TMV and Its Importance

7/31/2019 TMV and Its Importance

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The Time Value

of Money



The Time Value of Money

• Money grows over time when it earnsinterest.

• Therefore, money that is to be received atsome time in the future is worth less thanthe same dollar amount to be receivedtoday. Why?

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

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Some Examples

• Bought Oakland house for $29,500 in 1969

$23,600 mortgage, $244 mo. pymt

I bought my house in Los Altos in 1979 for $135,000$70,000 30 yr mortgage, $500 mo

In 2009, would still paying $500 mo!

House sold for over $1.25 million in 2006

Current owner paying $5,500 per monthI now own $935,000 home, no mortgage!

Time value of money



Indians – Manhattan Island

• In 1624, Indians got $24 for Manhattan island

• People think they were “taken”

• If invested at 8%, compounded annually, todaythey would have $206,635,300,000,000 (trillion)

• If compounded semiannually, $366 trillion

• If compounded quarterly, $493 trillion

• You could buy Manhattan today for around acouple $100 billion

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

• Time value of money!



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 justlets it ride, she will be worth $1,022,535

• If she waits just one more year to get started, shewould 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

• Suppose that you have $100 today andplan 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 = $108

Or• If k = 8%, then 1 + k = 1 + .08 or 1.08 Then,

$100 x (1.08)1 = $108

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The Future Value of a SingleAmount• 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?

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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:FV = PV (1 + k)n Table I, p. A-1

Appendix



The Future Value of a Single Amount

Calculator solution:

N = 15

I/Y = 8PV = -$100

PMT = 0

Compute (CPT) FV = $317.22



Present Value of a Single Amount

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

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

0 1 2

$100 PV = 100

(1.10)1=

PV = FVn

x 1

(1 + k)n

Table II, p. A-2, Appendix

$$100 x .9091 = $90.91



Present Value of a Single Amount

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

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

=PV =100

(1+.10)8

0 1 2 3 4 5 6 7 8

$100

PV = FVn x 1

(1 + k)n

$100 x .4665 = $46.65



Financial Calculator Solution - PV

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N I/YR PV PMT FV

- 46.65

PV =100

(1+.10)8

= 46.65Using Formula:

8 10 ? 100

Previous Example: Expect to receive $100 in EIGHTyears. If can invest at 10%, whatis it worth today?

Calculator Enter:N = 8I/YR = 10PMT = 0FV = 100

CPT PV = ?

0



Financial Calculator Solution - FV

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N I/YR PV PMT FV

322.10

5 10 -200 0 ?

Calculator Enter:N = 5

I/YR = 10PV = -200PMT = 0CPT FV = ?

Using Formula: FV = $200 (1.10)5

= $322.10

Previous Example: You invest $200 at 10%. Howmuch is it worth after 5 years?



Annuities

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

• For example, you pay your landlord anannuity since your rent is the sameamount, paid on the same day of themonth for the entire year.

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$500 $500 $500 $500 $500

Jan Feb Mar Dec



Future Value of an Annuity

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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 3years if interest were earned at a rate of 8% annually?




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$100(1.08)2 $100(1.08)1

$108.00$116.64$324.64

You deposit $100 each year (end of year) into asavings 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?

0 1 2 3

$0 $100 $100 $100

$100.00

$100(1.08)0




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

= 100(3.2464) = $324.64FVA = PMTx( ) (1+k) - 1

k

n = 100 (1+.08)3 - 1.08

( )

0 1 2 3

$0 $100 $100 $100 $100(1.08)2 $100(1.08)1

$108.00$116.64$324.64

$100.00

$100(1.08)0

Table III . A-3 A endix

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Future Value of an AnnuityCalculator Solution

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N I/YR PV PMT FV

3 8 0 -100 ?

Enter:N = 3

I/YR = 8PV = 0PMT = -100CPT FV = ?

0 1 2 3

$0 $100 $100 $100

324.64

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Present Value of an Annuity

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

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0 1 2 3

$0 $100 $100 $100

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$100 / (1.08)2

$92.60$85.73$79.38

$100/(1.08)1 $100 / (1.08)3

$257.71

How much would the following cash flows be worthto you today if you could earn 8% on yourdeposits?

0 1 2 3

$0 $100 $100 $100

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

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= 100(2.5771) = $257.71PVA = PMTx( )

1(1+k)n1 -

k

$100 / (1.08)2

$92.60$85.73$79.38

$100/(1.08)1 $100 / (1.08)3

$257.71

0 1 2 3

$0 $100 $100 $100

.08= 100

1 - 1(1.08)3( )

Table IV, p. A-4, Appendix

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Calculator Solution

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N I/YR PV PMT FV

3 8 ? 100 0

PV=?

Enter:N = 3I/YR = 8PMT = 100FV = 0CPT PV = ?

0 1 2 3

$0 $100 $100 $100

-257.71



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Future Value of an Annuity Due

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

savings account.


0 1 2 3

$100 $100 $100 FVA=?

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24

$100(1.08)2 $100(1.08)1$100(1.08)3

$108$116.64

$125.97

$350.61

0 1 2 3

$100 $100 $100

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


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25

$100(1.08)2 $100(1.08)1$100(1.08)3

$108

$116.64

$125.97

$350.61

0 1 2 3

$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)(1.08)=$350.61

FVA = PMTx( ) (1+k)(1+k)n - 1k

(1+.08)3 - 1.08

= 100 (1.08)( )



Calculator solution to annuity due

• Same as regular annuity, except

• Multiply your answer by (1 + k) to account for theadditional 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

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Present Value of an Annuity Due

• How much would the following cash flows be

worth to you today if you could earn 8% onyour deposits?

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PV=?

0 1 2 3

$100 $100 $100

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Present Value of an Annuity Due

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$100 / (1.08)2

$92.60$85.73

$100/(1.08)1

$278.33

0 1 2 3

$100 $100 $100

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

$100.00

$100/(1.08)0

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Present Value of an Annuity Due29

= 100(2.5771)(1.08) = 278.33PVA = PMTx( )

1(1+k)n1 -

k(1+k)

$100 / (1.08)2

$92.60$85.73

$100/(1.08)1

$278.33

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?

.08= 100

1 - 1(1.08)3

(1.08)( )

$100.00

$100/(1.08)0



Calculator solution to annuity due

• Same as regular annuity, except

• Multiply your answer by (1 + k) to account for theadditional 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

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Amortized Loans

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

• Solving for loan payments (PMT).

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

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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 thenext 5 years. If the interest rate on this loan is 6%, how much isyour annual payment?

3

N I/YR PV PMT FV

0 1 2 3 4 5

$5,000 $? $? $? $? $?

–1,186.98

5 6 5,000 ? 0

ENTER:N = 5

I/YR = 6PV = 5,000FV = 0CPT PMT = ?



Compounding more than once per Year

• If m = number of compounds, thenN = n x m and K = k / m

• Annual i.e. N = 4 K = 12%

• Semi-annual N = 4 x 2 = 8

• K = 12% / 2 = 6%

• Quarterly N = 4 x 4 = 16

• K = 12% / 4 = 3%• Monthly N = 4 x 12 = 48

• K = 12% / 12 = 1%

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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 forthe next 4 years. If the annual interest rate on this loan is 9%,how much is your monthly payment?

$20,000 = PMT(40.184782)PVA = PMTx( )

1

(1+k)n1 -k

PMT = 497.70

= PMT.0075

1 -1

(1.0075)48

$20,000 ( )

Note: Tables no longer work

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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 forthe next 4 years. If the annual interest rate on this loan is 9%,how much is your monthly payment?

ENTER:N = 48I/YR = .75PV = 20,000

FV = 0CPT PMT = ?

N I/YR PV PMT FV

– 497.70

48 .75 20,000 ? 0

Note:

N = 4 * 12 = 48

I/YR = 9/12 = .75

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Perpetuities

• A perpetuity is a series of equal paymentsat equal time intervals (an annuity) that willbe received into infinity.

PMTk

PVP =

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Perpetuities

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

Proof: $62.50 x .08 = $5.00

PMTkPVP =

A perpetuity is a series of equal payments at equaltime intervals (an annuity) that will be received intoinfinity (i.e., retirement payments)

Example: A share of preferred stock pays a constantdividend of $5 per year. What is the presentvalue if k =8%?

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Solving for k

Example: A $200 investment has grown to $230 overtwo years. What is the ANNUAL return onthis investment?

0 1 2

$230 $200

FV = PV(1+ k)n

230 = 200(1+ k)2

1.15 = (1+ k)2

1.0724 = 1+ k

1.15 = (1+ k)2

k = .0724 = 7.24%

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Solving for k - Calculator Solution

N I/YR PV PMT FV

Enter known values:

N = 2I/YR = ?PV = -200PMT = 0FV = 230

Solve for:

I/YR = ?

2 -200 230?

Example: A $200 investment has grown to $230 overtwo years. What is the ANNUAL return onthis investment?

7.24

0



Solving for N

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

• Enter known values:

• N = ?

• I/YR = 7.24

• PV = -200

• PMT = 0• FV = 230

N = 1.9995, or 2 years

N I/YR PV PMT FV

? 7.24 -200 0 230

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Compounding more than Once per Year

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

$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

Compounding

Frequency

TMV and Its Importance

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