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The Time Value of Money

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The Time Value

of Money

Chapter 8

October 3, 2012

- 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.

- 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?

- 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

- 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!

- 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.

- 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

FV = PV (1 + k)n

- 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

Calculator solution:

N = 15

I/Y = 8

PV = -$100

PMT = 0

Compute (CPT) FV = $317.22

1

(1 + k)n

PV = FVn x

0 1 2

100

(1.10)1

PV =

=

- 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

$

1

(1 + k)n

PV = FVn x

0 1 2 3 4 5 6 7 8

100

(1+.10)8

=

PV =

- 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

100

(1+.10)8

= 46.65

PV =

Using Formula:

N

I/YR

PV

PMT

FV

100

8 10 ?

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

Jan Feb Mar Dec

$500

$500

$500

$500

$500

- 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.

0 1 2 3

$0

$100

$100

$100

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?

0 1 2 3

$0

$100

$100

$100

$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?

$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( )

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

0 1 2 3

$0

$100

$100

$100

N

I/YR

PV

PMT

FV

Enter:

N = 3

I/YR = 8

PV= 0

PMT = -100

CPT FV = ?

324.64

3 8 0 -100 ?

0 1 2 3

$0

$100

$100

$100

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

0 1 2 3

$0

$100

$100

$100

- 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

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

- 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

0 1 2 3

$0

$100

$100

$100

N

I/YR

PV

PMT

FV

PV=?

Enter:

N = 3

I/YR = 8

PMT = 100

FV= 0

CPT PV = ?

-257.71

3 8 ? 100 0

- 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

0 1 2 3

$100

$100

$100

FVA=?

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

0 1 2 3

$100

$100

$100

$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.

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)

=100(3.2464)(1.08)=$350.61

- 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

0 1 2 3

$100

$100

$100

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

PV=?

- 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

$100/(1.08)1

$100 / (1.08)2

$100/(1.08)0

$100.00

$92.60

$85.73

$278.33

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

- 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

- 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

- 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)

0 1 2 3 4 5

$5,000

$?

$?

$?

$?

$?

N

I/YR

PV

PMT

FV

- 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

- 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%

1 -

)

(

= PMT

$20,000

1

(1.0075)48

.0075

1

(1+k)n

1 -

PVA = PMTx( )

k

- 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

N

I/YR

PV

PMT

FV

- 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

PMT

k

PVP =

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

PMT

k

PVP =

- 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%?

0 1 2

$200

$230

FV= PV(1+ k)n

1.15 = (1+ k)2

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%

N

I/YR

PV

PMT

FV

2

?

-200

230

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

N = 1.9995, or 2 years

N

I/YR

PV

PMT

FV

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

- $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