Download Time Value of Money

Introduction to Time Value of Money
Time Value of Money (TVM) refers to the notion that the value of money
depends upon its timing, i.e. when it is received or paid. A sum of $100
received today is worth more than $100 received a year from today.
The heart of a TVM calculation is the process of expressing the future
value of a payment made in the present (compounding) or the present
value of a payment made in the future (discounting). There are four
variables in a typical time value calculation, and if we know any three of
those variables, we can calculate the fourth.
Notation we will be using:
PV
present value
FVt
future value at time t
PMT
periodic payments made (annuity payments or bond
coupons)
i
effective annual rate of interest
N
number of effective interest rate periods (years in our
examples)
Most financial calculators have keys labeled as above (N for number of
periods, i or % for effective interest rate per period, PV for present value,
PMT for periodic payments, and FV for future value of a single future
payment). It must be remembered that an investment is a cash outflow,
so that for financial calculators an initial value should be entered as a
negative number.
We will cover four simple situations:
Future value of a single payment made today
Present value of a single payment made at some point in the future
Future value of a series of periodic payments (accumulated or
compounded value of annuity payments or bond coupons)
Present value of a series of periodic payments (discounted value of
annuity payments or bond coupons)
Future Value of a Single Payment made Today
If the single payment made today is denoted by PV, the future value after
t years is denoted by FVt, and the effective rate of interest is i, the
formula for FVt is:
FVt = PV0 (1 + i )t
That
after
after
.
after
is:
one year
two years
FV1 = PV + PV i = PV (1+ i)
FV2 = FV1 + FV1 i) = FV1 (1+i) =PV (1+ i) 2
t years
FVt = FVt-1 + FVt-1 i) =FVt-1 (1+ i) = PV (1+ i)t
Example: Future value after 10 years of a single payment of $1,000
today, if the effective annual interest rate is 4%:
FV10 = $1,000 x (1.04)10= $1,000 x 1.48024 = $1, 480.24
On a financial calculator, the calculation is done by making the following
key strokes:
N= 10, i (or %) = 4, PV = -1000, PMT = 0, FV = ? (ans. = 1480.24)
Once the values are entered for N, I, PV and PMT , pushing the FV
button will display the desired value.
Present Value of a Single Payment made in the future
Algebraically:
If FVt = PV (1 + i )t, then PV = FVt/ (1+i)t
i.e.:
PV0 = FV1/ (1+ i)
FV1 = FV2/ (1+ i) 
PV = FV2 /(1+i)2
.
.
FVt = FVt-1/ (1+ i) 
PV = FVt /(1+i)t
Example: Present value today of a single payment of $1,480.24 made ten
years in the future, if the effective annual interest rate is 4%
PV = $1,480.24 / (1.04)10= $1,480.24 / 1.48024 = $1, 000
On a financial calculator, the calculation is done by making the following
key strokes:
N = 10, i (or %) = 4, FV = -1480.24, PMT = 0, PV = ? (ans: 1000)
Once the values are entered for N, I, FV and PMT , pushing the PV
button will display the desired value.
Future value of a series of periodic payments
(Accumulated value of annuity or bond coupons)
The accumulated value of an annuity or bond coupons is nothing more
than the summation of a number of single payments, valued at the same
time (t) in the future, assuming payments are made at the end of the
year:
i.e. sum of : FV in t years of Payment made at time 1 FVt = PMT1 (1+ i)t-1
+ FV at year t of Payment made in year 2, FVt = PMT2 (1+i)t-2
+ FV at year t of Payment made in year t, FVt = PMTt)
e.g. Future Value after 4 years of annual of payments of $100, beginning
at the end of the first year, with interest at 5%:
FV = $100 (1.05)3 + $100 (1.05)2 + $100 (1.05) +$100
= $100 (1.1576) + $100 (1.1025) + $100 (1.05) + $100
= $431.01
On a financial calculator, the calculation is done by making the following
key strokes:
N= 4, i (or %) = 5, PV = 0, PMT = -100, FV = ? (ans. 431.01)
Once the values are entered for N, I, PV and PMT , pushing the FV
button will display the desired value (431.01).
Present value of a series of periodic payments
(Discounted value of annuity or bond coupons)
The discounted value of an annuity or bond coupons is nothing more
than the summation of the present value of a number of single
payments, valued today:
i.e. Sum: PV of payment at end of year 1: PV0 = PMT1/(1+i)
+ PV of payment in year 2
PV0 = PMT2/(1+i)2
.
+ PV of payment at time t
PV0 = PMTt/(1+i)t
e.g. Present value of an annuity of $100 per year, for 4 years at an
effective rate of 5%.
PV = $100/(1.05) + $100/ (1.05)2 + $100/(1.05)3 + $100/(1.05)4
= $100/1.05 + $100/1.1025 + $100/ 1.1576 + $100/1.2155
= $95.24 + $90.70+ $86.38+ $82.27 = $354.60
On a financial calculator, the calculation is done by making the following
key strokes:
N = 4, i (or %) = 5, FV = 0, PMT = -100, PV = ? (ans. 354.60)
Once the values are entered for N, I, FV and PMT , pushing the PV
button will display the desired value (354.60).
The following material is useful to keep in mind as you tackle more
involved problems.
Time Value of Money Rules:
(Prepared by an ex-FIN 5023 student)

Read the question carefully.

Draw the time line identifying the timing of the cash flows.
(Remember that the tick mark representing time 1 is both the end
of time period 1 and the beginning of time period 2.)

For single cash flows:
N= number of periods
PV= (in end mode) value at present time

For annuities (stream of equal cash flows):
N= number of payments
PV=(in end mode) value as of one period before first cash flow
(multiply that value by 1+ the interest rate to find the
present value as of the first cash flow)
FV= (in end mode)value as of the last cash flow

Both the perpetuity and growing perpetuity formulas give us the
present value as of one period before the first cash flow.

The Cash Flow Register gives us the value as of the first cash
flow. If you have streams of unequal cash flows, use the cash
flow register.

Nominal rates are never used on time lines or as calculator
inputs (unless I nom = periodic rate). Always use the periodic
interest rate.
Sample Problems
1. You just deposited $25,000 in a 5-year bank certificate of deposit (CD) that
pays 3% compounded annually. How much money will you receive when
the CD matures in five years?
PMT:
PV:
N:
I/YR:
FV:
0
25,000
5
3%
??? = $28,981.85
2. You are considering buying a zero coupon bond (no annual interest is paid)
that will be worth $50,000 when it matures in three years. If you want a rate
of return of at least 10%, what is the most you should be willing to pay for
this bond?
PMT:
FV:
N:
I/YR:
PV:
0
50,000
3
10%
??? = $37,565.74
3. Sissy plans to invest $2,000 and wants to triple her money within six years.
What rate of return must Sissy earn in order to accomplish this?
PMT:
FV:
N:
PV:
I/YR:
0
6,000
6
2,000
??? = 20.1%
4. The Justin Company has just made a big sale and wants to pay its rent for
the whole year in advance while it has the money to do so. Previously,
Justin has paid $200 in rent every month. Assuming an annual interest rate
of 6%, what single sum of money should Justin be willing to pay? (Note:
Since the rent payments are due at the beginning of the month, the
assumption of payments at the end of each period calls for an adjustment
to be made in the present value calculation.)
Set the calculator to read BGN, or beginning, to get an annuity due.
PMT:
FV:
N:
I/YR:
PV:
2,000
0
12
0.5% (6%/12 months = 0.5% per month)
??? = $23,354.05
5. You and your spouse are saving money to purchase your dream house
when you retire in 15 years. Currently, the house costs $150,000 and you
expect it to appreciate at a 3% annual rate of inflation. In order to pay cash
for the house when you retire in 15 years, you’ve set up a savings account
that pays 5% compounded annually. How much will you have to deposit in
the savings account at the end of each year in order to be able to buy the
house for cash when you retire fifteen years from now?
Be sure that the calculator does NOT read BGN.
Step 1: How much will the dream house cost in 15 years:
PMT:
PV:
N:
I/YR:
FV:
Step 2:
PV:
N:
I/YR:
FV:
PMT =
0
150,000
15
3%
??? = $233,695.11
How much will you have to deposit at the end of each of the
next three years in order to purchase the house:
0
15
5%
233,695.11
??? = $10,829.97
6. Johnny just purchased a new hybrid car costing $40,000. He put down
$10,000 of savings and borrowed the remaining $30,000 from his credit
union at a 6% annual rate of interest over a 5-year period. What is
Johnny’s monthly car loan payment?
PV:
N:
I/YR:
FV:
PMT =
30,000
60
0.5% (6% per year, or 6%/12 = 0.5% per month)
0
??? = $579.98
Practice Problems:
1. What is the Present Value of 10 annual annuity payments of $1000, at
8%?
Ans. $ 6,710.08
N=________, i ________, PV________, PMT________ , FV___________
2. What is the Future Value of 30 annual annuity payments of $100 at
5%?
Ans. $6,643.88
N=________, i ________, PV________, PMT________ , FV___________
3. What is the Present Value of a single payment of $500 made after 15
years, if interest is 6%?
Ans. $208.63
N=________, i ________, PV________, PMT________ , FV___________
4. What is the Future Value of at time 5 of a single payment today of
$20, at 6%?
Ans. $
26.76
N=________, i ________, PV________, PMT________ , FV___________
5. If interest is 12%, how many years does it take for a single payment of
$1000 today to accumulate to a Future Value of at least $2000? Ans. 7
(6.11 years > 6 years)
N=________, i ________, PV________, PMT________ , FV___________
6. What is the annual payment amount of an annuity for 10 years that
can be purchased today for $10,000, if interest is 8%? Ans. $1490.29
N=________, i ________, PV________, PMT________ , FV___________
ADDITIONAL PROBLEMS
Assume an interest rate of 10% for all three problems.
1. An annuity contract will pay $100/year each year from the
end of Year 8 to end of Year 21. What is the value of this
stream a) today (time 0) and b) 25 years from now.
2. A project requires an outlay of $100,000 today. It is
expected to yield net cash flows of $15,000 at the end of the
first year, $25,000 at the end of the second year and $
18,000 at the end of the third year. The project is expected
to give equal cash flows each year for the following 7 years.
What should be the minimum amount of these annual cash
flows so that the project breaks even?
3. You would like income of $50,000 ten years from now. For
each of the following 4 years (times 11-14), you would like
this income to grow at a rate of 4%. If you start with savings
of $38,000 today, how much should you deposit each year for
the next 10 years in order to be able to attain your desired
income in the future? (The first deposit will be 1 year from
now.)