公司理财chap009
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• A generalized formula for Present Value of Annuity: PVA = A × PVIFA
Where: PVA = Present value of the Annuity PVIFA = Annuity Factor = {1 – [1 ÷ (1+i)n] ÷ i} A = Annuity value i = Interest rate n = Number of periods
In the previous case, PV = $1,000, i = 10%, n = 4, hence;
9-5
Future Value of $1(FVIF)
Table 9–1
9-6
Future Value – Single Amount (Cont’d)
• In determining future value, the following can be used:
• The Future Value of an Annuity is the sum of the future values of single amounts receivable at the end of each period
9-17
Determining the Annuity Value
9-12
Compounding Process for Annuity
9-13
Future Value of an Annuity of $1(FVIFA)
Table 9–3
9-14
Present Value – Annuity
• Calculated by discounting each individual payment back to the present and then adding up all of these payments
– Assuming
9-8
Present Value of $1(PVIF)
Table 9–2
9-9
Relationship of Present and Future Value
9-10
Future Value – Annuity
• Annuity:
– A series of consecutive payments or receipts of equal amount
– Assuming n = 4, i = 6%:
9-20
Relationship of Present Value to Annuity
Annual interest is based on the beginning balance for each year as shown in the following table that shows flow of funds:
• A generalized formula for Future Value of Annuity: FVA = A × FVIFA
Where: FVA = Future value of the Annuity FVIFA = Annuity Factor = {[(1+i)n – 1] ÷ i} A = Annuity value i = Interest rate n = Number of periods; • Assuming, A = $1,000, n = 4, and i = 10%
– The relationship between present value and future value
• Inverse relationship exists between the present value and future value of a single amount
– The relationship between the Present Value of a single amount and the Present Value of an Annuity
9-4
Future Value – Single Amount (Cont’d)
A generalized formula for Future Value:
Where FV = Future value PV = Present value i = Interest rate n = Number of periods;
• The Present Value of an Annuity is the sum of the present values of single amounts payable at the end of each period
– The relationship between the Future Value and Future Value of Annuity
• Mathematical tools of the time value of money are used in making capital allocation decisions
9-3
Future Value – Single Amount
• Measuring value of an amount that is allowed to grow at a given interest over a period of time
Where
= the interest factor
• If $10,000 were invested for 10 years at 8%, the future value would be:
9-7
Present Value – Single Amount
• A sum payable in the future is worth less today than the stated amount
– The formula for the present value is derived from the original formula for future value:
– The present value can be determined by solving for a mathematical solution to the formula above, thus restating the formula as:
– Assuming that the worth of $1,000 needs to be calculated after 4 years at a 10% interest per year, we have:
1st year……$1,000 X 1.10 = $1,100 2nd year…...$1,100 X 1.10 = $1,210 3rd year……$1,210 X 1.10 = $1,331 4th year……$1,331 X 1.10 = $1,464
ChapterFra Baidu bibliotek
9
The Time Value of Money
McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter Outline
• Time value associated with money • Determining future value at given interest
rate • Present value based on current value of
funds to be received • Determining Yield on an Investment. • Compounding or discounting occurring on a
less than annual basis
– For n = 4, and i = 10%, $1,000 as below :
is 4.641. Thus, A equals
9-19
Annuity Equaling a Present Value
– Determining what size of an annuity can be equated to a given amount:
Table 9–5
9-21
Loan Amortization
• A mortgage loan to be repaid over 20 years at 8% interest:
9-22
Loan Amortization Table
•In such a case the part of the payments to the mortgage company will go toward the payment of interest, with the remainder applied to debt reduction, as indicated in the following table: Table 9–6
9-2
Relationship to The Capital Outlay Decision
• The time value of money is used to determine whether future benefits are sufficiently large to justify current outlays
• Future Value of an Annuity:
– Calculated by compounding each individual payment into the future and then adding up all of these payments
9-11
Future Value – Annuity (cont’d)
9-15
Present Value of an Annuity of $1(PVIFA)
Assuming that A = $1,000, n = 4, i = 10%, we have: Table 9–4
9-16
Time Value Relationships
• Comparisons include:
annuity
9-25
Yield – Present Value of a Single Amount
• To calculate the yield on an investment producing $1,464 after 4 years having a present value of $1,000:
9-23
Six Formulas
9-24
Determining the Yield on Investment
• Determining the unknown variable “ i “, given the following variables :
1. FV/PV : Future/Present value of money 2. N : no. of years 3. A : Annuity Value / payment per period in an
• Given the first three variables, and determining the fourth variable “A” (unknown ).
9-18
Annuity Equaling a Future Value
– Assuming that at a 10% interest rate, after 4 years, an amount of $4,641 needs to accumulated:
• A re-look at the variables involved in time value of money:
1. FV/PV : Future/Present value of money 2. N : no. of years 3. I : Interest or YIELD 4. A : Annuity Value / payment per period in an annuity
Where: PVA = Present value of the Annuity PVIFA = Annuity Factor = {1 – [1 ÷ (1+i)n] ÷ i} A = Annuity value i = Interest rate n = Number of periods
In the previous case, PV = $1,000, i = 10%, n = 4, hence;
9-5
Future Value of $1(FVIF)
Table 9–1
9-6
Future Value – Single Amount (Cont’d)
• In determining future value, the following can be used:
• The Future Value of an Annuity is the sum of the future values of single amounts receivable at the end of each period
9-17
Determining the Annuity Value
9-12
Compounding Process for Annuity
9-13
Future Value of an Annuity of $1(FVIFA)
Table 9–3
9-14
Present Value – Annuity
• Calculated by discounting each individual payment back to the present and then adding up all of these payments
– Assuming
9-8
Present Value of $1(PVIF)
Table 9–2
9-9
Relationship of Present and Future Value
9-10
Future Value – Annuity
• Annuity:
– A series of consecutive payments or receipts of equal amount
– Assuming n = 4, i = 6%:
9-20
Relationship of Present Value to Annuity
Annual interest is based on the beginning balance for each year as shown in the following table that shows flow of funds:
• A generalized formula for Future Value of Annuity: FVA = A × FVIFA
Where: FVA = Future value of the Annuity FVIFA = Annuity Factor = {[(1+i)n – 1] ÷ i} A = Annuity value i = Interest rate n = Number of periods; • Assuming, A = $1,000, n = 4, and i = 10%
– The relationship between present value and future value
• Inverse relationship exists between the present value and future value of a single amount
– The relationship between the Present Value of a single amount and the Present Value of an Annuity
9-4
Future Value – Single Amount (Cont’d)
A generalized formula for Future Value:
Where FV = Future value PV = Present value i = Interest rate n = Number of periods;
• The Present Value of an Annuity is the sum of the present values of single amounts payable at the end of each period
– The relationship between the Future Value and Future Value of Annuity
• Mathematical tools of the time value of money are used in making capital allocation decisions
9-3
Future Value – Single Amount
• Measuring value of an amount that is allowed to grow at a given interest over a period of time
Where
= the interest factor
• If $10,000 were invested for 10 years at 8%, the future value would be:
9-7
Present Value – Single Amount
• A sum payable in the future is worth less today than the stated amount
– The formula for the present value is derived from the original formula for future value:
– The present value can be determined by solving for a mathematical solution to the formula above, thus restating the formula as:
– Assuming that the worth of $1,000 needs to be calculated after 4 years at a 10% interest per year, we have:
1st year……$1,000 X 1.10 = $1,100 2nd year…...$1,100 X 1.10 = $1,210 3rd year……$1,210 X 1.10 = $1,331 4th year……$1,331 X 1.10 = $1,464
ChapterFra Baidu bibliotek
9
The Time Value of Money
McGraw-Hill/Irwin
Copyright © 2011 by The McGraw-Hill Companies, Inc. All rights reserved.
Chapter Outline
• Time value associated with money • Determining future value at given interest
rate • Present value based on current value of
funds to be received • Determining Yield on an Investment. • Compounding or discounting occurring on a
less than annual basis
– For n = 4, and i = 10%, $1,000 as below :
is 4.641. Thus, A equals
9-19
Annuity Equaling a Present Value
– Determining what size of an annuity can be equated to a given amount:
Table 9–5
9-21
Loan Amortization
• A mortgage loan to be repaid over 20 years at 8% interest:
9-22
Loan Amortization Table
•In such a case the part of the payments to the mortgage company will go toward the payment of interest, with the remainder applied to debt reduction, as indicated in the following table: Table 9–6
9-2
Relationship to The Capital Outlay Decision
• The time value of money is used to determine whether future benefits are sufficiently large to justify current outlays
• Future Value of an Annuity:
– Calculated by compounding each individual payment into the future and then adding up all of these payments
9-11
Future Value – Annuity (cont’d)
9-15
Present Value of an Annuity of $1(PVIFA)
Assuming that A = $1,000, n = 4, i = 10%, we have: Table 9–4
9-16
Time Value Relationships
• Comparisons include:
annuity
9-25
Yield – Present Value of a Single Amount
• To calculate the yield on an investment producing $1,464 after 4 years having a present value of $1,000:
9-23
Six Formulas
9-24
Determining the Yield on Investment
• Determining the unknown variable “ i “, given the following variables :
1. FV/PV : Future/Present value of money 2. N : no. of years 3. A : Annuity Value / payment per period in an
• Given the first three variables, and determining the fourth variable “A” (unknown ).
9-18
Annuity Equaling a Future Value
– Assuming that at a 10% interest rate, after 4 years, an amount of $4,641 needs to accumulated:
• A re-look at the variables involved in time value of money:
1. FV/PV : Future/Present value of money 2. N : no. of years 3. I : Interest or YIELD 4. A : Annuity Value / payment per period in an annuity