TIME
VALUE OF
MONEY
– CH 5
Dr. J. Cordeiro
All rights reserved
PURPOSE
• Finance focuses on managing cash: raising it, spending it,
managing it day to day
• Cash has at least four dimensions that need to be accounted for:
the amount, the currency, the risk, and the timing (CART)
• This chapter teaches us how to adjust for the timing dimesion by
“standardizing” time, i.e. converting cash flows at different points
in time into a common point in time:
• If we have $100 today, $200 next year, and $250 in four years,
we can find their value in today’s dollars by converting each
cash flow to today’s dollars and then adding them up!
• Cash has a “time value” because it earns interest – the sooner you
have the cash in hand, the more interest you can earn!
• Simple v/s compound interest
• Base is the dollar amount interest is calculated on
• We will be using compound interest
• Each period, we calculate $ interest = ($ Base) x (interest rate %)
• Where the compound interest base = ($Principal + all accumulated $
interest over time)
EXAMPLE OF COMPOUNDING
($100 PRINCIPAL INVESTED AT 10 PERCENT INTEREST) – GRAPH IS ON THE NEXT SLIDE FOR 25 YEARS
Period
Base (for
Compound
Interest)
Compound
interest (10
pct of Base)
“snowballs
over time”
Future Value =
(Base + all
accumulated
Interest)
FV if we had
used simple
interest instead
1 (1 year from now)
$100
(principal!)
$10 interest
$110 in year 1 =
FV1
$110 in year 1
2
$110
$11 interest
$121 in year 2
$120 in yr 2(keep
3
$121
$12.10
$133.10 in yr 3
$130 in yr 3
4
$133.10
$13.31
$146.41 in yr 4
$140 in yr 4
5
$146.41
$14.46
$158.87 in yr 5
$150 in yr5
(Base in simple
interest is the
principal always!)
picking up $10/yr in
interest)
$1 200
$1 083,5
$1 000
EFFECT OF TIME,
INTEREST RATES ON
COMPOUNDING AS
SEEN IN TWO
SEPARATE GRAPHS
$800
(EXPONENTIAL GROWTH)
$600
$985,0
$895,4
$814,0
$740,0
$672,7
$611,6
$556,0
$505,4
$459,5
$400
$200
$100
$417,7
$379,7
$350
$345,2
$330 $340
$320
$313,8
$300 $310
$285,3
$280 $290
$270
$259,4
$250 $260
$235,8
$230 $240
$220
$214,4
$200 $210
$194,9
$177,2$170 $180 $190
$161,1$160
$146,4$150
$133,1$140
$130
$121,0
$120
$110
$0
0
5
10
FV Simple
15 FV Compound
20
25
30
• The formula to capture the exponential growth: y = abx
FUTURE VALUE OF
A CASH FLOW
FV = PV(1+i)n
• Thus to find the FV of $100 invested now for 2 years at 10
percent
FV in 2 years = $121 in 2 years = ($100 now)(1.1)2
= ($100 now)*1.21
Let’s notice a few things:
• This is the same as ($100 now)(1.1)(1.1) which represents the
$100 growing at 10 percent for two periods in a row!
• The FV in 2 years = ($100 now)(1.1)2 =($100 now)(1.21) = $121 in
two years
• The 1.21 is called the FV interest factor (or FVIF) and it tells us
that each dollar invested now at 10 percent for two years will
grow TO $1.21 in two years. (Thus $100 invested now will grow to
$121 in 2 years!)
• EXAMPLE: Suppose a college education costs $10,000 a year
right now and is expected to grow at 8 percent a year. How
much will a college education cost in 18 years?
• FV in 18 years = ($10,000 now)*(1.08)18 = $39,960 in 18 years
FV in 18 years = ($10,000 now)*(1.04)18 = $20,258 in 18 years
Problem for you to try: Redo the problem but assume that tuition
grows at only 4 percent a year. How does this answer compare to
the $39,960 answer above? Explain why the value is not half of the
$39,960.
THE RULE OF 72
• This is a rule of thumb: it will take 72/i periods to
double an initial investment
• This explains why in the college tuition example, the
tuition quadrupled from $10,000 to roughly $40,000
after 18 years
• 72/8 pct = 9 years to double  18 years to quadruple
• Do NOT use this rule unless explicitly asked for!
• 72/3 = prices double in 24 years
• 72/7 = prices doube in 10 years
HOW WE PERFORM CALCULATIONS IN THIS COURSE
• Many introductory finance courses emphasize the use of financial calculators (such
as the TI BA ii or the HP 12 c) to calculate future values, present values, and so on.
• We DO NOT do this! (athough if you want to learn how to there are may guides
available online as well as free software emulator apps for the HP 12c, etc…)
• Using financial calculators often results in students practicing “push-button” or
voodoo finance without really understanding what they are doing or WHY the
numbers are coming out as they do. This course is NOT merely about calculations!!!
• We will use algebra first so that you understand the mathematics behind the
calculations
• We will then solve the same problems using Excel. Excel has many advantages over
financial calculators:
• The fact that Excel software is now available on tablets, phones, and of course laptops
means that it is just as portable as a financial calculator
• It is freely available (as a SUNY Brockport student you get it for free)
• You can document and save all the numbers, labels, and anything else you choose to
put into your spreadsheet  more clarity for the reader and for you when you come
back to your spreadsheet months or years later
• Excel has graphing capabilities, the ability to link spreadsheets, and many other tools of
great value for financial analysis
USING EXCEL TO FIND FUTURE VALUES
• For any problems you do in Excel, please provide a data table
as shown here. Also, format numbers appropriately using dollar
format, or number format or percent format (with at least two
decimals) as appropriate!
A B
3 Principal
• In this example, I am using Excel to find the future value of
$10,000 now if it is invested at 12 percent for 6 periods.
4 Interest rate:
• If you type in the formula = FV into a cell, it asks for the inputs
(rate, nper, pmt, (pv), [type]) where
5 Number of Periods:
• rate is the interest rate,
6 Future value:
• nper is the number of periods you are investing for, pmt indicates
whether this is a recurring amount (IGNORE pmt for now by just
leaving it blank) and
• pv is the amount invested today, represented as a negative
amount (since an investment you make is an outflow!)
• So, given the data table I am using, I type
= FV(C4,C5,,-C3) notice the space between commas after C5
to reflect that pmt is blank!)
C
$10,000
12.00%
6
$19,738.23
PRESENT VALUE OF A CASH FLOW
• Recall that the formula for future value is FV = PV (1+i)n
• By re-arranging tems, the PV = FV/ (1+i)n or PV = FV * (1/(1+i)n ) (remember c = a/b is c = a*1/b)
• This process is called discounting (the mirror image of compounding)
Example: Suppose we’d like to know how much $1 million in 40 years is worth today at 6 percent.
• PV today = ($1,000,000 in 40 years)*(1/(1.06)40) = $ 97,222 now worth today
• The 1/(1+i)n is called the Present value interest factor (PVIF) and is the inverse of the FVIF. For this
example it is .09722.
• This means that for every dollar you need in 40 years, if you earn 6 percent a year, you need to invest
$ 0.09722 today!
• The PVIF for period 0 (today) for ANY interest rate is always 1 (WHY?)
• PVIF and FVIF are the inverse of each other!
Problems for you to try:
(a) What would the PV be today if we could earn 12 percent interest instead of 6 percent? Why would it not be
half of the $97,222?
(b) Suppose you win a lottery that offers you the choice between receiving $2 million now or $5.5 million if you
wait 10 years. If you believe that 7 percent is a fair interest rate, should you take the money now or should
you wait? Ignore all tax effects.
(c) The PVIF for 3 years, 10 percent is 0.7513. Explain clearly what this number means. Next, find the FVIF of 3
years, 10 percent using only this PVIF. Finally, explain what the FVIF you just calculated means.
USING EXCEL TO FIND PRESENT VALUES
• If you type in the formula = PV into a cell, it asks for the inputs
(rate, nper, pmt, (fv), [type]) where
• rate is the interest rate,
• nper is the number of periods you are investing for, pmt
indicates whether this is a recurring amount (IGNORE pmt for
now by just leaving it blank), and
• fv is the amount to be received in the future,
• So, given the data table I am using, to find the PV of $2
million to be received in 45 years today, if I can earn 6
percent each year, I type
= PV(C4,C5,,C3) notice the space between commas to
capture that pmt is blank!)
A B
C
3 Future Value
$2,000,000
4 Interest rate:
6.00 %
Number of
5 Periods:
6 Present value:
45
-$145,300.15
SOME INTERESTING OBSERVATIONS
• Notice how FVs and PVs do NOT change linearly but
exponentially!
• Einstein reportedly claimed that compounding was the most
powerful force in the universe
• For a given period of time:
• The sooner you start saving the less you need to save to
reach a future goal! This is a HUGELY important lesson for
personal finance. Try the two examples below:
• Ex 1 : Suppose you have just inherited $50,000. You can blow it
today on a fancy car, cruise, etc… OR you can invest it at 7
percent. If you invest it, how much will you have in 45 years when
you retire?
• Ex 2: You want to have $1 million when you retire. If you can earn
7 percent, how much will you need to invest if you do so 30 years
before you retire, versus if you do so 15 years before?
• The higher the interest rate, the less you need to save to
reach a future goal!
• In general, PV = f (FV (+), i (-), n (-)) where the + or –
represents direct (+) or inverse (-) relationships, holding all
other variables constant!
• For FV, it is correspondingly FV = f(PV(+), i(+), n(+)
USING EXCEL TO FIND THE INTEREST RATE OR THE
NUMBER OF PERIODS
You can easily use Excel to solve for the interest rate
• Ex1: Suppose your friend will lend you $20,000 now if you agree to
repay her $30,000 in 10 years. You can find the interest rate she is
charging you If you type in the formula = RATE into a cell, it asks for
the inputs (nper, pmt, (pv), (fv), [type])
• So, given the data table I am using, to get the rate of 4.14 % I type
= rate(C5,,C4,C3) notice the space between commas to capture
that pmt is blank!)
You can easily use Excel to solve for the number of periods too
• Ex 2: Suppose your friend will lend you $20,000 now if you agree to
repay her $30,000 at 6 percent. You can find the number of
periods she is lending you the money for If you type in the formula
= NPER into a cell, it asks for the inputs (rate, pmt, (pv), (fv), [type])
• So, given the data table I am using, to get the 6,96 periods, I type
= nper (C5,,C4,C3) notice the space between commas to capture
that pmt is blank!)
A B
C
3 Future value:
4 Present Value
$30,000
(20,000)
$
Number of
5 Periods:
6 Rate:
A B
3 Future value:
4 Present Value
5 Rate:
6 Number of periods
10
4.14%
C
$
$30,000
(20,000)
6.00%
6.96
• So far we have been working with SINGLE cash flows
• The NPV is used when we have to deal with several cash
flows at one time
• Suppose I wanted to know how much first (e.g. euros) and
THE NET PRESENT
VALUE (NPV)
then add them up to get an an$100 dollars today, 50 euros
today, and 80 Indian rupees were worth today. I would need
to convert them to some common currency e.g.answer in
euros. (Note that I don’t HAVE TO use euros, I could use ANY
common currency).
• Similarly, if I wanted to know how much -$100 today, $50 in
year 1 and $200 in year 4 were worth in today’s dollars, I
would need to find the value of each cash flow at a
common point in time. For example, if I used today (the
present) as the common time period, I would calculate and
then add the three resulting present values up.
• The NPV is simply the sum of the PVs of a series of cash
flows.
• To find the NPV, therefore, we first find the PVs of the
individual cash flows (using the PV formula for a single
cash flow OR the PV formula for an annuity (we will see
this shortly!) and then we add the PVs up.
SOME SIMPLE NPV CALCULATIONS
Let’s find the NPV of $100 to be received in year 1 and another $100 to be received in year 2 if the
interest rate is 10 percent per year:
Step 1: I identify the solution “plan”: Find the PV of $100 in year 1 + PV of $100 in year 2
Step 2: Calculate the PVs and add them up: “carry out the plan!”: Treat as TWO SEPARATE C.FLOWS
NPV = PV of the $100 in year 1 + PV of the $100 in year 2
= ($100 in year 1)(1/(1.1)1) + ($100 in year 2) (1/(1.1)2)
= ($100 in year 1)(.9091) + ($100 in year 2) (.8264)
= $90.91 now + $82.64 now
= $173.55 now as the NPV
Always, make sure you can explain each number that you calculate!!! For example, the $82.64 is what
you need to invest today to get the $100 in year 2(Why?)
Problem for you to try: (a) Would the NPV be larger or smaller than $173.55 if the problem was to find
the NPV for $100 in year 2 and $100 in year 3 using 10 percent? WHY? (b) Confirm your intuition by
solving the problem.
ANNUITIES AND THEIR USE IN FINDING NPV
An annuity: is a series of equal cash flows, received in consecutive periods, with the same interest rate
applied to each period.
An annuity does NOT have to cover all the cash flows!
Thus, if I had $100 in period 1, $200 in period 2 and period 3, and $500 in period 6, I would have a single cashflow in
period 1, an annuity of $200/year in periods 2,3 and a single cashflow in period 6.
The formula for the PV of an annuity = ($ Equal cash flow)*(1/i – 1/(i*(1+i)n))
If we return to the problem where we had to find the NPV of $100 to be received in year 1 and another
$100 to be received in year 2 if the interest rate is 10 percent per year:
Step 1:We can identify a different solution plan – this time using the annuity concept: Find the PV of an
annuity of $100 to be received in years 1 and 2. TREAT AS ONE SINGLE ANNUITY
Step 2: Calculate the PVs using the annuity formula :
($100 annuity in years 1, 2) * ((1/.1) – (1/(.1)(1.1)2)) =
($100 in years 1, 2)*(1.7355)
(The 1.7355 is called the PVIFA for years 1,2 at 10 %)
= $173.55 now
We get the same answer as before! This is because the 1.7355 is the PVIFA (present value interest factor for
an annuity)and it is simply the sum of the PVIF of 0.9509 in year 1 and the PVIF of 0.8264 in year 2.
Note that the PVIFA for period 0 is always zero! Remember, the PVIF for period 0 is always 1 (for ANY rate!)
ANOTHER NPV CALCULATION
• We can find the NPV of ANY series of cash flows using a combination of present values for single
cash flows or present values of annuities (if annuities are present). Using annuities often speeds
up calculations.
• Example: Find the NPV of -$300 now, $100 in years one to three, $400 in year 5.
• Step 1: Come up with a plan. Note that there are different possible plans – I could find the PV of $300 now and then the PV of an annuity of $100 for years 1 to 3 and then the PV of the single
cash flow of $400 in year 5 and then add them all up OR I could ignore the fact that the $100 in
years 1 to 3 are an annuity and just treat them as three separate cash flows (a more tedious
approach)!
• Step 2: Calculate the PVs based on your plan. Let’s assume we go with the first plan. If so, then
NPV = PV of $-300 now + PV of an annuity of $100/year in years 1 to 3 + PV of $400 in year 5
= (-$300 now) + ($100/year in years 1-3 )(1/.1)-(1/(.1)(1.1)3)) + ($400 in yr 5)(1/(1.1)5)
= -$300 now + ($100/year in years 1-3)(2.4869) + ($400 in year 5)(.6209)
= -$300 now + $248.69 now + $248.36 now = $197.05 now is the NPV
Problem for you to try: Find the NPV of $-500 now, $100 in years 1-4, $300 in year 5, $800 in year 6
using a discount rate of 10 percent per year. TRY ALSO IN EXCEL!!!!!
USING EXCEL TO FIND THE NPV
There are two key things to remember when using the NPV function in
Excel:
a. The NPV function starts discounting with the very first cell named,
so you want to make sure you do NOT accidentally discount
period 0
b. You cannot leave blank cells in the series of cash flows – you must
physically insert a zero if there is no cash flow for the period.
The NPV function is = NPV (rate, value1, value 2, …) where rate is the
discount rate and value 1, value 2 are the cash flows. You should
make sure that value 1 is indeed the first cash flow you want to
discount – typically the cash flow in period one, NOT zero!
B
Period
0
2
1
3
2
4
3
5
6
7
So, I have used = NPV(C10,C3:C8)+C2
8
Note:
9
a. I have put a zero in cell C6 and not left it blank!
b. I have discounted starting with the cash flow in Cell C3 and
accounted for cashflows in cell C3 to C8, and made sure that the
blank cell C6 has a zero in it (Excel has replaced it with a dash as
part of its formatting). I also need to add in the cash flow in period
zero by adding C2 to the NPV.
C
4
5
6
Cash Flow
$
(500.00)
$
100.00
$
100.00
$
100.00
$
$
$
Discount
10 Rate:
NPV:
300.00
600.00
10.00%
$
273.65
MISCELLANEOUS OTHER TIME VALUE IDEAS
• You can find the Future Value of an Annuity using the FV function which is
= FV(rate, nper, pmt, [pv],[type])
• =FV(.06, 20, -300) gives you the future value of $300 per period for 20 periods at 6 %
• When dealing withannuities (PV or FV), the “type” is set to 0 indicates if the annuity is an
“ordinary annuity” (paid at the end of each period or to 1 if it is an “annuity due” paid a
the beginning of each period)
• The PV of a perpetuity (an annuity that goes on forever!) = Amount per period/i
• To calculate the Payment on a loan you can use the PMT function which is
= PMT(rate, nper, pv, [fv],[type])
• =PMT(.06, 15, -100,000) gives you the payment of $10, 296.28 each period for 36 periods
for a $10,000 loan today. The fv is used here if you need some money at the end of the
loan period and is not typically used.
• You do not always have to use annual periods. To convert to monthly periods, you divide the
rate by 12 and multiply the number of periods by 12
• Thus, = PMT(.005, 180, -100000) gives you the monthly payment of $843.86 on the same
loan above
SOLVING FOR UNKNOWNS
• We use the idea that NPV = 0 to solve for unknowns
• This works only for financing arrangements where there are two parties
exchanging cash (NOT other assets or labor)at different points in time.
LOAN AMORTIZATION
To amortize a mortgage, car loan, etc… we need to consider that:
(a)We first need to calculate a payment per period using the PMT function
(b) Next we calculate Interest per period: this is paid on the unpaid loan balance using the PMT
function
(c ) The rest of the payment is used to reduce the loan balance
The example below is for a $25,000 loan for 4 years at 8 percent:
Loan Amount:
Loan period (years):
Interest rate:
Payment:
$ 25,000
4
8.00%
$7,548.02
Period
Payment
Interest
Principal
Repaid
0
Unpaid
Balance
$ 25,000.00
1
$7,548.02
$ 2,000.00
$5,548.02
$19,451.98
2
$7,548.02
$ 1,556.16
$5,991.86
$13,460.12
3
$7,548.02
$ 1,076.81
$6,471.21
$6,988.91
4
$7,548.02
$
$6,988.91
$0.00
559.11