1 Time Value of Money (TVM) - the Intuition A cash flow today is worth more than a cash flow in the future since: uIndividuals prefer present consumption

1

Time Value of Money (TVM) - the Intuition

A cash flow today is worth more than a cash flow

in the future since:

Individuals prefer present consumption to future consumption.

Monetary inflation will cause tomorrow’s dollars to be worth less than today’s.

Any uncertainty associated with future cash flows reduces the value of the cash flow.

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The Time-Value-of-Money

The Basic Time-Value-of-Money Relationship:

FVt+T = PVt X (1 + r)T

where r is the interest rate per period T is the duration of the investment, stated in the

compounding time unit PVt is the value at period t (beginning of the

investment) FVt+T is the value at period t+T (end of the investment)

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Future Value and CompoundingCompounding:

How much will $1 invested today at 8% be worth in two years?(The Time Line)

Year 0 1 2

$1.1664

Future Value: FV2 = $1 x 1.082 = $1.1664

$1.1664

$1

$1

$1.08$1.08 x 1.08

$1 x 1.08

Or:

4

Housekeeping functions:

1. Set to 8 decimal places:

2. Clear previous TVM data:

3. Set payment at Beginning/End of Period:

3. Set # of times interest is calculated (compounded) per year to 1:

TVM in your HP 10B Calculator

Yellow =

DISP

8

INPUT

CLEAR ALL

0

BEG/END

Yellow

Yellow

PMT

P/YR

Yellow1

5

First, clear previous data, and check that your calculator is set to 1 P/YR:

The display should show: 1 P_Yr

Input data (based on above FV example)

FV in your HP 10B Calculator

INPUT

CLEAR ALL

+/-

Yellow

1 PV

8 I/YR

2 N

FV

Key in PV (always -ve)

Key in interest rate

Key in number of periods

Compute FV Display should show: 1.1664

6

Q. Deposit $5,000 today in an account paying 12%. How much will you have in 6 years? How much is simple interest? How much is compound interest?

A. Multiply the $5000 by the future value interest factor:

$5000 x(1 + r)T= $5000 x ( )6

= $5000 x 1.9738227

= $__________

At 12%, the simple interest is ___ x $5000 = $ ___ peryear. After 6 years, this is 6 x $____ = $ ______;

the compound interest is thus:

$4869.11 - $3600 = $1,269.11

An Example - Future Value for a Lump Sum

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(The Time Line)

Year 0 1 2

$1

$1$0.9259$1 / 1.08

$0.9259 / 1.08

Or:

Present Value and DiscountingDiscounting: How much is $1 that we will receive in two years worth today (r = 8%)?

$0.8573

$0.8573

Present Value: PV0 = $1 / 1.082 = $0.8573

8



Input data (based on above PV example)

PV in your HP 10B Calculator

INPUT

CLEAR ALL

Yellow

1 FV

8 I/YR

2 N

PV

Key in FV



Compute PV Display should show: -0.85733882

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Q. Suppose you need $20,000 in three years to pay your university tuition. If you can earn 8% annual interest on your money, how much do you need to invest today?

A. We know the future value ($20,000), the rate (8%), and the number of periods (3). We are looking for the present amount to be invested (present value). We first define the variables:

FV3 = $20,000 r = 8 percent

T= 3 years PV0 = ?

Set this up as a TVM equation and solve for the present value:

________ = PV0 x (_____)--

Solve for PV:

PV0 = $_________________ = $15,876.64

$15,876.64 invested today at 8% annually, will grow to $20,000 in three years.

Example 1 - Present Value of a Lump Sum

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Q. Suppose you are currently 21 years old, and can earn 10 percent on your money. How much must you invest today in order to accumulate $1 million by the time you reach age 65?

A. We first define the variables:

FV65 = _______ r = _______

T= __________ PV21 = ?

Set this up as a TVM equation and solve for the present value:

_________ = PV21 x (_________)44

Solve for PV:

PV21 = _____________ = $15,091.13

If you invest $15,091.13 today at 10% annually, you will have $1 million by the time you reach age 65

Example 2 - Present Value of a Lump Sum

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How Long is the Wait?

If we deposit $5000 today in an account paying 10%, how long do we have to wait for it to grow to $10,000?

Solve for T:

FVt+T = PVt x (1 + r)T

$10000 = $5000 x (1.10)T

(1.10)T = 2

T = ln(2) / ln(1.10)

= 7.27 years

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Input data (based on above example)

T in your HP 10B Calculator

INPUT

CLEAR ALL

Yellow

FV

I/YR

N

Key in FV


Compute T Display should show: 7.27254090

10,000

Key in PV +/- PV5,000

10

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An Example - How Long is the Wait?

Q. You have $70,000 to invest. You decided that by the time this investment grows to $700,000 you will retire. Assume that you can earn 14 percent annually. How long do you have to wait for your retirement?

A. We first define the variables:

FV? = $700,000 r = 14 percent

PV0 = $70,000 T= ?

Set this up as a TVM equation and solve for T:

_______________________

Solve for T:

T = ln(10)/ln(1.14) = 17.57 years

If you invest $70,000 today at 14% annually, you will reach your goal of $700,000 in 17.57 years

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Assume the total cost of a University education will be $50,000 when your child enters college in 18 years.

You have $5,000 to invest today.

What rate of interest must you earn on your investment to cover the cost of your child’s education?

Solve for r :FVt+T = PVt x (1 + r)T

$50000 = $5000 x (1 + r)18

(1 + r)18 = 10____________________r = 0.13646 = 13.646% per year

What Rate Is Enough?

15




Interest Rate (r) in your HP 10B Calculator

INPUT

CLEAR ALL

Yellow

FV

N

I/YR

Key in FV

Key in T

Compute r Display should show: 13.64636664

50,000

Key in PV +/- PV5,000

18

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An Example - Finding the Interest Rate (r):

Q. In December 1937, the market price of an ABC company common stock was $3.37. According to The Financial Post, the price of an ABC company common stock in December 1999 is $7,500. What is the annually compounded rate of increase in the value of the stock?

A. Set this up as a TVM problem.

Future value = ________ Present value = _________

T = _________ r = _________

FV1999 = PV1937 x (1 + r)T so,

Solve for r:

r = _____________________ = .1324 = 13.24%

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Net Present Value (NPV)Example for NPV:

You can buy a property today for $3 million, and sell it in 3 years for

$3.6 million. The annual interest rate is 8%.

Qa. Assuming you can earn no rental income on the property, should

you buy the property?

Aa. The present value of the cash inflow from the sale is:

PV0 = $3,600,000/(1.08)3 = $2,857,796.07

Since this is less than the purchase price of $3 million - don’t buy

We say that the Net Present Value (NPV) of this investment is

negative:

NPV = -C0 + PV0(Future CFs)

= -3,000,000+2,857,796.07

= -142,203.93 < 0

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Example for NPV (continued):

Qb. Suppose you can earn $200,000 annual rental income (paid at the

end of each year) on the property, should you buy the

property now?

Ab. The present value of the cash inflow from the sale is:

PV0 = [200,000 /1.08] + [200,000 /1.082] + [3,800,000/1.083]

= $3,373,215.47

Since this is more than the purchase price of $3 million - buy

We say that the Net Present Value (NPV) of this investment is

positive:

NPV = -C0 + PV0(Future CFs)

= -3,000,000+ 3,373,215.47

= 373,215.47 > 0

The general formula for calculating NPV:

NPV = -C0 + C1/(1+r) + C2/(1+r)2 + ... + CT/(1+r)T

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Simplifications Perpetuity

A stream of constant cash flows that lasts forever

Growing perpetuityA stream of cash flows that grows at a constant

rate forever

AnnuityA stream of constant cash flows that lasts for a

fixed number of periods

Growing annuityA stream of cash flows that grows at a constant

rate for a fixed number of periods

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PerpetuityA Perpetuity is a constant stream of cash flows without end.

Simplification: PVt = Ct+1 / r

0 1 2 3 …forever...|---------|--------|---------|--------- (r = 10%)

$100 $100 $100 ...forever…

PV0 = $100 / 0.1 = $1000

The British consol bond is an example of a perpetuity.

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Q1. ABC Life Insurance Co. is trying to sell you an investment policy that will pay you and your heirs $1,000 per year (starting next year) forever. If the required annual return on this investment is 13 percent, how much will you pay for the policy?

A1. The most a rational buyer would pay for the promised cash flows is

C/r = $1,000/0.13 = $7,692.31

Q2. ABC Life Insurance Co. tells you that the above policy costs $9,000. At what interest rate would this be a fair deal?

A2. Again, the present value of a perpetuity equals C/r. Now solve the following equation:

$9,000 = C/r = $1,000/r

r = 0.1111 = 11.11%

Examples - Present Value for a Perpetuity

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Growing Perpetuity

A growing perpetuity is a stream of cash flows that grows at a constant rate forever.

Simplification: PVt = Ct+1 / (r - g)

0 1 2 3 …forever...

|---------|---------|---------|--------- (r = 10%)

$100 $102 $104.04 … (g = 2%)

PV0 = ________________ = $1250

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Q. Suppose that ABC Life Insurance Co. modifies the policy, such that it will pay you and your heirs $1,000 next year, and then increase each payment by 1% forever. If the required annual return on this investment is 13 percent, how much will you pay for the policy?

A. The most a rational buyer would pay for the promised cash flows is

__________________ = $8,333.33

Note: Everything else being equal, the value of the growing perpetuity is always higher than the value of the simple perpetuity, as long as g>0.

An Example - Present Value for a Growing Perpetuity

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Annuity An annuity is a stream of constant cash flows that lasts for a fixed number of

periods.

Simplification: PVt = Ct+1 (1/r){1 - [1 / (1 + r)T]}

FVt+T = Ct+1 (1/r){[(1 + r)T] - 1}

0 1 2 3 years|----------|---------|---------| (r = 10%)

$100 $100 $100

PV0 = 100 (1/0.1){1 - [ 1/(1.13)]} = $248.69

FV3 = 100 (1/0.1){[1.13 ] - 1} = $331

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Input data (based on above PV example)

PV and FV of Annuity in your HP 10B Calculator

INPUT

CLEAR ALL

Yellow

PMT

I/YR

3 N

PV

Key in payment



Compute PV Display should show: -248.68519910

100

FVCompute FV * Display should show: -331.00000000

PV0

* Note: you can calculate FV directly, by following first 3 steps, and replacing

PV with FV in the fourth step.

10

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Q. A local bank advertises the following: “Pay us $100 at the end of the next 10 years. We will pay you (or your beneficiaries) $100, starting at the eleventh year forever.” Is this a good deal if if the effective annual interest rate is 8%?

A. We need to compare the PV of what you pay with the present value of what you get:

- The present value of your annuity payments:

PV0 = 100 (1/0.08){1 - [ 1/(1.0810)]} = $671.01

- The present value of the bank’s perpetuity payments at the

end of the tenth year (beginning of the eleventh year):

PV10 = C11/r = (100/0.08) = $1,250

The present value of the bank’s perpetuity payments today:

PV0 = PV10 /(1+r)10 = (100/0.08)/(1.08)10 = = $578.99

Present Value of an Annuity - Example 1

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Q. You take $20,000 five-year loan from the bank, carrying a 0.6% monthly interest rate. Assuming that you pay the loan in equal monthly payments, what is your monthly payment on this loan?

A. Since payments are made monthly, we have to count our time units in months. We have: T = 60 monthly time periods in five years, with a monthly interest rate of: r = 0.6%, and PV0 = 20,000

With the above data we have:

20,000 = C (1/0.006){1 - [ 1/(1.00660)]}

Solving for C, we get a monthly payment of: $397.91.

Note: you can easily solve for C in your calculator, by keying:

Present Value of an Annuity - Example 2

PV1) Key in the PV

+/_ 20,000

I/YR2) Key in interest

rate0.6

N3) Key in # of

payments60

PMT4) Compute PMT

Display should show: 397.91389639

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Annuity Due –Self Study An annuity due is a stream of constant cash flows that is paid at the beginning

of each period and lasts for a fixed number of periods (T).

Simplification: PVt = Ct + Ct+1 (1/r){1 - [1 / (1 + r)T-1]}

FVt+T = Ct (1/r){(1 + r)T+1 - (1+r)}

0 1 2 3 years (T = 3)

|----------|---------|---------| (r = 10%)

$100 $100 $100

PV0 = 100 + 100 (1/0.1){1 - [ 1/(1.12)]} = $273.55

FV3 = 100 (1/0.1){[1.14 ] - 1.1} = $364.10

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PV and FV of Annuity Due in your HP 10B Calculator

INPUT

CLEAR ALL

Yellow

PMT

I/YR

3 N

PV

Key in payment


Key in number of PAYMENTS

Compute PVDisplay should show:

-273.55371901

100

FVCompute FVDisplay should show:

-364.10000000PV0

0YellowSet payment to beginning of period

When finished -

don’t forget to set your

payment to End of period

When finished -

don’t forget to set your

payment to End of period

BEG/END

10

30

Growing Annuity A Growing Annuity is a stream of cash flows that grows

at a constant rate over a fixed number of periods.

Simplification for PV:

PVt = Ct+1 [1/(r-g)]{1 - [(1+g)/(1+r)]T}

0 1 2 3

|---------|----------|---------| (r = 10%)

$100 $102 $104.04 (g = 2%)

PV0 = 100 [1/(0.10-0.02)]{1 - (1.02/1.10)3} = $253

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Q. Suppose that the bank rewords its advertisement to the following: “Pay us $100 next year, and another 9 annual payments such that each payment is 4% lower than the previous payment. We will pay you (or your beneficiaries) $100, starting at the eleventh year forever.” Is this a good deal if if the effective annual interest rate is 8%?

A. Again, we need to compare the PV of what you pay with the present value of what you get:

- The present value of your annuity payments (note: g = -4%):

PV0 = C1 [1/(r-g)]{1- [(1+g)/(1+r)]T}

= 100[1/(0.08-(-0.04))]{1-[(1+(-0.04))/(1.08)]10}

= [100/0.12]{1-[0.96/1.08]10} = $576.71

- The present value of the bank’s perpetuity payments today:

$578.99 (see example above)

An Example - Present Value of a Growing Annuity

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A special case - when r < g, we still use the above formula

Example:

0 1 2 3

|------------------|-------------------|------------------| (r = 4%)

$100 $100x1.07 $100x1.072 (g = 7%)

PV0 = 100 [1/(0.04-0.07)]{1 - (1.07/1.04)3} = $296.86

Growing Annuity - Special Cases – Self Study

(-) (-) (+)

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Growing Annuity - Special Cases – Self Study A special case - when r = g, we cannot use the above formula

Example-1:

0 1 2 3

|------------------|-------------------|------------------| (r = 5%)

$100 $100x1.05 $100x1.052 (g = 5%)

In general, when cashflow starts at time t+1, use:

71.285$05.1

1003

05.1100

05.1100

05.1100

05.105.1100

05.105.1100

05.1100

3

2

20

PV

rC

TPV tt 1

1

34

Example-2:

0 1 2 3

|------------------|-------------------|------------------| (r = 5%)

$100 $1001.05 $1001.052 $1001.053 (g = 5%)

In general, when cashflow starts at time t, use:

400$1004

10010010010005.1

05.110005.1

05.110005.1

05.1100100 3

3

2

2

0

PV

tt CTPV 1

Growing Annuity - Special Cases –Self Study

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Growing Annuity Simplification for FV:

FVt+T = Ct+1 [1/(r-g)]{(1+r)T - (1+g)T}

0 1 2 3

|---------|----------|---------| (r = 10%)

$100 $102 $104.04 (g = 2%)

FVt+T = ____ [1/(___-___)]{___3 - ____3} = $337.24

Documents

1 Time Value of Money (TVM) - the Intuition A cash flow today is worth more than a cash flow in the future since: uIndividuals prefer present consumption