TheTime Value of Money

1
Chapter 5

• TheTime Value of Money

2
Time Value of Money

• One of the most effective tools a student (of any
  major) can learn.
• Professor McDermott feels so strongly about it,
  he has decided to expand this module to provide
  more theory, and applications than provided by
  the book.

3
A Little Theory . . .
Assume we invest a lump sum of 100 in time
period zero.
Money
The interest rate is 10 per year.
300 200 100
Time
0 1 2 3
4
A Little Theory . . .
We let it grow for three years.
Money
In one year it is worth 100 x 1.10 110 In
two years it is worth 110 x 1.1 121 In
three years it is worth 121 x 1.10 133.10
133.10
130 120 100
Time
0 1 2 3
5
A Little Theory . . .
These values can be calculated from lump
sum present value and future values in your book.
Money
They can also be calculated using a financial
calculator.
Again we are talking about lump sums!
The future value of 100 for three periods at 10
is 133.10.
133.10
130 120 100
The present value of 133.10 for three periods is
100.
This represents the future value and present
value of a lump sum.
Time
0 1 2 3
6
How do we do it on a financial calculator?
7
How do we calculate this future value on a
financial calculator?
Calculator
N
I/Yr
PV
PMT
FV
N number of time periods, in this case 3
I/Yr interest rates per year, in this case 10.
PV is the lump sum we invest now, in this case
100.
PMT is only used if we have an annuity (discussed
in a moment. We dont so Put 0 here.
Having put all the values in, punch FV and you
will get the answer 133.10.
8
A Little More Theory . . .

• A lump sum is one sum of money invested at some
  point in time.
• We can also have annuities.
• An annuity is a series of payments of the same
  amount received or paid at equal periods of time.
• 100 invested for 3 periods is an annuity.

9
To Illustrate . . .
The first year we make a payment of 100. That
amount grows with interest until we make a second
payment which in turn grows with interest until
we make a third payment.
Money
New Axis
331
At the end of three years we have 331 from the
annuity.
300 200 100
The future value of 3 payments of 100 at 10
interest per period is 331.
Again, we could calculate this from annuity
tables or using a financial calculator.
Time
0 1 2 3
10
How do we calculate the future value of this
annuity on a financial calculator?
Calculator
N
I/Yr
PV
PMT
FV
N number of time periods, in this case 2
I/Yr interest rates per year, in this case 10.
PV is the lump sum we invest now (in time period
0) in this case 0.
Since this is an annuity we put 100 here.
Having put all the values in, punch FV and you
will get the answer 331.
11
How do we calculate the present value of this
annuity on a financial calculator?
Calculator
N
I/Yr
PV
PMT
FV
N number of time periods, in this case 2
I/Yr interest rates per year, in this case 10.
PV is now what we are solving for so put nothing
here.
Since this is an annuity we put 100 here.
Having put all the values in, punch PV and you
will get the answer 246.69.
12
What does this 246.69 mean?

• A person who could earn 10 a year would be just
  as well off financially by taking 246.69 now as
  an annuity of 300 a year for three years.

13
All of this illustrates the theory behind time
value of money or discounted cash flow
14
Net Present Value MethodS

• The discounted cash flow technique is generally
  recognized as the best conceptual approach to
  making capital budgeting decisions.
• This technique considers both the estimated total
  cash inflows and the time value of money.
• Two methods are used with the discounted
  cash flow technique
• 1) net present value and
• 2) internal rate of return

15
Net Present Value Method

• Under the net present value method, cash inflows
  are discounted to their present value and then
  compared with the capital outlay required by the
  investment.
• The interest rate used in discounting the future
  cash inflows is the required minimum rate of
  return.
• A proposal is acceptable when NPV is zero or
  positive.
• The higher the positive NPV, the more attractive
  the investment.

16
Net Present Value Decision Criteria
The author assumes the capital investment takes
place in time period 0. If not, then we must
discount the capital investment also.
17
Present Value of Annual Cash Inflows-Equal
Annual Cash Flows
Stewart Soup Companys annual cash inflows are
24,000.
If we assume this amount is uniform over the
assets useful life, the present value of the
annual cash inflows can be computed by using the
present value of an annuity for 10 periods.
Calculate the present value at rates of return of
12 and 15.
18
Lets do the calculation at 12 just using
present value buttonsstep one calculate the
present value of 24,000 for 10 periods at 12
19
Lets do the calculation at 12 just using
present value buttonsfirst calculate the present
value of 24,000 for 10 periods at 12
Calculator
N
I/Yr
PV
PMT
FV
N Number of periods so enter 10
I/Yr We will assume a 12 rate so enter 12
Pmt is the cash flow in or out each period so
enter 24,000
FV is the future value is 0 since there is no
salvage value, enter 0
Having entered all the values push PV to
calculate the present value which is 135,605
20
Step Two

• Add the present value of the stream of cash
  coming in to with present value of the initial
  investment.
• The initial investment is 135,000.
• Since it is invested in period 0, the present
  value is 135,000.
• 135,605 - 130,000 5,605 NPV

21
Professor McDermotts Recommendation

• Buy a cheap financial calculator (you can get one
  for about 24) to do the calculations for the
  homework and tests.
• It will save you time and is a valuable asset for
  anyone who handles money (everyone).

22
New Question . . .

• Okay, we have talked about finding the present
  value and future value of a . . .
• Lump sum
• Annuity
• What if we have a series of unequal cash payments
  received at uniform spaces of time. Can we solve
  for a present value of this stream of cash flows?

23
Yup!

• Either using tables or a financial calculator.
• Using tables, we use the lump sum table and
  discount each payment individually for the
  appropriate number of periods.
• We do this by finding the appropriate discount
  rate for each period and discounting the payment
  by that rate.

24
Present Value of Annual Cash Inflows-Unequal
Annual Cash Flows

• 260,000
  155,667 140,061

These factors come from the lump sum tables.
25
Present Value of Annual Cash Inflows-Unequal
Annual Cash Flows
For example, look at the present value lump sum
table on page A8. The discount factor for a lump
sum 4 years out at 12 is 63.552 as shown here.

• 260,000
  155,667 140,061

26
Present Value of Annual Cash Inflows-Unequal
Annual Cash Flows
Multiply the discount rate (for example 36,000)
by the discount rate for that year (for example
.89286 if we are discounting at 12) to get the
present value of this figure (in this case
32,143.

• 260,000
  155,667 140,061

27
Present Value of Annual Cash Inflows-Unequal
Annual Cash Flows

• 260,000
  155,667 140,061

Then sum each payment to get the present value
for the stream of cash payments.
28
How do we find the present value of the stream of
cash flows with a financial calculator?
Calculator
N
I/Yr
PV
FV
PMT
CFj
This button stands for cash flow in the jth
period.
29
Calculation
Orange Button
Clear out previous data (on my calculator hold
the orange button down, then the c all
button). Make sure the calculator is set to one
period per year. On my calculator press 1, then
hold the orange button and press P/Yr. Enter
-135,000, then hit CFj button. It will give a
read out of 0 meaning the 0 time
period. Enter 36,000 and CFj button, it will
give a read out of 1 Enter 32,000 and the CFj
button, it will give a read out of 2 Continue
until all cash flows have been entered. Then hit
orange button and NPV and you will get a readout
of 155,667.66. The same answer as with the
tables.
30
Analysis of Proposal Using Net Present Value
Method
Therefore, the analysis of the proposal by the
net present value method is as follows

• Positive (negative) net present value
  25,667 10,061

In this example, the present values of the cash
inflows are greater than the 130,000 capital
investment. Thus, the project is acceptable at
both a 12 and 15 required rate of return.
31
Compute the net present value of a 260,000
investment with a 10-year life, annual cash
inflows of 50,000 and a discount rate of 12.
Review Question

• (9,062).
• 22,511.
• 9,062.
• (22,511).

32
Compute the net present value of a 260,000
investment with a 10-year life, annual cash
inflows of 50,000 and a discount rate of 12.
Review Question

• Calculator solution
• Clear calculator and set periods per year to one.
• Enter discount rate of 12 into I/Yr button.
• Enter 260,000 into CFj button. The screen should
  read back 0.
• Enter -50,000 into the CFj button ten times
  (there is a shorter way, but it involves more
  buttons, I will let you look the method up in
  your financial calculator book if you wish).
• Push orange button and then NPV and you should
  get the answer 22,511.

• (9,062).
• 22,511.
• 9,062.
• (22,511).

33
Additional Considerations

• The previous NPV example relied on tangible costs
  and benefits that can be relatively easily
  quantified.
• By ignoring intangible benefits, such as
  increased quality, improved safety, etc. capital
  budgeting techniques might incorrectly eliminate
  projects that could be financially beneficial to
  the company.

34
Additional Considerations

• To avoid rejecting projects that actually should
  be accepted, two possible approaches are
  suggested
• 1. Calculate net present value ignoring
  intangible benefits. Then, if the NPV is
  negative, ask whether the intangible benefits are
  worth at least the amount of the negative NPV.
• 2. Project rough, conservative estimates of
  the value of the intangible benefits, and
  incorporate these values into the NPV calculation.

35
New Problem

• You invest 200,000 today in a truck and get the
  following cash flows over the next four years
• Year 1 50,000
• Year 2 60,000
• Year 3 70,000
• Year 4 80,000
• Your discount rate is 10, the truck is worth
  50,000 at the end of the four years.
• What is the net present value?

36
New Problem

• To work problem on the calculator
• Clear out registers and make sure P/YR is set to
  1.
• Enter 10 in I/Yr
• Enter -200,000 in CFj
• Enter 50,000 into CFj, then enter 60,000, 70,000,
  and (80,000 50,000 130,000).
• Hit orange key and NPV
• The answer should be 36,425.11

37
New Problem

• Since the number is positive, we will earn 10
  plus 36,245.11.
• What is the actual rate we will earn?
• That is called the Internal Rate of Return.
• It is the number that will give us a present
  value of zero.
• Hit the orange button and then IRR/YR and you
  should get 16.92

38
New Problem

• Now if we entered in the same cash flow and
  entered 16.92, then our present value would be
  very very close to 0.
• Try it!

39
Internal Rate of Return Decision Criteria

• The decision rule is Accept the project when
  the internal rate of return is equal to or
  greater than the required rate of return. Reject
  the project when the internal rate of return is
  less than the required rate.

40
Comparison of Discounted Cash Flow Methods

• In practice, the internal rate of return and cash
  payback methods are most widely used.
• A comparative summary of the two discounted cash
  flow methods-net present value and internal rate
  of return- is presented below

41
A 60,000 project has net cash inflows for 10
years of 9,349. Compute the internal rate of
return from this investment.
Review Question

• 8.
• 10.
• 9.
• 11.

42
A 60,000 project has net cash inflows for 10
years of 9,349. Compute the internal rate of
return from this investment.
Review Question

• 8.
• 10.
• 9.
• 11.

43
Annual Rate of Return Formula

• The annual rate of return technique is based on
  accounting data. It indicates the profitability
  of a capital expenditure. The formula is

The annual rate of return is compared with its
required minimum rate of return for investments
of similar risk. This minimum return is based on
the companys cost of capital, which is the rate
of return that management expects to pay on all
borrowed and equity funds.
44
Formula for Computing Average Investment
Expected annual net income (13,000) is obtained
from the projected income statement. Average
investment is derived from the following formula

• For Reno, average investment is 65,000
• (130,000 0)/2

45
Solution to Annual Rate of Return Problem
The expected annual rate of return for Reno
Companys investment in new equipment is
therefore 20, computed as follows

• 13,000 65,000 20

The decision rule is A project is acceptable if
its rate of return is greater than managements
minimum rate of return. It is unacceptable when
the reverse is true. When choosing among several
acceptable projects, the higher the rate of
return for a given risk, the more attractive the
investment.
46
Bear Company computes an expected annual net
income from an investment of 30,000. The
investment has an initial cost of 200,000 and a
terminal value of 20,000. Compute the annual
rate of return.
Review Question

• 15.
• 30.
• 25.
• 27.3.

47
Bear Company computes an expected annual net
income from an investment of 30,000. The
investment has an initial cost of 200,000 and a
terminal value of 20,000. Compute the annual
rate of return.
Review Question

• 15.
• 30.
• 25.
• 27.3.

48
Interesting Computations with Financial
Calculators

• The backup for the calculations are on the
  website in an Excel Format entitled Financial
  Planning with Time Value of Money.

49
Mortgages

• Almost everyone during their life has one or more
  home mortgages.
• Understanding how mortgages work, and the impact
  of time and compounding of interest on the amount
  you eventually pay for a home using a mortgage
  can save tens of thousands of dollars.

50
First a Little Theory

• A mortgage payment is a form of annuity
• Equal payments
• Equally spaced
• Payment payoff periods vary considerable
  (typically from 15 to 30 years).

51
Lets Calculate a Mortgage with Our Calculator

• Clear the register
• Set payments to 12 per year
• Assume we have a 30 year mortgagetype 360 (30
  x 12) and hit the N button.
• Assume a 12 yearly interest ratetype 12 and hit
  I/Yr button.

52
Mortgage Calculation

• Lets assume a small mortgage of 100,000.
• Type 100,000 and hit PV button.
• When the mortgage is paid off, the FV will be
  zero. Type 0 and hit FV button.
• Now we will solve for the monthly payment by
  hitting the PMT button.
• The answer is 1,028.61.

53
It is important to remember that interest is paid
first, then principal.
Knowing this, lets figure out how much is
applied to the loan the first month. Since we owe
100,000 and the interest rate is 12 or 1 per
month, the interest paid 1 x 100,000 or 1,000
as shown below.
Great! We have paid 1,028.61 but only reduced
our loan by 28.61.
Dont worry. Things get better, next month it is
28.90.
The first year amortization schedule for this
loan is shown on the following page.
Happy Banker
54
Amortization Table
What if when we pay the first payment of
1,028.61, we enclose an additional amount for
28.90? Will jump from Januarys payment to
Marchs payment. For and additional 28.90 we
will never make that 1,028.61 payment.
Not a bad investment! Pay 28.90, save 1,028.61!
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Amortization Table
What if we pay the sum of the remaining principal
payments for the year (red)? This totals to
334.24. Just include that with the first check
of 1,028.61 and you will skip so that next month
instead of making the February payment, you will
now be on the January 2008 payment, a savings of
10,980.47 in interest.
56
Lets look at how little difference in monthly
payments a change in the length of the mortgage
makes.
The difference in the monthly payment from
cutting your length of mortgage in half is
only 171.56.
However, the difference in the amount you wind up
paying for the house is 151,209!
57
Lets look at how little difference in monthly
payments a change in the length of the mortgage
makes.
I have actually hear (on a radio interview)
bankers complaining about how hard it is for
young people to make mortgage payments with
higher interest rates (I have seen 17 In my
lifetime) and advocate going to 40 or 50 year
mortgages.
Who do you think that proposal is designed to
benefit. The poor young couples or the bankers?
58
Objective

• Remember the purpose of this exercise is not to
  tell you what to do, only to show you how, with a
  financial calculator, you can determine the
  actual impact of different decision options.

59
Lets Have Some Fun . . .

• Assume there are two twin brothers, Fred and
  Frank.

They have the income, the same taste in houses.
60
Dream Home

• They both have plans for the same dream home, a
  rambler costing 250,000.
• To simplify calculations assume no down payment
• Interest rate of 8
• No inflation

61
Freds Decision

• Fred has to have it NOW.
• He borrows the money and incurs an 1,834.41
  monthly payment for 30 years.

62
Franks Decision

• Frank is a little more patient.
• He has the same payment to make on a house.
• He takes his financial calculator and determines
  how much house he can buy with a 10 year mortgage
  for1,834.41.
• He an buy a modes 151,195 home.

63
Jump Ahead 10 years

• Franks home is paid for. He has 151,195 in
  equity.
• Fred, having made the same number of payments in
  the same amount still owes 219,312. He has
  250,000 - 219,312 30,688 in equity.

64
Jump Ahead 10 years

• Frank takes his 151,195 equity and makes a down
  payment on the 250,000 home.
• His mortgage is for 98,805.
• He continues making the 1,834.41 payment each
  month.

65
We are now 187 months out

• It takes 67 months (5 years 7 months for Frank to
  retire mortgage).
• He owns the home outright.
• Fred still owes 187,993. He still has 173
  payments to make.

66
Investment

• Since Frank no longer has to make a mortgage
  payment, he invests the amount he would pay each
  month the stock market.
• The historical return on the stock market is 10
  a year.

67
30 Years After Their Initial Purchase

• Fred finally finishes paying for his 30 year
  home.
• At that time he has a 250,000 home.
• Frank also has the same home, but in addition he
  has a savings account worth 710,850. His net
  worth is 960,851.
• Both have made the same payments for the same
  amount of years!

68
Another Illustration

• Young couples often say they dont have a lot of
  money to save for retirement.
• That may be true, but what they do have is a lot
  of time, and the earlier you start the better.
• The following illustration was taken from an
  insurance company brochure.

69
Rob and Rich

• Fred and Frank have twin cousins, Rob and Rich.
• Both are concerned about retirement.

70
Rob and Rich

• At age 25, Rob makes five yearly deposits a
  mutual fund earning 12.
• He never makes another deposit.

71
Rob and Rich

• Rich during those six years deposits nothing.
• He spends his money on wine, women, and song.
• The rest of it he plain wastes.

72
Rob and Rich

• It takes Rich almost 25 years to catch his
  brother.
• When they both retire at age 65
• Rob who made six 2,000 deposits has 856,957.79
  in his savings account.
• Rob who has made thirty-four 2,000 deposits has
  861,326.99 in his account.
• How important is time when you are compounding
  interest?

73
New Question

• How much do you have to deposit monthly at 10 to
  have 1,000,000 when you retire?
• Age 25--158.12
• Age 30--161.69
• Age 35--446.07
• Age 40757.49
• Age 50--2,417.23
• Age 55--4,887.39

74
Last Example

• You decide you want to surprise your great-grand
  daughter with a 1,000,000 inheritance 100 years
  from now.
• How much do you have to deposit today, assuming
  you can get 10 a year, compounded monthly, to
  reach that goal?

75
The End

• What other problems can you come up with to work
  on your financial calculator?