Time is an important variable in business decisions. Money value of time not only in business in but in life also. Financial decisions take today exercise their impact for a number of years to come. Every individual has time preference of money.

Reasons :

Reasons Uncertainty of receiving the money later. Preference for consumption today. Loss of investment opportunities. Loss of value because of inflation.

The Time Value of Money:

The Time Value of Money The Interest Rate Simple Interest Compound Interest Amortizing a Loan Compounding More Than Once per Year

The Interest Rate:

Obviously, $10,000 today . You already recognize that there is TIME VALUE TO MONEY !! The Interest Rate Which would you prefer -- $10,000 today or $10,000 in 5 years ?

Why TIME?:

TIME allows you the opportunity to postpone consumption and earn INTEREST . Why TIME? Why is TIME such an important element in your decision?

Types of Interest:

Types of Interest Compound Interest Interest paid (earned) on any previous interest earned, as well as on the principal borrowed (lent). Simple Interest Interest paid (earned) on only the original amount, or principal, borrowed (lent).

Simple Interest Formula:

Simple Interest Formula Formula SI = P 0 ( i )( n ) SI : Simple Interest P 0 : Deposit today (t=0) i : Interest Rate per Period n : Number of Time Periods

Simple Interest Example:

SI = P 0 ( i )( n ) = 1,000 ( .07 )( 2 ) = 140 Rs. Simple Interest Example Assume that you deposit Rs. 1,000 in an account earning 7% simple interest for 2 years. What is the accumulated interest at the end of the 2nd year?

Simple Interest (FV):

FV = P 0 + SI = 1,000 + $140 = 1,140 Future Value is the value at some future time of a present amount of money, or a series of payments, evaluated at a given interest rate. Simple Interest (FV) What is the Future Value ( FV ) of the deposit?

Simple Interest (PV):

The Present Value is simply the 1,000 you originally deposited. That is the value today! Present Value is the current value of a future amount of money, or a series of payments, evaluated at a given interest rate. Simple Interest (PV) What is the Present Value ( PV ) of the previous problem?

Future Value Single Deposit (Formula):

FV 1 = P 0 (1+ i ) 1 = $1,000 (1 .07 ) = $1,070 Compound Interest You earned Rs.70 interest on your Rs.1,000 deposit over the first year. This is the same amount of interest you would earn under simple interest. Future Value Single Deposit (Formula)

Slide 13:

FV 1 = P 0 (1+ i ) 1 = 1,000 (1 .07 ) = 1,070 FV 2 = FV 1 (1+ i ) 1 = P 0 (1+ i )(1+ i ) = 1,000 (1 .07 )(1 .07 ) = P 0 (1+ i ) 2 = 1,000 (1 .07 ) 2 = 1,144.90 You earned an EXTRA 4.90 in Year 2 with compound over simple interest. Future Value Single Deposit (Formula)

General Future Value Formula:

FV 1 = P 0 (1+ i ) 1 FV 2 = P 0 (1+ i ) 2 General Future Value Formula: FV n = P 0 (1+ i ) n or FV n = P 0 ( FVIF i , n ) -- See Table I General Future Value Formula etc.

Valuation Using Table I:

FVIF i , n is found on Table I at the end of the book. Valuation Using Table I

Using Future Value Tables:

FV 2 = $1,000 ( FVIF 7% , 2 ) = $1,000 ( 1.145 ) = $1,145 [Due to Rounding] Using Future Value Tables

Double Your Money!!!:

We will use the “ Rule-of-72 ” . Double Your Money!!! Quick! How long does it take to double $5,000 at a compound rate of 12% per year (approx.)?

The “Rule-of-72”:

Approx. Years to Double = 72 / i % 72 / 12% = 6 Years [Actual Time is 6.12 Years] The “Rule-of-72” Quick! How long does it take to double Rs.5,000 at a compound rate of 12% per year (approx.)?

Solving the Period Problem:

The result indicates that a Rs. 1,000 investment that earns 12% annually will double to Rs. 2,000 in 6.12 years . Note: 72/12% = approx. 6 years Solving the Period Problem N I/Y PV PMT FV Inputs Compute 12 1,000 0 +2,000 6.12 years

Present Value Single Deposit (Graphic):

Assume that you need $1,000 in 2 years. Let’s examine the process to determine how much you need to deposit today at a discount rate of 7% compounded annually. 0 1 2 $1,000 7% PV 1 PV 0 Present Value Single Deposit (Graphic)

Present Value Single Deposit (Formula):

PV 0 = FV 2 / (1+ i ) 2 = $1,000 / (1 .07 ) 2 = FV 2 / (1+ i ) 2 = $873.44 Present Value Single Deposit (Formula) 0 1 2 $1,000 7% PV 0

General Present Value Formula:

PV 0 = FV 1 / (1+ i ) 1 PV 0 = FV 2 / (1+ i ) 2 General Present Value Formula: PV 0 = FV n / (1+ i ) n or PV 0 = FV n ( PVIF i , n ) -- See Table II General Present Value Formula etc.

Valuation Using Table II:

PVIF i , n is found on Table II at the end of the book. Valuation Using Table II

Using Present Value Tables:

PV 2 = $1,000 ( PVIF 7% , 2 ) = $1,000 ( .873 ) = $873 [Due to Rounding] Using Present Value Tables

Solving the PV Problem:

N: 2 Periods (enter as 2) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: Compute (Resulting answer is negative “deposit”) PMT: Not relevant in this situation (enter as 0) FV: $1,000 (enter as positive as you “receive $”) Solving the PV Problem N I/Y PV PMT FV Inputs Compute 2 7 0 +1,000 -873.44

Story Problem Example:

Julie Miller wants to know how large of a deposit to make so that the money will grow to $10,000 in 5 years at a discount rate of 10% . Story Problem Example 0 1 2 3 4 5 $10,000 PV 0 10%

Story Problem Solution:

Calculation based on general formula: PV 0 = FV n / (1+ i ) n PV 0 = $10,000 / (1+ 0 .10 ) 5 = $6,209.21 Story Problem Solution

Solving the PV Problem:

Solving the PV Problem N I/Y PV PMT FV Inputs Compute 5 10 0 +10,000 -6,209.21 The result indicates that a $10,000 future value that will earn 10% annually for 5 years requires a $6,209.21 deposit today (present value).

Types of Annuities:

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. An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.

Examples of Annuities:

Examples of Annuities Student Loan Payments Car Loan Payments Insurance Premiums Mortgage Payments Retirement Savings

Parts of an Annuity:

Parts of an Annuity 0 1 2 3 $100 $100 $100 (Ordinary Annuity) End of Period 1 End of Period 2 Today Equal Cash Flows Each 1 Period Apart End of Period 3

Parts of an Annuity:

Parts of an Annuity 0 1 2 3 $100 $100 $100 (Annuity Due) Beginning of Period 1 Beginning of Period 2 Today Equal Cash Flows Each 1 Period Apart Beginning of Period 3

Overview of an Ordinary Annuity -- FVA:

FVA n = R (1+ i ) n-1 + R (1+ i ) n-2 + ... + R (1+ i ) 1 + R (1+ i ) 0 Overview of an Ordinary Annuity -- FVA R R R 0 1 2 n n+1 FVA n R = Periodic Cash Flow Cash flows occur at the end of the period i% . . .

Example of an Ordinary Annuity -- FVA:

FVA 3 = $1,000 (1 .07 ) 2 + $1,000 (1 .07 ) 1 + $1,000 (1 .07 ) 0 = $1,145 + $1,070 + $1,000 = $3,215 Example of an Ordinary Annuity -- FVA $1,000 $1,000 $1,000 0 1 2 3 4 $3,215 = FVA 3 7% $1,070 $1,145 Cash flows occur at the end of the period

Hint on Annuity Valuation:

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.

Valuation Using Table III:

FVA n = R ( FVIFA i% , n ) FVA 3 = $1,000 ( FVIFA 7% , 3 ) = $1,000 ( 3.215 ) = $3,215 Valuation Using Table III

Solving the FVA Problem:

N: 3 Periods (enter as 3 year-end deposits) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: Not relevant in this situation (no beg value) PMT: $1,000 (negative as you deposit annually) FV: Compute (Resulting answer is positive) Solving the FVA Problem N I/Y PV PMT FV Inputs Compute 3 7 0 -1,000 3,214.90

Overview View of an Annuity Due -- FVAD:

FVAD n = R (1+ i ) n + R (1+ i ) n-1 + ... + R (1+ i ) 2 + R (1+ i ) 1 = FVA n (1+ i ) Overview View of an Annuity Due -- FVAD R R R R R 0 1 2 3 n-1 n FVAD n i% . . . Cash flows occur at the beginning of the period

Example of an Annuity Due -- FVAD:

FVAD 3 = $1,000 (1 .07 ) 3 + $1,000 (1 .07 ) 2 + $1,000 (1 .07 ) 1 = $1,225 + $1,145 + $1,070 = $3,440 Example of an Annuity Due -- FVAD $1,000 $1,000 $1,000 $1,070 0 1 2 3 4 $3,440 = FVAD 3 7% $1,225 $1,145 Cash flows occur at the beginning of the period

Valuation Using Table III:

FVAD n = R ( FVIFA i% , n )(1+ i ) FVAD 3 = $1,000 ( FVIFA 7% , 3 )(1 .07 ) = $1,000 ( 3.215 )(1 .07 ) = $3,440 Valuation Using Table III

Overview of an Ordinary Annuity -- PVA:

PVA n = R /(1+ i ) 1 + R /(1+ i ) 2 + ... + R /(1+ i ) n Overview of an Ordinary Annuity -- PVA R R R 0 1 2 n n+1 PVA n R = Periodic Cash Flow i% . . . Cash flows occur at the end of the period

Example of an Ordinary Annuity -- PVA:

PVA 3 = $1,000 /(1 .07 ) 1 + $1,000 /(1 .07 ) 2 + $1,000 /(1 .07 ) 3 = $934.58 + $873.44 + $816.30 = $2,624.32 Example of an Ordinary Annuity -- PVA $1,000 $1,000 $1,000 0 1 2 3 4 $2,624.32 = PVA 3 7% $934.58 $873.44 $816.30 Cash flows occur at the end of the period

Hint on Annuity Valuation:

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 future value of an annuity due can be viewed as occurring at the end of the first cash flow period.

Valuation Using Table IV:

PVA n = R ( PVIFA i% , n ) PVA 3 = $1,000 ( PVIFA 7% , 3 ) = $1,000 ( 2.624 ) = $2,624 Valuation Using Table IV

Solving the PVA Problem:

N: 3 Periods (enter as 3 year-end deposits) I/Y: 7% interest rate per period (enter as 7 NOT .07) PV: Compute (Resulting answer is positive) PMT: $1,000 (negative as you deposit annually) FV: Not relevant in this situation (no ending value) Solving the PVA Problem N I/Y PV PMT FV Inputs Compute 3 7 -1,000 0 2,624.32

Overview of an Annuity Due -- PVAD:

PVAD n = R /(1+ i ) 0 + R /(1+ i ) 1 + ... + R /(1+ i ) n-1 = PVA n (1+ i ) Overview of an Annuity Due -- PVAD R R R R 0 1 2 n-1 n PVAD n R : Periodic Cash Flow i% . . . Cash flows occur at the beginning of the period

Example of an Annuity Due -- PVAD:

PVAD n = $1,000 /(1 .07 ) 0 + $1,000 /(1 .07 ) 1 + $1,000 /(1 .07 ) 2 = $2,808.02 Example of an Annuity Due -- PVAD $1,000.00 $1,000 $1,000 0 1 2 3 4 $2,808.02 = PVAD n 7% $ 934.58 $ 873.44 Cash flows occur at the beginning of the period

Valuation Using Table IV:

PVAD n = R ( PVIFA i% , n )(1+ i ) PVAD 3 = $1,000 ( PVIFA 7% , 3 )(1 .07 ) = $1,000 ( 2.624 )(1 .07 ) = $2,808 Valuation Using Table IV

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