Time Value of Money
The Starting Point
• NPV analysis allows us to compare monetary
amounts that differ in timing. We can also
incorporate risk into the analysis, however we
will not concern ourselves with this complication
at this time.
• Two items need to be determined before you start
the NPV analysis, future cash flows and interest
rates. Forecasting these is often more an art than
a science, however in many situations these are
either known or can be estimated.
Items needed to solve these
problems
• You will need to know all but one of the following:
•
•
•
•
•
interest rate
# of periods
future value
present value
cash flow
i
n
FV
PV
PMT
Methods to solve the
problems
• A decent business calculator (e.g.,
HP10BII)
• A formula
• Tables
• A spreadsheet package (e.g., excel)
The following are useful
formulas
• Future value of a single sum
FV = PV * (1+i)**n
• Present value of a single sum
PV = FV * 1/(1+i)**n
Simple versus compound
interest
• Simple interest involves computing interest
only on the original principal, not on any
accrued interest. Compound interest
involves calculating interest on interest.
Future Value – Simple Interest
Example 1
Invest $1 for 3 years @ 12% per annum.
Period
1
2
3
Beg. Amt.
1.00
1.00
1.00
InterestEnd. Amt
0.12
1.12
0.12
1.24
0.12
1.36
Future Value – Compound
Interest
Example 2
Period
1
2
3
Beg. Amt. Interest End. Amt
1.0000
0.12
1.1200
1.1200
0.13
1.2544
1.2544
0.15
1.4049
n =3, i = 12, PV = 1, FV = ?
Formula
1.12**1
1.12**2
1.12**3
Future Value
Example 3
Invest $5 at the end of each year for 4 years @ 12%. What
is the FV?
5
5
5
5
now
1
2
5 x 1.00 = 5.00
5 x 1.12 = 5.60
5 x 1.2544 = 6.2720
5 x 1.4049 = 7.0245
4.779 23.897
3
4
This is the same as the future value of an
ordinary annuity
n =5, i = 12, Pmt = 5, FV = ?
Present Value
In each of the cases so far we wished to
determine what a dollar would be worth in
the future. We can also go the other
direction. Often we wish to know what
future sums are worth today. This is called
present value (PV)
Present Value
Example 4
What is the PV of a 10 dollars received 1 year from
today assuming 12% interest?
?
Now
$10
1
Note that $8.93 grows to $10 in 1 year @ 12%
8.93 x 1.12 = 10
n =1, i = 12, V = FV, PV = ?
Present Value
Example 5
What is the PV of $4 received 3 years from today and $4
received 2 and 1 year from today at 5% interest?
Now
4
1
4
2
4
3
4 x .9524 = 3.810
4 x .9070 = 3.628
4 x .8638 = 3.455
2.7232 10.893
n =3, i = 5, PMT = 4, PV = ?
Non- Annual Periods
So far we have computed FV of a single sum
and an annuity and also PV of a single sum
and an annuity. Each are basically the
reverse of the other. Each has been
computed with one compounding period
per year. Often the compounding period is
shorter.
Future values with non-annual
deposits
Example 6
What is the FV of a $75,000 deposit made every 6 months
for 3 years using an annual rate of 10%?
0
1
75
2
75
3
75
4
75
75
5
6
7
8
9
10
75
[((1.05**6)-1)/.05] x 75,000
6.80191 x 75,000 = 510,143
n=6, i = 10, pmt = 75,000, FV = ?
Note: Be sure to set your calculator to 2 payments per year.
Other Items to Solve For
• N = how long will it take a sum to grow to
a certain FV at a given interest rate
• i = what interest rate is required to grow a
certain sum to a given FV in a given length
of time
• PMT = what payment is required to pay off
a loan at a given interest rate in a set
amount of time
Solving for n
Example 7
How many periods does it take for $130 to
grow to $261.48 @ 15% per annum?
n = ?, i = 15, PV = 130, FV = -261.48
Solving for i
Example 8
At what annual interest rate will $175 grow
to $377.81 in ten years?
n = 10, i = ?, PV = 175, FV = -377.81
Find the required payment
Example 9
Compute the required semi-annual payment in order to have
$14,000 at the end of 5 years @ 8%
14,000
0
1
2
3
4
5
6
x
x
x
x
x
x
n=10, i=8, PMT = ?, FV = -14,000
7
8
x x
9
x
10
x
Car payments
Example 10
What would be your monthly car
payment on a $15,000 4 year loan @
10%. Payments are made at the end
of each month.
PV = 15,000 n = 48 i = 10%;
pmt =
Car payment
Example 10 (continued)
Instead of a 4 year loan, compute the
payment for a 5 year (60 payment) loan.
PV = 15,000 n = 60 i = 10%;
pmt =
Car payment
Example 10 (continued)
Leave the loan at 5 years, but lower the
interest rate to 8%. Compute the payment.
PV = 15,000 n = 60 i = 8%;
pmt =
Car payment
Example 10 (continued)
With the 5 year, 8 % loan, assume the
maximum payment you can afford is $275.
How much of a loan can you afford?
n = 60 i = 8% pmt = 275;
PV =
Car payment
Example 10 (continued)
Go back to the $15,000, 5 year, 10% loan.
How much of the 12th payment applies toward
principal? Interest? What is the remaining
balance?
Do the same for the 36th payment?
Do the same for the 13th – 24th payments combined?
Present Value of an Annuity
Example 11
You win a $4,000,000 lottery that pays
$200,000 per year for 20 years. What is the
present value of the lottery assuming a rate of
10%?
n = 20, i = 10, PMT = 200,000; PV =
Uneven cash flows
• Up to this point we have assumed cash
flows are the same each period. This is
common for mortgage and lease payments.
Things are not nearly as tidy when you
need to determine if a project makes
financial sense. Typically you will
experience cash flows from revenues and
expenses that vary each period.
Uneven cash flows
• This is the situation firms face when
attempting to decide if a new location
makes economic sense. Luckily this
situation can still be handled with your
financial calculator. You will be using a
few new keys:
• CFj, Nj, IRR/YR, and NPV
Internal rate of return
• IRR/YR is used to compute the internal
rate of return. This represents the interest
rate that the project is earning over its life.
This is similar to solving for i in the
previous problems.
Net present value
• Sometimes you may know that you need a
minimum return (internal rate of return) to take
on a new location. You can then use this interest
rate in the calculation and then compute the
present value of all the combined cash flows.
The summary number is the net present value of
the project. If the project is earning a return
greater the the required IRR, the NPV will be
positive, otherwise it will be negative.
IRR and NPV example
Example 16
You wish to determine the IRR and NPV of a project with the following
projected cash flows:
At inception: -10,000
End of year 1: 3,000
End of year 2: 1,000
End of year 3: 3,000
End of year 4: 8,000
Determine the IRR and then the NPV if the required return is 15%
Tips
1.Draw time lines
2.Put in all the knowns
3.Be sure to use the period interest rate
4.Make sure the answer passes the smell test
(e.g., is the present value < the future
value?)