2. 1- 2
Some questions that we will answer…
FV ? What is the future value of a single sum? (single sum)
PV ? What is the present value of a single sum? (single sum)
FVA ? What is the future value of an annuity? (annuity)
PVA ? What is the present value of an annuity? (annuity)
FVA(Due) ? What is the future value of an annuity due? (annuity due)
PVA(Due)? What is the present value of an annuity due? (annuity
due)
PV of an Uneven Cash Flow Series ? How do we answer this?
PVP (Present Value of a Perpetuity) ?
…
3. 1- 3
We have $100 and the interest rate is 10%.
0 1
|__________________|
$100 ----------------------------> ?
$110
0 1
|__________________|
? <- ---------------------------- $110
$100
4. 1- 4
Cash Flow Time Lines
CF0 CF1 CF3CF2
0 1 2 3
r%
Tick marks at ends of periods, so t=0 is today;
t=1 is the end of Period 1; or the beginning of
Period 2.
Graphical representations used to
show the timing of cash flows.
5. 1- 5
Future Value
The amount to which a cash flow or series of
cash flows will grow over a period of time
when compounded at a given interest rate.
How much would you have at the end of one year if you
deposited $100 in a bank account that pays 10 percent
interest each year?
FVt = FV1 = PV + INT
= PV + (PV x r)
= PV (1 + r)
= $100(1+0.10) = $100(1.10) = $110
6. 1- 6
Finding FVs is Compounding.
What’s the FV of an initial $100
after 3 years if r = 10%?
FV = ?
0 1 2 3
r=10%
100
7. 1- 7
In general, FVt = PV (1 + r)t
After 3 years:
FV3 = PV(1 + r)3
= 100 (1.10)3
= $133.10.
After 2 years:
FV2 = PV(1 + r)2
= $100 (1.10)2
= $121.00.
After 1 year:
FV1 = PV + INT = PV + PV (r)
= PV(1 + r)
= $100 (1.10)
= $110.00.
Future Value
8. 1- 8
What is the PV of $100 due
in 3 years if r = 10%?
100
0 1 2 3
10%
PV = ?
9. 1- 9
Solve FVt = PV (1 + r )t for PV:
( )
t
r+1
1
FV=
tr+1
t
FV
=PV t
( )
$75.13.=
0.7513$100=
3
1.10
1
$100=PV
What is the PV of $100 due
in 3 years if r = 10%?
10. 1- 10
Present Value
Present value is the value today of a
future cash flow or series of cash flows.
Discounting is the process of finding
the present value of a cash flow or
series of cash flows, the reverse of
compounding.
11. 1- 11
Consider the following two questions:
"What is the present value of $500 to be received in 20 years
if the interest rate is 7 percent?"
and
"How much would I have to invest now in order to receive
$500 after 20 years, given an interest rate of 7 percent?"
12. 1- 12
Time line for an ordinary annuity of $100 for
3 years.
100 100100
0 1 2 3
r%=10
13. 1- 13
What’s the FV of a 3-year
Ordinary Annuity of $100 at 10%?
100 100100
0 1 2 3
10%
110
121
FV = 331
14. 1- 14
What is the PV of this Ordinary
Annuity?
248.69 = PV
100 100100
0 1 2 3
10%
90.91
82.64
75.13
18. 1- 18
Find the FV and PV if the
Annuity were an Annuity Due.
100 100
0 1 2 3
r =10%
100
19. 1- 19
What is the PV of this Annuity Due?
100 100
0 1 2 3
r%=10
100
90.91
82.64
PV = 273.55
20. 1- 20
What’s the FV of a 3-year Annuity Due of $100 at
10%?
100 100
0 1 2 3
10%
110
121
133.10
FV = 364.10
100
21. 1- 21
PVAdue = PVAordinary x (1+r)]
PVA(due)3 = 100[(2.4869) x 1.10] = 273.55
FVAdue = FVAordinary x 1 + r)]
FVA(due)3 = 100[(3.31) x 1.10] = 364.10
22. 1- 22
Uneven Cash Flow Streams
A series of cash flows in which the
amount varies from one period to the
next.
Payment (PMT) designates constant
cash flows
Cash flow (CF) designates cash flows
in general, including uneven cash
flows
23. 1- 23
What is the PV of this
Uneven Cash Flow Stream?
0
100
1
300
2
300
3
10%
-50
4
90.91
247.93
225.39
-34.15
530.08 = PV
24. 1- 24
Amortized Loans
Amortized Loan: A loan that is repaid in
equal payments over its life.
Amortization tables are widely used-- for
home mortgages, auto loans, business
loans, retirement plans, etc.
They are very important!
25. 1- 25
Construct an amortization schedule
for a $1,000, 10% annual rate loan
with 3 equal payments.
Step 1: Find the required payments
PMT PMTPMT
0 1 2 3
10%
1000
PVA3 = PMT (PVIFA I,N)
1000 = PMT (2.4869) what is PMT?
PMT = 1000/2.4869 = 402.11
26. 1- 26
Step 2: Find interest charge for Year 1
INTt = Beg balt (i)
INT1 = 1000(0.10) = $100.
Step 3: Find repayment of principal in Year 1
Repmt. = PMT - INT
= 402.11 - 100
= $302.11.
Step 4: Find ending balance after Year 1
End bal = Beg bal - Repmt
= 1000 - 302.11 = $697.89.
Repeat these steps for Years 2 and 3
to complete the amortization table.
27. 1- 27
Loan Amortization Table
10 Percent Interest Rate
YR Beg Bal PMT INT Prin PMT End Bal
1 $1000.00 $402.11 $100.00 $302.11 $697.89
2 697.89 402.11 69.79 332.32 365.57
3 365.57 402.11 36.56 365.55 0.02
Total 1,206.33 206.35 999.98 *
* Rounding difference
31. 1- 31
$100 (1 + r )3 = $125.97.
What interest rate would cause $100 to
grow to $125.97 in 3 years?
Solve for r,
r = (125.97/100)1/3 - 1 = 8 %
32. 1- 32
$100 (1 + r )3 = $125.97.
What interest rate would cause $100 to
grow to $125.97 in 3 years?
INPUTS
OUTPUT
3 ? -100 0 125.97
8%
N I/YR PV PMT FV
33. 1- 33
250 250
0 1 2 3
r = ?
- 846.80
4
250 250
You pay $846.80 for an investment that promises
to pay you $250 per year for the next four years,
with payments made at the end of each year.
What interest rate will you earn on this
investment?
Solving for Interest Rates
with Annuities
34. 1- 34
Use trial-and-error by substituting different values
of r into the following equation until the right side
equals $846.80.
Numerical Solution
= +
r
-1
$250$846.80
4
)(1
1
r
36. 1- 36
What if we want find n?
What oh what can we do?
37. 1- 37
Solve for n:
FVn = 1(1 + r)n
2 = 1(1.20)n
If your money grows at 20% per
year,how long before your money
doubles?
The numerical solution is somewhat difficult.
39. 1- 39
Semiannual and Other
Compounding Periods
Annual compounding is the process of
determining the final value of a cash flow
or series of cash flows when interest is
added once a year.
Semiannual compounding is the process of
determining the final value of a cash flow
or series of cash flows when interest is
added twice a year.
40. 1- 40
LARGER! If compounding is more
frequent than once a year--for example,
semi-annually, quarterly, or daily--
interest is earned on interest more often.
Will the FV of a lump sum
be larger or smaller if we
compound more often,
holding the stated r% constant?
Why?
42. 1- 43
FV of $100 after 3 years under 10% semi-
annual compounding? Quarterly?
FV = $100 1 +
0.10
2
3s
2x3
nxm
m
.10
+1PV=FV
= $100(1.05)6 = $134.01
FV3Q = $100(1.025)12 = $134.49
43. 1- 44
Effective (or equivalent) annual rate
(EAR): the annual rate of interest actually
being earned, considering compounding.
– EAR% for 10% semiannual interest
EAR% = (1 + Quoted Rate/M)M – 1
= (1 + 0.10/2)2 – 1 = 10.25%
– Should be indifferent between
receiving 10.25% annual interest and
receiving 10% interest, compounded
semiannually.
5-44
44. 1- 45
Why is it important to consider
effective rates
of return?
Investments with different compounding
intervals provide different effective
returns.
To compare investments with different
compounding intervals, you must look at
their effective returns (EAR).
5-45
45. 1- 46Why is it important to consider
effective rates
of return?
See how the effective return varies
between investments with the same
nominal rate, but different compounding
intervals.
EARANNUAL 10.00%
EARSEMIANNUALLY 10.25%
EARQUARTERLY 10.38%
EARMONTHLY 10.47%
EARDAILY (365) 10.52%
5-46
46. 1- 47
What is the FV of $100 after 3 years under 10%
semiannual compounding? Quarterly compounding?
$134.49)$100(1.025FV
$134.01$100(1.05)FV
2
0.10
1$100FV
M
RateQuoted
1PVFV
12
3Q
6
3S
32
3S
NM
N
==
==
+=
+=
5-47
