Time Value of Money
Time Value of Money
Time Value of Money
An amount of money received today is worth more than the same dollar
value received a year from now. Why?
Do you prefer a $100 today or a $100 one year from now? why?
Now,
Do you prefer a $100 today or $110 one year from now? Why?
Note:
Two elements are important in valuation of cash flows:
- What interest rate (opportunity rate, discount rate, required rate of
return) do you want to evaluate the cash flow based on?
- At what time do these the cash flows occur and at what time do you need
to evaluate them?
1
Time Lines:
0 1 2 3
i%
0 1 2
i
100
Example 2 : $10 repeated at the end of next three years (ordinary annuity )
0 1 2 3
i
10 10 10
2
Calculations of the value of money problems:
The value of money problems may be solved using
1- Formulas.
2- Interest Factor Tables. (see p.684)
3- Financial Calculators (Basic keys: N, I/Y, PV, PMT, FV).
I use BAII Plus calculator
4- Spreadsheet Software (Basic functions: PV, FV, PMT, NPER,RATE).
I use Microsoft Excel.
3
FV & PV PV = FV/(1+i)^n FV = $1000 (1+0.03)^1
FV = PV (1+i)^n FV = $1030
Examples:
• You deposited $1000 today in a saving account at BancFirst that pays
you 3% interest per year. How much money you will get at the end of the
first year ?
i=3% FV1
0 1
$1000
• You lend your friend $500 at 5% interest provided that she pays you
back the $500 dollars plus interest after 2 years. How much she should
pay you?
FV = ?
i=5% FV2 FV = $500(1+0.05)^2
FV = 551.25
0 1 2
$500
• You borrowed $10,000 from a bank and you agree to pay off the loan after
5 years from now and during that period you pay 13% interest on loan.
$10,000
0 1 2 3 4 5
FV5
i=13%
Investment
Present Future
Value of Compounding Value of
Money Money
4
Detailed calculation:
Simple example:
Invest $100 now at 5%. How much will you have after a year?
FV1 = PV + INT
= PV + (PV × i)
= PV × (1 + i)
5
Another example: Invest $100 at 5% (per year) for 4 years.
0 1 2 3 4
6
To solve for FV, You need
1- Present Value (PV)
2- Interest rate per period (i)
3- Number of periods (n)
1- By Formula FV n = PV 0 (1 + i ) n
2- By Table I FV n = PV 0 (FV IFi ,n )
⇒ FVIFi ,n = (1 + i )n
INPUTS 3 10 -100 0
N I/Y PV PMT
OUTPUT 133.10
CPT FV
Notes:
- To enter (i) in the calculator, you have to enter it in % form.
- To solve the problems in the calculator or excel, PV and FV cannot have the
same sign. If PV is positive then FV has to be negative.
7
Example:
Time line
0 1 2 3
6%
?
1000
Solution:
8
By calculator:
INPUTS 3 6 -1000 0
N I/Y PV PMT
OUTPUT 1,191.02
CPT FV
By Excel:
=FV (0.06, 3, 0,-1000, 0)
9
PRESENT VALUE OF A SINGLE CASH FLOW
Examples:
• You need $10,000 for your tuition expenses in 5 years how much should
you deposit today in a saving account that pays 3% per year?
$10,000
0 1 2 3 4 5
PV0 FV5
i=3%
• One year from now, you agree to receive $1000 for your car that you sold
today.
How much that $1000 worth today if you use 5% interest rate?
$1000
0 i=5% 1 FV1
PV0
Discounting
Present
Future
Value of
Value of
Money
Money
10
Detailed calculation
FV n = PV (1 + i ) n
FV n
⇒ PV 0 =
(1 + i ) n
1
⇒ PV 0 = FV n ×
(1 + i ) n
Example:
0 1 2 3 4
11
Or
PV2= FV3× [1/ (1+i)]
= $115.76× [1/ (1.05)]
= $110.25
Or
PV1= FV2× [1/ (1+i)]
= $110.25× [1/ (1.05)]
= $105
In general, the present value of an initial lump sum is: PV0 = FVn× [1/(1+i) n]
12
To solve for PV, You need
4- Future Value (FV)
5- Interest rate per period (i)
6- Number of periods (n)
1
1- By Formula PV 0 = FV n ×
(1 + i ) n
2- By Table II PV 0 = FV n (PV IFi ,n )
1
⇒ PV IFi , n =
(1 + i ) n
INPUTS 3 10 133.10 0
N I/Y FV
PV PMT
OUTPUT -100
CPT PV
13
Example:
Jack needed a $1191 in 3 years to be off some debt. How much should jack
put in a saving account that earns 6% today?
Time line
0 1 2 3
$1191
6%
?
Before solving the problem, List all inputs:
I = 6% or 0.06
N= 3
FV= $1191
PMT= 0
Solution:
14
By calculator:
INPUTS 3 6 1191 0
N I/Y FV
PV PMT
OUTPUT -1000
CPT PV
By Excel:
=PV (0.06, 3, 0, 1191, 0)
15
Solving for the interest rate i
You can buy a security now for $1000 and it will pay you $1,191 three years from
now. What annual rate of return are you earning?
By Formula: ⎡
i=⎢
FVn ⎤ n
−1
⎣ PV ⎦⎥
1
⎡ 1191 ⎤ 3
i =⎢ − 1 = 0.06
⎣1000 ⎥⎦
By Table: FV n = PV 0 ( FV IFi , n )
FV n
⇒ FV IFi ,n =
PV 0
1191
FV IFi ,3 = = 1.191
1000
From the Table I at n=3 we find that the interest rate that yield 1.191 FVIF is 6%
Or PV 0 = FV n ( PV IFi ,n )
PV 0
⇒ PV IFi ,n =
FV n
1000
PV IFi ,3 = = 0.8396
1191
From the Table II at n=3 we find that the interest rate that yield 0.8396 PVIF is 6%
16
By calculator:
17
Solving for n:
Your friend deposits $100,000 into an account paying 8% per year. She wants
to know how long it will take before the interest makes her a millionaire.
n=
( Ln FV n ) − ( ln PV )
By Formula:
Ln (1 + i )
FV n = $1, 000, 000 PV = $100,000 1 + i = 1.08
13.82 − 11.51
= = 30 years
0.077
By Table: FV n = PV 0 ( FV IFi , n )
FV n
⇒ FV IFi ,n =
PV 0
1, 000, 000
FV IF8,n = = 10
100, 000
From the Table I at i=8 we find that the number of periods that yield 10 FVIF is 30
Or PV 0 = FV n ( PV IFi , n )
PV 0
⇒ PV IFi , n =
FV n
100, 000
PV IF8, n = = 0.1
1, 000, 000
From the Table II at i=8 we find that the number of periods that yield 0.1 PVIF is 30
18
By calculator:
19
Ordinary
0 1 2 3
i
PM PM PM
Due
0 1 2 3
i
PM PM PM
Example: Suppose you deposit $100 at the end of each year into a savings
account paying 5% interest for 3 years. How much will you have in the
account after 3 years?
0 1 2 3
5%
Time 0 1 2 3 4 n-1 n
PMT PMT PMT PMT PMT PMT
FV A N n = PMT (1 + i ) + PMT (1 + i )
n −1 n −2
+ .... + PMT
(Hard to use this formula)
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⎡ (1 + i )n − 1 ⎤
FV AN n = PMT ⎢ ⎥
⎢⎣ i ⎥⎦
= PMT (FV IFA i ,n )
Future Value Interest
Factor for an Annuity
Note: For an annuity due, simply multiply the answer above by (1+i).
21
Annuity Due:
22
Remark:
1-BY Formula:
⎡ (1 + i )n − 1 ⎤
FV AN n = PMT ⎢ ⎥ ==Î Ordinary Annuity
⎢⎣ i ⎥⎦
⎡ (1 + i )n − 1 ⎤
FV AND n = PMT ⎢ ⎥ (1 + i ) ==Î Annuity Due
⎢⎣ i ⎥⎦
FVANDn = FVAN n (1 + i )
2- BY Table III:
FV A N n = PMT ( FV IFA i ,n ) ==Î Ordinary Annuity
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3- BY calculator:
Ordinary Annuity:
1- Clean the memory: CLR TVMÎ CE/C 2nd FV
3- Make sure you can see END written on the screen then press CE/C
NOTE: If you do not see BGN written on the upper right side of the screen,
you can skip
Step 2 and 3.
INPUTS 3 5 0 -100
N I/Y PMT
PV
OUTPUT FV 315.25
CPT
24
Annuity Due:
Clean the memory: CLR TVM Î CE/C 2nd FV
Set payment mode to BGN of period: BGN Î 2nd PMT
SET Î
2nd ENTER
Make sure you can see BGN written on the screen then press CE/C
INPUTS 3 5 0 -100
N I/Y PMT
PV
OUTPUT FV 331.10
CPT
25
Example:
You agree to deposit $500 at the end of every year for 3 years in an investment
fund that earns 6%.
Time line
0 1 2 3
$500 $500 $500
6%
FV=?
Solution:
⎡ (1 + i )n − 1 ⎤
By formula:
FV AN n = PMT ⎢ ⎥
⎢⎣ i ⎥⎦
⎡ (1 + 0.06)3 − 1 ⎤ ⎡1.191 − 1 ⎤
= 500 ⎢ ⎥ = 500 ⎢ ⎥ = 1,591.80
⎣ 0.06 ⎦ ⎣ 0.06 ⎦
26
By calculator:
INPUTS 3 6 0 -500
N I/Y PMT
PV
OUTPUT FV 1,591.80
CPT
27
Now assume that you deposit the $500 at the beginning of the year not at the
end of the year.
Time line
0 1 2 3
$500 $500 $500 FV=?
6%
Solution:
⎡ (1 + i )n − 1 ⎤
By formula: FV AND n = PMT ⎢ ⎥ (1 + i )
⎢⎣ i ⎥⎦
⎡ (1 + 0.06 )n − 1 ⎤
FV AND 3 = 500 ⎢ ⎥ (1 + 0.06)
⎢⎣ 0.06 ⎥⎦
⎡ 0.191 ⎤
= 500 ⎢ ⎥ (1.06) = 1, 687.30
⎣ 0.06 ⎦
28
By calculator:
Make sure you can see BGN written on the screen then press CE/C
INPUTS 3 6 0 -500
N I/Y PMT
PV
OUTPUT FV 1,687.31
CPT
29
PRESENT VALUE OF ANNUTIES
i = 5%, since you would invest the money at this rate if you had it.
How big does the lump sum have to be to make the choices equally good?
0 1 2 3
Time
100 100 100
÷1.05
95.24 ÷1.052
90.70 ÷1.053
86.38
PVAN3 = 272.32
Formula:
30
⎡1 − 1 ⎤
⎢ 3⎥
PVA 3 = $100⎢ 1.05 ⎥
.05
⎢ ⎥
⎣ ⎦
= $100(2.7232) = $272.32
Note: For annuities due, simply multiply the answer above by (1+i)
PVANDn (annuity due) = PMT (PVIFAi,n) (1+i)
1- BY Formula:
⎡ 1 ⎤
⎢ 1 − n ⎥
PVAN n = PMT ⎢
(1 + i ) ⎥
⎢ i ⎥ ==Î Ordinary Annuity
⎢ ⎥
⎣ ⎦
⎡ 1 ⎤
⎢ 1 − ⎥
( + )
n
1 i ⎥ (1 + i )
PVANDn = PMT ⎢
⎢ i ⎥ ==Î Annuity Due
⎢ ⎥
⎣ ⎦
PVANDn = PVAN n (1 + i )
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2- BY Table IV:
3- BY calculator:
Ordinary Annuity:
Clean the memory: CLR TVMÎ CE/C 2nd FV
Make sure you do not see BGN written on the upper right side of the screen.
INPUTS 3 5 0 -100
N I/Y FV PMT
OUTPUT PV 272.32
CPT
32
Annuity Due:
Clean the memory: CLR TVM Î CE/C 2nd FV
Set payment mode to BGN of period: BGN Î 2nd PMT
SET Î
2nd ENTER
Make sure you can see BGN written on the screen then press CE/C
INPUTS 3 5 0 -100
N I/Y FV PMT
OUTPUT PV 285.94
CPT
33
Example:
You agree to receive $500 at the end of every year for 3 years in an investment
fund that earns 6%.
Time line
0 1 2 3
PV=? $500 $500 $500
6%
Solution:
⎡ 1 ⎤
⎢ 1 − ⎥
( + )
n
1 i
PVAN n = PMT ⎢ ⎥
By formula: ⎢ i ⎥
⎢ ⎥
⎣ ⎦
⎡ 1 ⎤
⎢ 1 − ⎥ ⎡ 1 ⎤
( + )
3
1 0.06 1−
PVAN n = 500 ⎢ ⎥ ⎢ 1.191 ⎥
⎢ 0.06 ⎥ = 500 ⎢ ⎥ = $1, 336.51
⎢ 0.06 ⎥
⎢ ⎥
⎣ ⎦ ⎣ ⎦
34
By Table: PVAN n = PMT ( PVIFAi , n )
PVAN 3 = 500( PVIFA6,3 )
= 500(2.673) = 1, 336.51
By calculator:
Clean the memory: CLR TVMÎ CE/C 2nd FV
Make sure you do not see BGN written on the upper right side of the screen.
INPUTS 3 6 0 -500
N I/Y FV PMT
OUTPUT PV 1,336.51
CPT
35
Now assume that you receive the $500 at the beginning of the year not at the
end of the year.
Time line
0 1 2 3
$500 $500 $500
6%
PV=?
Solution
⎡ 1 ⎤
⎢ 1 − ⎥
( + )
n
1 i ⎥ (1 + i )
PVANDn = PMT ⎢
By formula: ⎢ i ⎥
⎢ ⎥
⎣ ⎦
⎡ 1 ⎤ ⎡ 1 ⎤
⎢ 1 − 3 ⎥ −
PVANDn = 500 ⎢
(1 + 0.06 ) ⎥ (1 + 0.06) ⎢ 1 ⎥
= 500 ⎢ 1.191 ⎥ (1.06)
⎢ 0.06 ⎥
⎢ ⎥ ⎢ 0.06 ⎥
⎣ ⎦ ⎣ ⎦
= 1, 416.70
36
By Table: PVANDn = PMT ( PVIFAi ,n ) (1 + i )
PVAND3 = 500( PVIFA6,3 ) (1 + 0.06 )
= 500(2.673)(1.06) = 1, 416.69
By calculator:
Make sure you can see BGN written on the screen then press CE/C
INPUTS 3 6 0 -500
N I/Y FV PMT
OUTPUT PV 1,416.69
CPT
37
Perpetuities
⎡1 − 0 ⎤ = PMT × ⎛ 1 ⎞ = PMT
PVPER0 ( perpetuity ) = PMT × ⎢ ⎥ ⎜ ⎟
⎣ i ⎦ ⎝i⎠ i
Formula:
PMT
PVPER0 =
i
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UNEVEN CASH FLOWS
Present Value
0 1 2 3
100 50 200
95.24 ÷1.05 ÷1.05 2
45.35
÷1.053
172.77
$313.36
Future Value
0 1 2 3
5%
100 50 200.00
×1.05
52.50
×1.052
110.25
$362.75
39
Example:
40
By Calculator:
Press NPV , then the it will ask you to enter the Interest rate (I)
Enter I = 10 Î 10 ENTER
NOTE:
To calculate the future value of uneven cash flows, it is much easier to start by
calculating the Present value of the cash flows using NPV function then
calculate the future value using the future value of a single cash flow rules. The
single cash flow in this case will be the present value.
41
Simple and Compound Interest
Simple Interest
¾ Interest paid on the principal sum only
Compound Interest
¾ Interest paid on the principal and on interest
Example:
Calculate the future value of $1000 deposited in a saving account for 3 years earning
6% . Also, calculate the simple interest, the interest on interest, and the compound
interest.
Principal = PV = $1000
Compound interest = FV – PV = 1191.02 – 1000 = 191.02
Simple Interest = PV * i * n =1000 * 0.06 * 3 = $180
Interest on interest = Compound interest - Simple Interest = 191.02 – 180 =
11.02
42
Effect of Compounding over Time
43
Example: You invest $100 today at 5% interest for 3 years.
= $100(1.05)3
= $100(1.1576)
= $115.76
What if interest is compounded semi-annually (twice a year)?
Then the periods on the time line are no longer years, but half-years!
6 months
Time: 0 1 2 3 4 5 6
2.5%
PV=100
FV6=?
5%
i = Periodic interest rate = = 2.5%
2
n = No. of periods = 3 × 2 = 6
FVn = PV (1 + i ) n
FV6 = $100(1.025)6
= $100(1.1597)
= $115.97
Note: the final value is slightly higher due to more frequent compounding.
44
Will the FV of a lump sum be larger or
smaller if compounded more often,
holding the stated I% constant?
LARGER, as the more frequently compounding
occurs, interest is earned on interest more often.
0 1 2 3
10%
100 133.10
Annually: FV3 = $100(1.10)3 = $133.10
0 1 2 3
0 1 2 3 4 5 6
5%
100 134.01
Semiannually: FV6 = $100(1.05)6 = $134.01
6-24
Important: When working any time value problem, make sure you keep
straight what the relevant periods are!
n = the number of periods
i = the periodic interest rate
The effective annual rate is the interest rate actually being earned per year.
To compare among different nominal rates or to know what is the actual rate
that you’re getting on any investment you have to use the Effective annual
interest rate.
m
⎛ i ⎞
Effective Annual Rate:
i eff = ⎜1 + ⎟ − 1
⎝ m⎠
To compare the two rates in the example,
1
⎛ 0.11 ⎞
1- i eff = ⎜1 + ⎟ − 1 = 0.11 or 11% (Nominal and Effective rates are equal in annual
⎝ 1 ⎠
compounding)
12
⎛ 0.10 ⎞
2- i eff = ⎜1 + ⎟ − 1 = 0.1047 or 10.47 %
⎝ 12 ⎠
46
To compute effective rate using calculator:
ICONV Î 2nd 2
Enter Nominal Rate Î NOM 10 ENTER
(1 + rf ) = (1 + rf′ )(1 + i n )
rf = rf′ + i n + rf′i n
rf ≈ rf′ + i n
47
Amortized Loans
Example: You borrow $10,000 today and will repay the loan in equal installments at
the end of the next 4 years. How much is your annual payment if the interest rate is
9%?
Time 0 9% 1 2 3 4
PVA N= $10,000 PMT PMT PMT PMT
Inputs:
48
Interest amount = Beginning balance * i
Principal reduction = annual payment - Interest amount
Ending balance = Beginning balance - Principal reduction
Beginning balance: Start with principal amount and then equal to previous
year’s ending balance.
• later on, less of each payment is used for interest, and more of it is applied
to paying off the principal.
49