The Time Value
This presentation is the property of its rightful owner.
Sponsored Links
1 / 41

The Time Value of Money PowerPoint PPT Presentation


  • 40 Views
  • Uploaded on
  • Presentation posted in: General

The Time Value of Money. Chapter 8. October 3, 2012. Learning Objectives. The “time value of money” and its importance to you and business decisions The future value and present value of a single amount. The future value and present value of an annuity.

Download Presentation

The Time Value of Money

An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -

Presentation Transcript


The time value of money

The Time Value

of Money

Chapter 8

October 3, 2012


Learning objectives

Learning Objectives

  • The “time value of money” and its importance to you and business decisions

  • The future value and present value of a single amount.

  • The future value and present value of an annuity.

  • The present value of a series of uneven cash flows.


The time value of money

The Time Value of Money

  • Money grows in amount over time as it earns from investments.

  • However, money that is to be received at some time in the future is worth less than the same dollar amount to be received today. Why?

  • Similarly, a debt of a given amount to be paid in the future is less burdensome than that debt to be paid now. Why?


Some examples

Some Examples

  • Bought Oakland house for $29,500 in 1969

    $23,600 mortgage, $175 mo. pymt

    I bought my house in Los Altos in 1979 for $135,000

    $40,000 30 yr mortgage, $300 mo

    In 2009, would still paying $300 mo!

    House sold for over $1.25 million in 2006

    Current owner paying $5,500 per month

    I now own $935,000 home, no mortgage!

    Time value of money


Indians manhattan island

Indians – Manhattan Island

  • In 1624, Indians got $24 for Manhattan island

  • People think they were “taken”

  • If invested at 8%, compounded annually, today they would have $223,166,200,000,000 (trillion)

  • If compounded semiannually, $396 trillion

  • If compounded quarterly, $534 trillion

  • You could buy Manhattan Island today for around $500 billion

  • They could pay off the nat’l debt/buy back US!

  • Time value of money!


16 year old saves for retirement

16 year old saves for retirement!

  • Earns $2,000 per year for 6 years/stops

  • Reinvests at 10% per year

  • At 21 years old, she is worth $15,431

  • At age 65, with no add’l investment, if she just lets it ride, she will be worth $1,022,535

  • If she waits just one more year to get started, she would be worth only $929,578

  • She loses $92,957! (final years earnings)

  • So start saving now! You’ll never miss it.


The future value of a single amount

The Future Value of a Single Amount

  • Suppose that you have $100 today and plan to put it in a bank account that earns 8% (k) per year.

  • How much will you have after 1 year?

  • After one year:$100 + (.08 x $100) = $100 + $8 = $108Or

  • If k = 8%, then 1 + k = 1 + .08 or 1.08Then, $100 x (1.08)1 = $108


The future value of a single amount1

FV = PV (1 + k)n

The Future Value of a Single Amount

  • Suppose that you have $100 today and plan to put it in a bank account that earns 8% per year.

  • How much will you have after 1 year? 5? 15?

  • After one year:$100 x (1.08)1 = $100 x 1.08 = $108

  • After five years:

    $100 x 1.08 x 1.08 x 1.08 x 1.08 x 1.08 = $146.93

    $100 x (1.08)5 = $100 x 1.4693 = $146.93

  • After fifteen years:

    $100 x (1.08)15 = $100 x 3.1722* = $317.22

  • Equation:

*Table I, p. A-1

Appendix


The future value of a single amount2

The Future Value of a Single Amount

Calculator solution:

N = 15

I/Y = 8

PV = -$100

PMT = 0

Compute (CPT) FV = $317.22


Present value of a single amount

1

(1 + k)n

PV = FVn x

0 1 2

100

(1.10)1

PV =

=

Present Value of a Single Amount

  • Value today of an amount to be received or paid in the future.

*Table II, p. A-2, Appendix

Example: Expect to receive $100 in one year. If can invest at 10%, what is it worth today?

$100

$100 x .9091* = $90.91

$


Present value of a single amount1

1

(1 + k)n

PV = FVn x

0 1 2 3 4 5 6 7 8

100

(1+.10)8

=

PV =

Present Value of a Single Amount

  • Value today of an amount to be received or paid in the future.

Example: Expect to receive $100 in EIGHT years. If can invest at 10%, what is it worth today?

$100

$100 x .4665* = $46.65

*Table II, p. A-2, Appendix


Financial calculator solution pv

100

(1+.10)8

= 46.65

PV =

Using Formula:

N

I/YR

PV

PMT

FV

100

8 10 ?

Financial Calculator Solution - PV

Previous Example: Expect to receive $100 in EIGHT years. If can invest at 10%, what is it worth today?

Calculator Enter:

N = 8

I/YR= 10

PMT = 0

FV= 100

CPT PV = ?

- 46.65

0


Annuities

Jan Feb Mar Dec

$500

$500

$500

$500

$500

Annuities

  • An annuity is a series of equal cash flows spaced evenly over time.

  • For example, you pay your landlord an annuity since your rent is the same amount, paid on the same day of the month for the entire year.


Future value of an annuity

0 1 2 3

$0

$100

$100

$100

Future Value of an Annuity

You deposit $100 each year (end of year) into a savings account (saving up for an IPad).

How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?


Future value of an annuity1

0 1 2 3

$0

$100

$100

$100

Future Value of an Annuity

$100(1.08)2

$100(1.08)1

$100(1.08)0

$100.00

$108.00

$116.64

$324.64

You deposit $100 each year (end of year) into a savings account.

How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?


Future value of an annuity2

$100(1.08)2

$100(1.08)1

$100(1.08)0

$100.00

$108.00

$116.64

$324.64

)

(1+.08)3 - 1

.08

(

= 100

n

(1+k) - 1

k

FVA = PMTx( )

Future Value of an Annuity

0 1 2 3

$0

$100

$100

$100

How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?

= 100(3.2464*) = $324.64

*Table III, p. A-3, Appendix


Future value of an annuity calculator solution

0 1 2 3

$0

$100

$100

$100

N

I/YR

PV

PMT

FV

Future Value of an AnnuityCalculator Solution

Enter:

N = 3

I/YR = 8

PV= 0

PMT = -100

CPT FV = ?

324.64

3 8 0 -100 ?


Present value of an annuity

0 1 2 3

$0

$100

$100

$100

Present Value of an Annuity

  • How much would the following cash flows be worth to you today if you could earn 8% on your deposits?


Present value of an annuity1

0 1 2 3

$0

$100

$100

$100

Present Value of an Annuity

  • How much would the following cash flows be worth to you today if you could earn 8% on your deposits?

$100/(1.08)1

$100 / (1.08)2

$100 / (1.08)3

$92.60

$85.73

$79.38

$257.71


Present value of an annuity2

0 1 2 3

$100/(1.08)1

$100 / (1.08)2

$100 / (1.08)3

$92.60

$0

$100

$100

$100

$85.73

$79.38

1

(1.08)3

$257.71

1 -

(

)

= 100

1

(1+k)n

1 -

.08

PVA = PMTx( )

k

Present Value of an Annuity

  • How much would the following cash flows be worth to you today if you could earn 8% on your deposits?

= 100(2.5771*) = $257.71

*Table IV, p. A-4, Appendix


Present value of an annuity calculator solution

0 1 2 3

$0

$100

$100

$100

N

I/YR

PV

PMT

FV

Present Value of an AnnuityCalculator Solution

PV=?

Enter:

N = 3

I/YR = 8

PMT = 100

FV= 0

CPT PV = ?

-257.71

3 8 ? 100 0


Annuity due

Annuity Due

  • An annuity is a series of equal cash payments spaced evenly over time.

  • Ordinary Annuity: The cash payments occur at the END of each time period.

  • Annuity Due: The cash payments occur at the BEGINNING of each time period.

  • Lotto is an example of an annuity due


Future value of an annuity due

0 1 2 3

$100

$100

$100

FVA=?

Future Value of an Annuity Due

You deposit $100 each year (beginning of year) into a savings account.

How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?


Future value of an annuity due1

0 1 2 3

$100

$100

$100

Future Value of an Annuity Due

$100(1.08)3

$100(1.08)2

$100(1.08)1

$108

$116.64

$125.97

$350.61

You deposit $100 each year (beginning of year) into a savings account.

How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?


Future value of an annuity due2

0 1 2 3

(1+k)n - 1

k

FVA= PMTx( )

$108

$100(1.08)3

$100(1.08)2

(1+k)

$100(1.08)1

$100

$100

$100

$116.64

$125.97

$350.61

)

(

(1+.08)3 - 1

.08

= 100

(1.08)

Future Value of an Annuity Due

How much would this account have in it at the end of 3 years if interest were earned at a rate of 8% annually?

=100(3.2464)(1.08)=$350.61


Calculator solution to annuity due

Calculator solution to annuity due

  • Same as regular annuity, except

  • Multiply your answer by (1 + k) to account for the additional year of compounding or discounting

  • Future value of an annuity due:

    n = 3, i/y = 8%, pmt = -100, PV = 0

    CPT FV = 324.64 (1.08) = 350.61


Present value of an annuity due

0 1 2 3

$100

$100

$100

Present Value of an Annuity Due

  • How much would the following cash flows be worth to you today if you could earn 8% on your deposits?

PV=?


Present value of an annuity due1

  • How much would the following cash flows be worth to you today if you could earn 8% on your deposits?

0 1 2 3

$100

$100

$100

Present Value of an Annuity Due

$100/(1.08)1

$100 / (1.08)2

$100/(1.08)0

$100.00

$92.60

$85.73

$278.33


Present value of an annuity due2

0 1 2 3

$100

$100

$100

1

(1.08)3

1 -

(

)

(1.08)

= 100

1

(1+k)n

1 -

.08

(1+k)

PVA = PMTx()

k

Present Value of an Annuity Due

  • How much would the following cash flows be worth to you today if you could earn 8% on your deposits?

$100/(1.08)1

$100 / (1.08)2

$100/(1.08)0

$100.00

$92.60

$85.73

$278.33

= 100(2.5771)(1.08) = 278.33


Calculator solution to annuity due1

Calculator solution to annuity due

  • Same as regular annuity, except

  • Multiply your answer by (1 + k) to account for the additional year of compounding or discounting

  • Present value of an annuity due:

    N = 3, i/y = 8%, PMT = 100, FV = 0,

    CPT PV = -257.71 (1.08) = -278.33


Amortized loans

Amortized Loans

  • A loan that is paid off in equal amounts that include principal as well as interest.

  • Solving for loan payments (PMT).

  • Note: The amount of the loan is the present value (PV)


Amortized loans1

0 1 2 3 4 5

$5,000

$?

$?

$?

$?

$?

N

I/YR

PV

PMT

FV

Amortized Loans

  • You borrow $5,000 from your parents to purchase a used car. You agree to make payments at the end of each year for the next 5 years. If the interest rate on this loan is 6%, how much is your annual payment?

ENTER:

N = 5

I/YR = 6

PV= 5,000

FV= 0

CPT PMT = ?

–1,186.98

5 6 5,000 ? 0


Compounding more than once per year

Compounding more than once per Year

  • If m = number of compounds, then

    N = n x m and K = k / m

  • Annual i.e.N = 4K = 12%

  • Semi-annualN = 4 x 2 = 8

  • K = 12% / 2 = 6%

  • QuarterlyN = 4 x 4 = 16

  • K = 12% / 4 = 3%

  • MonthlyN = 4 x 12 = 48

  • K = 12% / 12 = 1%


Amortized loans2

1 -

)

(

= PMT

$20,000

1

(1.0075)48

.0075

1

(1+k)n

1 -

PVA = PMTx( )

k

Amortized Loans

  • You borrow $20,000 from the bank to purchase a used car. You agree to make payments at the end of each month for the next 4 years. If the annual interest rate on this loan is 9%, how much is your monthly payment?

$20,000 = PMT(40.184782)

PMT = 497.70

Note: Tables no longer work


Amortized loans3

N

I/YR

PV

PMT

FV

Amortized Loans

  • You borrow $20,000 from the bank to purchase a used car. You agree to make payments at the end of each month for the next 4 years. If the annual interest rate on this loan is 9%, how much is your monthly payment?

ENTER:

N = 48

I/YR = .75

PV= 20,000

FV= 0

CPT PMT = ?

– 497.70

Note:

N = 4 * 12 = 48

I/YR = 9/12 = .75

48 .75 20,000 ? 0


Perpetuities

PMT

k

PVP =

Perpetuities

  • A perpetuity is a series of equal payments at equal time intervals (an annuity) that will be received into infinity.


Perpetuities1

PMT

k

PVP =

Perpetuities

  • A perpetuity is a series of equal payments at equal time intervals (an annuity) that will be received into infinity (i.e., retirement payments)

If k = 8%: PVP = $5/.08 = $62.50

Proof: $62.50 x .08 = $5.00

Example:A share of preferred stock pays a constant dividend of $5 per year. What is the present value if k =8%?


Solving for k

0 1 2

$200

$230

FV= PV(1+ k)n

1.15 = (1+ k)2

Solving for k

Example: A $200 investment has grown to $230 over two years. What is the ANNUAL return on this investment?

230 = 200(1+ k)2

1.15 = (1+ k)2

1.0724 = 1+ k

k = .0724 = 7.24%


Solving for k calculator solution

N

I/YR

PV

PMT

FV

2

?

-200

230

Solving for k - Calculator Solution

Example: A $200 investment has grown to $230 over two years. What is the ANNUAL return on this investment?

Enter known values:

N=2

I/YR=?

PV= -200

PMT = 0

FV= 230

Solve for:

I/YR= ?

7.24

0


Solving for n

N = 1.9995, or 2 years

N

I/YR

PV

PMT

FV

Solving for N

Example: A $200 investment has grown to $230. If the ANNUAL return on this investment is 7.24%, how long would it take?

  • Enter known values:

  • N= ?

  • I/YR= 7.24

  • PV= -200

  • PMT = 0

  • FV= 230

? 7.24 -200 0 230


Compounding more than once per year1

Compounding more than Once per Year

  • $500 invested at 9% annual interest for 2 years. Compute FV.

Compounding Frequency

$500(1.09)2= $594.05 Annual

$500(1.045)4= $596.26 Semi-annual

$500(1.0225)8= $597.42 Quarterly

$500(1.0075)24= $598.21 Monthly

$500(1.000246575)730= $598.60 Daily


  • Login