2. The “Rule-of-72”
Approx. Years to Double = 72 / i%
QUICK!!
How long does it take to double 150,000 Taka at a
compound interest rate of 12% per year?
72 / 12% = 6 years
[Actual time is 6.12 years]
4. PV: Compound interest
•Assume that you need 1,000 Taka in 2 years.
How much do you need to deposit today at a
discount rate of 7% compounded annually?
0 1 2
7%
Taka 1,000
PV1PV0
5. PV: Compound interest formula
Formula PV0 = FVn / (1+i)n
PV0: Present value (at time 0)
FVn: Future value after n time periods
i: Interest rate per period
n: The number of time periods
7. General PV compound interest formula
Formula
PV0 = FV1 / (1+i)1
PV0 = FV2 / (1+i)2
etc
General present value formula
PV0 = FVn / (1+i)n
or PV0 = FVn (PVIFi,n) -- See Table II
8. Valuation using PV table
•PVIFi,n is found in this table.
– You can find this table in your text book.
– I will also provide you with one during
tests/midterm etc.
Period 6% 7% 8%
1 .9434 .9346 .9259
2 .8900 .8734 .8573
3 .8396 .8163 .7938
4 .7921 .7629 .7350
5 .7473 .7130 .6806
9. Valuation using PV table
PV2 = Taka 1,000 (PVIF7%,2)
= Taka 1,000 (.8734)
= Taka 873.40
Period 6% 7% 8%
1 .9434 .9346 .9259
2 .8900 .8734 .8573
3 .8396 .8163 .7938
4 .7921 .7629 .7350
5 .7473 .7130 .6806
10. PV table example #1
Shovon wants to know how large a deposit to make
so that the money will grow to 10,000 Taka in 5 years
at a discount rate of 6%.
0 1 2 3 4 5
10,000 Taka
PV0
6%
11. PV table solution #1
Shovon wants to know how large a deposit to make
so that the money will grow to 10,000 Taka in 5 years
at a discount rate of 6%.
Calculation based on general formula:
PV0 = FVn / (1+i)n
PV0 = Taka 10,000 / (1+ 0.06)5
= Taka 7,472.58
Calculation based on table:
PV0 = Taka 10,000 (PVIF6%, 5)
= Taka 10,000 (.7473)
= Taka 7,473.00
12. PV table example #2
Marjan wants to know how large a deposit to make
so that the money will grow to 10,000 Taka in 3 years
at a discount rate of 8%.
Calculation based on general formula:
PV0 = FVn / (1+i)n
PV0 = Taka 10,000 / (1+ 0.08)3
= Taka 7,938.32
Calculation based on table:
PV0 = Taka 10,000 (PVIF8%, 3)
= Taka 10,000 (.7938)
= Taka 7,938.00
13. PV table example #3
Galib wants to know how large a deposit to make so
that the money will grow to 10,000 Taka in 4 years at
a discount rate of 7%.
Calculation based on general formula:
PV0 = FVn / (1+i)n
PV0 = Taka 10,000 / (1+ 0.07)4
= Taka 7,628.95
Calculation based on table:
PV0 = Taka 10,000 (PVIF7%, 4)
= Taka 10,000 (.7629)
= Taka 7,629.00
14. Annuities
An Annuity represents a series of equal
payments (or receipts) occurring over a
specified number of equidistant periods.
15. Types of annuities
•Ordinary annuity: Payments or receipts
occur at the end of each period.
•Annuity due: Payments or receipts occur at
the beginning of each period.
17. Parts of an annuity
0 1 2 3
Tk. 100 Tk. 100 Tk. 100
End of
Period 1
End of
Period 2
Today Equal Cash Flows
Each 1 Period Apart
End of
Period 3
Ordinary Annuity
18. Parts of an annuity
0 1 2 3
Tk.100 Tk.100 Tk.100
Beginning of
Period 1
Beginning of
Period 2
Today Equal Cash Flows
Each 1 Period Apart
Beginning of
Period 3
Annuity Due
19. Overview of an ordinary annuity - FVA
FVAn = R(1+i)n-1 + R(1+i)n-2 +
... + R(1+i)1 + R(1+i)0
R R R
0 1 2 n n+1
FVAn
R = Periodic
Cash Flow
Cash flows occur at the end of the period
i% . . .
20. Example of an ordinary annuity - FVA
FVA3 = $1,000(1.07)2 + $1,000(1.07)1 +
$1,000(1.07)0
= $1,145 + $1,070 + $1,000
= $3,215
$1,000 $1,000 $1,000
0 1 2 3 4
$3,215 = FVA3
7%
$1,070
$1,145
Cash flows occur at the end of the period
21. Hint on annuity valuation
The future value of an ordinary annuity can
be viewed as occurring at the end of the last
cash flow period, whereas the future value of
an annuity due can be viewed as occurring at
the beginning of the last cash flow period.
23. Overview of an annuity due - FVAD
FVADn = R(1+i)n + R(1+i)n-1 +
... + R(1+i)2 + R(1+i)1
= FVAn (1+i)
R R R R R
0 1 2 3 n-1 n
FVADn
i% . . .
Cash flows occur at the beginning of the period
24. Example of an annuity due – FVAD
FVAD3 = $1,000(1.07)3 +
$1,000(1.07)2 +
$1,000(1.07)1
= $1,225.04 + $1,144.90 + $1,070.00
= $3,439.94
$1,000 $1,000 $1,000 $1,070.00
0 1 2 3 4
$3,439.94 = FVAD3
7%
$1,225.04
$1,144.90
Cash flows occur at the beginning of the period
26. Overview of an ordinary annuity – PVA
PVAn = R/(1+i)1 + R/(1+i)2
+ ... + R/(1+i)n
R R R
0 1 2 n n+1
PVAn
R = Periodic
Cash Flow
i% . . .
Cash flows occur at the end of the period
27. Example of an ordinary annuity – PVA
PVA3 = $1,000/(1.07)1
+ $1,000/(1.07)2
+ $1,000/(1.07)3
= $934.58 + $873.44 + $816.30
= $2,624.32
$1,000 $1,000 $1,000
0 1 2 3 4
$2,624.32 = PVA3
7%
$ 934.58
$ 873.44
$ 816.30
Cash flows occur at the end of the period
28. Hint on annuity valuation
The present value of an ordinary annuity can
be viewed as occurring at the beginning of the
first cash flow period, whereas the present
value of an annuity due can be viewed as
occurring at the end of the first cash flow
period.
30. Overview of an annuity due - PVD
PVADn = R/(1+i)0 + R/(1+i)1 + ... + R/(1+i)n-1
= PVAn (1+i)
R R R R
0 1 2 n-1 n
PVADn
R: Periodic
Cash Flow
i% . . .
Cash flows occur at the beginning of the period
31. Example of an annuity due – PVAD
PVADn = $1,000/(1.07)0 + $1,000/(1.07)1
+ $1,000/(1.07)2 = $2,808.02
$1,000.00 $1,000 $1,000
0 1 2 3 4
$2,808.02 = PVADn
7%
$ 934.58
$ 873.44
Cash flows occur at the beginning of the period
33. Steps to solve TVM problems
1. Read problem thoroughly
2. Determine if it is a PV or FV problem
3. Create a time line
4. Put cash flows and arrows on time line
5. Determine if solution involves a single CF,
annuity stream(s), or mixed flow
6. Solve the problem
7. Check with financial calculator (optional)
![The “Rule-of-72”
Approx. Years to Double = 72 / i%
QUICK!!
How long does it take to double 150,000 Taka at a
compound interest rate of 12% per year?
72 / 12% = 6 years
[Actual time is 6.12 years]](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)