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FIN 301 Class Notes
Chapter 4: Time Value of Money
The concept of 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?
- Consumption forgone has value
- Investment lost has opportunity cost
- Inflation may increase and purchasing power decrease
Now,
Do you prefer a $100 today or $110 one year from now? Why?
You will ask yourself one question:
- Do I have any thing better to do with that $100 than lending it for $ extra?
- What if I take $100 now and invest it, would I make more or less than $110 in one year?
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?
Time Lines:
Show the timing of cash flows. Tick marks occur at the end of periods, so Time 0 is today; Time 1 is the end of the first period (year, month, etc.) or the beginning of the second period.
Example 1 : $100 lump sum due in 2 years
Today End of End of Period 1 Period 2 (1 period (2 periods form now) form now)
Example 2 : $10 repeated at the end of next three years (ordinary annuity )
CF 0 CF 1 CF 2 CF (^3)
(^0 1 2 )
i%
100
0 1 2 i
10 10 10
0 1 2 3 i
FUTUR VALUE OF A SINGLE CASH FLOW
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% FV
- 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?
i=5% FV
- 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.
FV
i=13%
Present
Value of
Money
Future
Value of
Money
Investment
Compounding
Detailed calculation:
Simple example:
Invest $100 now at 5%. How much will you have after a year?
FV 1 = PV + INT
= PV + (PV × i) = PV × (1 + i)
FV 1 = $100 + INT
= $100 + ($100 × .05)
Or
FV 1 = $100 × (1+0.05)
= $100 × (1.05)
To solve for FV, You need 1- Present Value (PV) 2- Interest rate per period (i) 3- Number of periods (n)
Remarks: As PVÇ, FV nÇ.
As iÇ, FV nÇ.
As nÇ, FVnÇ.
1- By Formula 0 (1^ )
n FV (^) n = PV + i
2- By Table I FV^ n = PV^ 0 (^ FV IFi n , )
, (1^ )
n
⇒ FVIFi n = + i
3- By calculator (BAII Plus)
Clean the memory: CLR TVM Î
Notes:
- To enter (i) in the calculator, you have to enter it in % form.
- Use To change the sign of a number. For example, to enter -100: 100
- 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.
INPUTS
OUTPUT
N I/Y PV PMT
FV
3 10 0
133.
-
CPT
PV
2nd FV
CE/C
Example:
Jack deposited $1000 in saving account earning 6% interest rate. How much will jack money be worth at the end of 3 years?
Time line
Before solving the problem, List all inputs: I = 6% or 0. N= 3 PV= 1000 PMT= 0
Solution:
By formula: FVn = PV × (1+i)n
FV 3 = $1000 × (1+0.06)^3 = $1000 ×(1.06)^3 = $1000 ×1. = $ 1,
By Table: FVn= PV × FVIF (^) i,n
FV 3 = $1000 × FVIF (^) 6%, = $1000 × 1. = $ 1,
1000
(^0 1 2 ) ? 6%
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,
PV0 FV
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?
0 i=5% 1 FV
PV
Present
Value of
Money
Future
Value of
Money
Discounting
Detailed calculation
FV n = PV (1 + i ) n
0
n n
FV
PV
i
0
PV FV n n
i
⇒ = ×
Example:
PV 4 = FV 4 = $121.
PV 3 = FV 4 × [1/(1+i)]
= $121.55× [1/(1.05)] = $115.
PV 2 = FV 4 × [1/(1+i)(1+i)]
= $121.55× [1/(1.05)(1.05)] = $121.55× [1/(1.05)^2 ] = $110.
$100 $105 $110.25 $115.76 = $121.
÷ 1.05 ÷ 1.05 ÷ 1.05 ÷ 1.
To solve for PV, You need 4- Future Value (FV) 5- Interest rate per period (i) 6- Number of periods (n)
Remarks: As FVn Ç, PVÇ As iÇ, PVÈ As nÇ, PVÈ
1- By Formula 0
PV FV n n
i
= ×
2- By Table II PV^ 0 = FV^ n (^ PV IFi n , )
,
PV IF i n n
i
3- By calculator (BAII Plus)
Clean the memory: CLR TVM Î
INPUTS
OUTPUT
N I/Y PV PMT
PV
3 10 0
-
133.
CPT
FV
CE/C 2nd FV
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
Before solving the problem, List all inputs: I = 6% or 0. N= 3 FV= $ PMT= 0
Solution:
By formula: PV 0 = FV 3 × [1/(1+i) n]
PV 0 = $1,191 × [1/(1+0.06) 3 ] = $1,191 × [1/(1.06) 3 ] = $1,191 × (1/1.191) = $1,191 × 0. = $
By Table: = FVn × PVIFi,n
PV 0 = $1,191 × PVIF6%, = $1,191 × 0. = $ 1000
?
0 1 2
6%
3 $
Solving for the interest ratei
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: (^1)
PV
FV i
n
1 n − ⎥⎦
⎤ ⎢⎣
⎡
1 1191 3 1 1000
i
⎡ ⎤ = − ⎢ ⎥ ⎣ ⎦
=0.
By Table: FV^ n = PV^ 0 (^ FV IFi n , )
, 0
n i n
FV FV IF PV
⇒ =
,
FV IF i = =
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 , )
0 i n , n
PV PV IF FV
⇒ =
,
PV IF i = =
From the Table II at n=3 we find that the interest rate that yield 0.8396 PVIF is 6%
By calculator:
Clean the memory: CLR TVM Î
INPUTS
OUTPUT
N (^) PV PV PMT
I/Y
(^3) -1000 0
5.
1191
CPT
FV
CE/C 2nd FV
By calculator:
Clean the memory: CLR TVM Î
FUTURE VALUE OF ANNUTIES
An annuity is a series ofequal payments at fixed intervals for a specified number of periods.
PMT = the amount of periodic payment
Ordinary (deferred) annuity: Payments occur at the end of each period.
Annuity due: Payments occur at thebeginning of each period.
INPUTS
OUTPUT
I/Y (^) PV PMT
N
(^8) -100,000 0
29.
1,000,
CPT
FV
CE/C 2nd FV
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?
1 2
n n
FV A N n PMT i PMT i PMT
− −
(Hard to use this formula)
PM PM
i
PM
Due
Ordinary
PM PM PM
i
0 1 2 3 4 n-1 n
PMT PMT PMT PMT PMT PMT
Time
