ending balance after Year 1

  • To find the balance at the end of the period, subtract the amount paid toward principal from the beginning balance.

    END BAL = BEG BAL – PRIN

    = $1,000 - $302.11

    = $697.89


Constructing an amortization table repeat steps 1 4 until end of loan
Constructing an amortization table: perpetuity?Repeat steps 1 – 4 until end of loan

  • Interest paid declines with each payment as the balance declines. What are the tax implications of this?


Illustrating an amortized payment where does the money go
Illustrating an amortized payment: perpetuity?Where does the money go?

$

  • Constant payments.

  • Declining interest payments.

  • Declining balance.

402.11

Interest

302.11

Principal Payments

0

1

2

3


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