Future value annuity formula pdf
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future value annuity formula pdf

Future Value of Annuity Due (Formula) Calculation with. Three approaches exist to calculate the present or future value of an annuity amount, known as a time-value-of-money calculation. You can use a formula and either a regular or financial calculator to figure out the present value of an ordinary annuity., So, future value of an annuity due always greater than ordinary annuity Future value of an ordinary annuity can be calculated using same method as a mixed stream FV = PMT x { [ ( 1 + r ) - 1 ] / r} S Finding the Future Value of an Annuity Due Slight change to those for an ordinary annuity Payment made at beginning of period, instead of end.

Present Value New York University

Future Value of Annuity Calculator FVA Calculator. Formula. Following is the formula for finding future value of an ordinary annuity: FVA = P * ((1 + i) n - 1) / i) where, FVA = Future value P = Periodic payment amount n = Number of payments i = Periodic interest rate per payment period, See periodic interest calculator for conversion of nominal annual rates to periodic rates., Future Value Annuities: Chapter 10.1 l) Future Value Annuities l. Instead of depositing one lump sum, waiting for compound interest to increase the value, and then withdrawing the FV amount, we will make a series of equal deposits or payments made at regular time intervals, wait for compound interest to increase the value, and then withdrawing the.

Finally, this comes up to be the compound value of Rs. 1 for four years at 6% interest rate. Formula. Hence, if “A” is the periodic payment, then the annuity of the future value A(n,i) is: A(n,i) = A[(1+i) n – 1/i] Perpetuity. Perpetuity is nothing but a special form of an annuity. PRESENT VALUE OF AN ANNUITY DEFINITIONS: Present value of an annuity: lump sum amount that equals the value now of a set of equal periodic payments to be paid in the future. Formulas and Examples: PV =.(PMT)K, where Example: Find the present value of an annuity with periodic payments of $2000,

Calculate the future value of an annuity due, ordinary annuity and growing annuities with optional compounding and payment frequency. Annuity formulas and derivations for future value based on FV = (PMT/i) [(1+i)^n - 1](1+iT) including continuous compounding The future value of an annuity due is higher than the future value of an (ordinary) annuity by the factor of one plus the periodic interest rate. This is because due to the advance nature of cash flows, each cash flow is subject to compounding effect for one additional period.

04/11/2017В В· In this video, I will explain how to calculate SI, CI, different types of Rates, and the most important- how to calculate Annuity using Formula and using Calculator. explanation by Aman Barnwal If Future value annuity due tables are used to provide a solution for the part of the future value of an annuity due formula shown in red, this is sometimes referred to as the future value annuity due factor. FV = Pmt x Future value annuity due factor Annuity Due Tables Future Value Example

Annuities Due (Simple and General) Therefore, the future value at the end of the last payment period is $3310.13 . Example 2: A four-year lease agreement requires payments of $10,000 at the beginning of every year. If the interest rate is 6% compounded monthly, what is the cash value of the lease? (Focal Date) • The accumulated value of the annuity at time n is denoted by snei or sne. • This is the future value of ane at time n.Thus,wehave sne = ane ×(1+i) n = (1+ i)n −1 i. (2.2) • If the annuity is of level payments of P, the present and future values of the annuity are Pane and Psne, respectively.

The future value of an annuity due is higher than the future value of an (ordinary) annuity by the factor of one plus the periodic interest rate. This is because due to the advance nature of cash flows, each cash flow is subject to compounding effect for one additional period. LIST OF FORMULAS 133 Ordinary interest: I 0 = Ie 1+ 1 72 or I 0 = 1.014Ie Exact interest: Ie = I 0 1+ 1 73 or Ie = I 0 1.014 Equivalent time: n = Pini Pi Interest rate by the dollar-weighted method:

>>> Practice Future Value of Annuity MCQs. Example # 7: If 10 annual payments of Rs. 900 are made into saving account that pays 6 percent interest per year. What is the future value of this annuity, if compounding take place semi-annually. Solve this problem by factor formula and table? Future Value of Annuity Table Download . Example # 8: Future Value Annuities: Chapter 10.1 l) Future Value Annuities l. Instead of depositing one lump sum, waiting for compound interest to increase the value, and then withdrawing the FV amount, we will make a series of equal deposits or payments made at regular time intervals, wait for compound interest to increase the value, and then withdrawing the

PRESENT VALUE OF AN ANNUITY DEFINITIONS: Present value of an annuity: lump sum amount that equals the value now of a set of equal periodic payments to be paid in the future. Formulas and Examples: PV =.(PMT)K, where Example: Find the present value of an annuity with periodic payments of $2000, • The accumulated value of the annuity at time n is denoted by snei or sne. • This is the future value of ane at time n.Thus,wehave sne = ane ×(1+i) n = (1+ i)n −1 i. (2.2) • If the annuity is of level payments of P, the present and future values of the annuity are Pane and Psne, respectively.

Annuities Due (Simple and General) George Brown College

future value annuity formula pdf

Annuity Wikipedia. • The accumulated value of the annuity at time n is denoted by snei or sne. • This is the future value of ane at time n.Thus,wehave sne = ane ×(1+i) n = (1+ i)n −1 i. (2.2) • If the annuity is of level payments of P, the present and future values of the annuity are Pane and Psne, respectively., pmt - the value from cell C6, 100000. fv - 0. type - 0, payment at end of period (regular annuity). With this information, the present value of the annuity is $116,535.83. Note payment is entered as a negative number, so the result is positive. Annuity due. With an annuity due, payments are made at the beginning of the period, instead of the end..

Future Value of An Annuity Formula Definition. Calculate the future value of an annuity due, ordinary annuity and growing annuities with optional compounding and payment frequency. Annuity formulas and derivations for future value based on FV = (PMT/i) [(1+i)^n - 1](1+iT) including continuous compounding, Calculating a life annuity The calculation an annuity payable for the remaining lifetime of an annuitant needs to take account of four primary factors: The Pattern of Income The present value of all future income payments must always equal the investment lump sum. The graph below illustrates the resulting change in.

Future Value and Perpetuity Definition Formulas with

future value annuity formula pdf

Present Value of Annuity I Ordinary & Due Annuity I. Finally, this comes up to be the compound value of Rs. 1 for four years at 6% interest rate. Formula. Hence, if “A” is the periodic payment, then the annuity of the future value A(n,i) is: A(n,i) = A[(1+i) n – 1/i] Perpetuity. Perpetuity is nothing but a special form of an annuity. Future value annuity due tables are used to provide a solution for the part of the future value of an annuity due formula shown in red, this is sometimes referred to as the future value annuity due factor. FV = Pmt x Future value annuity due factor Annuity Due Tables Future Value Example.

future value annuity formula pdf


Annuities Due (Simple and General) Therefore, the future value at the end of the last payment period is $3310.13 . Example 2: A four-year lease agreement requires payments of $10,000 at the beginning of every year. If the interest rate is 6% compounded monthly, what is the cash value of the lease? (Focal Date) LIST OF FORMULAS 133 Ordinary interest: I 0 = Ie 1+ 1 72 or I 0 = 1.014Ie Exact interest: Ie = I 0 1+ 1 73 or Ie = I 0 1.014 Equivalent time: n = Pini Pi Interest rate by the dollar-weighted method:

Future Value of a Growing Annuity. The future value of a growing annuity can be calculated by working out each individual cash flow by (a) growing the initial cash flow at g; (b) finding future value of each cash flow at the interest rate r and (c) then summing up all the component future … Formula. Following is the formula for finding future value of an ordinary annuity: FVA = P * ((1 + i) n - 1) / i) where, FVA = Future value P = Periodic payment amount n = Number of payments i = Periodic interest rate per payment period, See periodic interest calculator for conversion of nominal annual rates to periodic rates.

To get the present value of an annuity, you can use the PV function. In the example shown, the formula in C9 is: = PV ( C5 , C6 , C4 , 0 , 0 ) Explanation An annuity is a series of equal cash flows, spaced equally in time. To get the present value of an annuity, you can use the PV function. In the example shown, the formula in C9 is: = PV ( C5 , C6 , C4 , 0 , 0 ) Explanation An annuity is a series of equal cash flows, spaced equally in time.

Future value of a lump sum investment is explained on the future value of a single sum page. In this article future value or sum of an annuity is determined. Formula: The following formula is used to calculate future value of an annuity: 3.3 Future value annuities (EMCFZ) For future value annuities, we regularly save the same amount of money into an account, which earns a certain rate of compound interest, so that we have money for the future.

The future value of annuity grows based on the stated discount rate, as such the higher discount rate the higher will be the future value of the annuity. Recommended Articles. This has been a guide to Future Value of Annuity Due. Here we discuss how to calculate Future Value of Annuity Due using its formula along with some practical examples 21/05/2018 · Future value of an annuity due table May 21, 2018 / Steven Bragg. An annuity is a series of payments that occur at the same intervals and in the same amounts. An example of an annuity is a series of payments from the buyer of an asset to the seller, where the buyer promises to make a series of regular payments. Thus, Hobo Clothiers buys a warehouse from Marlowe …

The formula for calculating the future value of an annuity must take into account the fact that cash received today is more valuable than cash in the future. pmt - the value from cell C6, 100000. fv - 0. type - 0, payment at end of period (regular annuity). With this information, the present value of the annuity is $116,535.83. Note payment is entered as a negative number, so the result is positive. Annuity due. With an annuity due, payments are made at the beginning of the period, instead of the end.

To calculate the future value of the annuity, we have to calculate the future value of each cash flow. Let us assume that you are receiving $1,000 every … pmt - the value from cell C6, 100000. fv - 0. type - 0, payment at end of period (regular annuity). With this information, the present value of the annuity is $116,535.83. Note payment is entered as a negative number, so the result is positive. Annuity due. With an annuity due, payments are made at the beginning of the period, instead of the end.

Future Value Formula Calculator (Excel template)

future value annuity formula pdf

Future Value of Annuity Due Formula & Example. Example: PV of an Annuity n The present value of an annuity of $1,000 for the next five years, assuming a discount rate of 10% is - n The notation that will be used in the rest of these lecture notes for the present value of an annuity will be PV(A,r,n). PV of $1000 each year for next 5 years = $1000 1 - 1 (1.10) 5.10, Future Value of a Growing Annuity. The future value of a growing annuity can be calculated by working out each individual cash flow by (a) growing the initial cash flow at g; (b) finding future value of each cash flow at the interest rate r and (c) then summing up all the component future ….

Present value and Future value tables Table 1 Future

What Is the Future Value of an Annuity? SmartAsset. Future value annuity tables are used to carry out annuity calculations without using a financial calculator. Examples and free PDF download are available., Finally, this comes up to be the compound value of Rs. 1 for four years at 6% interest rate. Formula. Hence, if “A” is the periodic payment, then the annuity of the future value A(n,i) is: A(n,i) = A[(1+i) n – 1/i] Perpetuity. Perpetuity is nothing but a special form of an annuity..

04/11/2017В В· In this video, I will explain how to calculate SI, CI, different types of Rates, and the most important- how to calculate Annuity using Formula and using Calculator. explanation by Aman Barnwal If Future Value Annuities: Chapter 10.1 l) Future Value Annuities l. Instead of depositing one lump sum, waiting for compound interest to increase the value, and then withdrawing the FV amount, we will make a series of equal deposits or payments made at regular time intervals, wait for compound interest to increase the value, and then withdrawing the

The future value of an annuity formula assumes that 1. The rate does not change 2. The first payment is one period away 3. The periodic payment does not change. If the rate or periodic payment does change, then the sum of the future value of each individual cash flow would need to be calculated to determine the future value of the annuity. If the first cash flow, or payment, is … The Present Value of an Ordinary Annuity •The present value of an ordinary annuity measures the value today of a stream of cash flows occurring in the future. •For example, we will compute the PV of ordinary annuity if we wish to answer the question: what is the value today equivalent of receiving every year for the

Finally, this comes up to be the compound value of Rs. 1 for four years at 6% interest rate. Formula. Hence, if “A” is the periodic payment, then the annuity of the future value A(n,i) is: A(n,i) = A[(1+i) n – 1/i] Perpetuity. Perpetuity is nothing but a special form of an annuity. To get the present value of an annuity, you can use the PV function. In the example shown, the formula in C9 is: = PV ( C5 , C6 , C4 , 0 , 0 ) Explanation An annuity is a series of equal cash flows, spaced equally in time.

Future Value Annuities: Chapter 10.1 l) Future Value Annuities l. Instead of depositing one lump sum, waiting for compound interest to increase the value, and then withdrawing the FV amount, we will make a series of equal deposits or payments made at regular time intervals, wait for compound interest to increase the value, and then withdrawing the So, future value of an annuity due always greater than ordinary annuity Future value of an ordinary annuity can be calculated using same method as a mixed stream FV = PMT x { [ ( 1 + r ) - 1 ] / r} S Finding the Future Value of an Annuity Due Slight change to those for an ordinary annuity Payment made at beginning of period, instead of end

Calculate the future value of an annuity due, ordinary annuity and growing annuities with optional compounding and payment frequency. Annuity formulas and derivations for future value based on FV = (PMT/i) [(1+i)^n - 1](1+iT) including continuous compounding The future value of an annuity is the accumulated amount, including payments and interest, of a stream of payments made to an interest-bearing account. For an annuity-immediate, it is the value immediately after the n-th payment. The future value is given by: ВЇ = (+) в€’.

Future value of annuity formula; Annuity Tables. Tables are a common feature used in time value of money annuity formula calculations, they provide a quick method of performing calculations without the need for a financial calculator. We provide downloadable tables in PDF format for the most common functions, including present value, future Present Value and Future Value Tables Table A-1 Future Value Interest Factors for One Dollar Compounded at k Percent for n Periods: FVIF k,n = (1 + k)

04/11/2017В В· In this video, I will explain how to calculate SI, CI, different types of Rates, and the most important- how to calculate Annuity using Formula and using Calculator. explanation by Aman Barnwal If Formula Sheet for Financial Mathematics - S is the future value (or maturity value). It is equal to the principal plus the interest earned. COMPOUND INTEREST FV = PV (1 + i) n. i = рќђЈ рќђ¦ j = nominal annual rate of interest m = number of compounding periods . i = periodic rate of interest . PV = FV (1 + i)в€’n OR PV = рќђ…рќђ• (рќџЏ + рќђў)рќђ§. ANNUITIES Classifying rationale Type of

Future value of annuity formula; Annuity Tables. Tables are a common feature used in time value of money annuity formula calculations, they provide a quick method of performing calculations without the need for a financial calculator. We provide downloadable tables in PDF format for the most common functions, including present value, future The Present Value of an Ordinary Annuity •The present value of an ordinary annuity measures the value today of a stream of cash flows occurring in the future. •For example, we will compute the PV of ordinary annuity if we wish to answer the question: what is the value today equivalent of receiving every year for the

This future value of annuity calculator estimates the value (FV) of a series of fixed future annuity payments at a specific interest rate and for a no. of periods the interest is compounded (either ordinary or due annuity). There is more info on this topic below the form. PRESENT VALUE OF AN ANNUITY DEFINITIONS: Present value of an annuity: lump sum amount that equals the value now of a set of equal periodic payments to be paid in the future. Formulas and Examples: PV =.(PMT)K, where Example: Find the present value of an annuity with periodic payments of $2000,

Future value annuity tables are used to provide a solution for the part of the future value of an annuity formula shown in red, this is sometimes referred to as the future value annuity factor. FV = Pmt x Future value annuity factor Annuity Tables Future Value Example. What is the future value of 6,000 received at the end of each year for 8 The Present Value of an Ordinary Annuity •The present value of an ordinary annuity measures the value today of a stream of cash flows occurring in the future. •For example, we will compute the PV of ordinary annuity if we wish to answer the question: what is the value today equivalent of receiving every year for the

Annuity formula An ordinary annuity is a stream of N equal cash flows paid at regular intervals. … The mathematical derivation of the PV formula The present value of an N-period annuity A with payment C and interest r is given by: + = 1+ + 1+ + 1+ +⋯+ 1+ , + =∗ 1 1+ , You may recognize this, from Calculus classes, as a finite geometric Future value of a lump sum investment is explained on the future value of a single sum page. In this article future value or sum of an annuity is determined. Formula: The following formula is used to calculate future value of an annuity:

Future Value of a Growing Annuity. The future value of a growing annuity can be calculated by working out each individual cash flow by (a) growing the initial cash flow at g; (b) finding future value of each cash flow at the interest rate r and (c) then summing up all the component future … Calculate the future value of an annuity due, ordinary annuity and growing annuities with optional compounding and payment frequency. Annuity formulas and derivations for future value based on FV = (PMT/i) [(1+i)^n - 1](1+iT) including continuous compounding

The Present Value of an Ordinary Annuity •The present value of an ordinary annuity measures the value today of a stream of cash flows occurring in the future. •For example, we will compute the PV of ordinary annuity if we wish to answer the question: what is the value today equivalent of receiving every year for the Present Value and Future Value Tables Table A-1 Future Value Interest Factors for One Dollar Compounded at k Percent for n Periods: FVIF k,n = (1 + k)

Annuity Wikipedia. So, future value of an annuity due always greater than ordinary annuity Future value of an ordinary annuity can be calculated using same method as a mixed stream FV = PMT x { [ ( 1 + r ) - 1 ] / r} S Finding the Future Value of an Annuity Due Slight change to those for an ordinary annuity Payment made at beginning of period, instead of end, Once (1+r) is factored out of future value of annuity due cash flows, it becomes equal to the cash flows from an ordinary annuity. Therefore, the future value of an annuity due can be calculated by multiplying the future value of an ordinary annuity by (1+r), which is the formula ….

Future Value of Annuity Due Formula (with Calculator)

future value annuity formula pdf

Future Value of Annuity Due Formula (with Calculator). The Present Value of an Ordinary Annuity •The present value of an ordinary annuity measures the value today of a stream of cash flows occurring in the future. •For example, we will compute the PV of ordinary annuity if we wish to answer the question: what is the value today equivalent of receiving every year for the, 04/11/2017 · In this video, I will explain how to calculate SI, CI, different types of Rates, and the most important- how to calculate Annuity using Formula and using Calculator. explanation by Aman Barnwal If.

Annuity Wikipedia

future value annuity formula pdf

What Is the Future Value of an Annuity? SmartAsset. The future value of an annuity is the accumulated amount, including payments and interest, of a stream of payments made to an interest-bearing account. For an annuity-immediate, it is the value immediately after the n-th payment. The future value is given by: ВЇ = (+) в€’. 04/11/2017В В· In this video, I will explain how to calculate SI, CI, different types of Rates, and the most important- how to calculate Annuity using Formula and using Calculator. explanation by Aman Barnwal If.

future value annuity formula pdf

  • 120803 Derivation of the formulas of annuities and
  • Future Value and Perpetuity Definition Formulas with
  • Present value and Future value tables Table 1 Future

  • pmt - the value from cell C6, 100000. fv - 0. type - 0, payment at end of period (regular annuity). With this information, the present value of the annuity is $116,535.83. Note payment is entered as a negative number, so the result is positive. Annuity due. With an annuity due, payments are made at the beginning of the period, instead of the end. • Future Value Interest Factor for an Annuity 97-98 • Present Value Interest Factor 99-100 • Present Value Interest Factor for an Annuity 101-102 2. Standard Normal Probability Distribution Table 103 3. t Distribution Table 104 4. Area in the Right Tail of a Chi-Square (П‡2) Distribution Table 105-106 5. F Distribution Table 107-108 6. Control Chart Factors Table 109 7. Table for Value …

    18/09/2017В В· Business Math - Finance Math (10 of 30) Future Value of an Annuity (End of Pay Period) - Duration: 6:50. Michel van Biezen 24,825 views Present Value and Future Value Tables Table A-1 Future Value Interest Factors for One Dollar Compounded at k Percent for n Periods: FVIF k,n = (1 + k)

    Calculating the Future Value of a Regular Annuity. As noted above, according to the principle of value additivity, we can treat an annuity as a series of lump sum cash flows. Well, we have already seen how to calculate the future value of a lump sum. All that we need to do is apply this formula to each of the cash flows individually, and then Future value annuity tables are used to provide a solution for the part of the future value of an annuity formula shown in red, this is sometimes referred to as the future value annuity factor. FV = Pmt x Future value annuity factor Annuity Tables Future Value Example. What is the future value of 6,000 received at the end of each year for 8

    Once (1+r) is factored out of future value of annuity due cash flows, it becomes equal to the cash flows from an ordinary annuity. Therefore, the future value of an annuity due can be calculated by multiplying the future value of an ordinary annuity by (1+r), which is the formula … pmt - the value from cell C6, 100000. fv - 0. type - 0, payment at end of period (regular annuity). With this information, the present value of the annuity is $116,535.83. Note payment is entered as a negative number, so the result is positive. Annuity due. With an annuity due, payments are made at the beginning of the period, instead of the end.

    At the end of period 9 what is the value of these future payments? Here the answer is 3a 8j = 4a 8j = 11s 8j = 12 s nj Adeferred annuity is one that begins payments at some time in the future. Using the setting above, we could describe this stream of payments from the time t = 0 as 12ja 8j = (8 payment annuity immediate deferred 12 periods.) Present Value and Future Value Tables Table A-1 Future Value Interest Factors for One Dollar Compounded at k Percent for n Periods: FVIF k,n = (1 + k)

    • The accumulated value of the annuity at time n is denoted by snei or sne. • This is the future value of ane at time n.Thus,wehave sne = ane ×(1+i) n = (1+ i)n −1 i. (2.2) • If the annuity is of level payments of P, the present and future values of the annuity are Pane and Psne, respectively. Future value annuity due tables are used to provide a solution for the part of the future value of an annuity due formula shown in red, this is sometimes referred to as the future value annuity due factor. FV = Pmt x Future value annuity due factor Annuity Due Tables Future Value Example

    Future value annuity tables are used to carry out annuity calculations without using a financial calculator. Examples and free PDF download are available. The formula for calculating the future value of an annuity must take into account the fact that cash received today is more valuable than cash in the future.

    • The accumulated value of the annuity at time n is denoted by snei or sne. • This is the future value of ane at time n.Thus,wehave sne = ane ×(1+i) n = (1+ i)n −1 i. (2.2) • If the annuity is of level payments of P, the present and future values of the annuity are Pane and Psne, respectively. The future value of an annuity due is higher than the future value of an (ordinary) annuity by the factor of one plus the periodic interest rate. This is because due to the advance nature of cash flows, each cash flow is subject to compounding effect for one additional period.

    Formula Sheet for Financial Mathematics - S is the future value (or maturity value). It is equal to the principal plus the interest earned. COMPOUND INTEREST FV = PV (1 + i) n. i = 𝐣 𝐦 j = nominal annual rate of interest m = number of compounding periods . i = periodic rate of interest . PV = FV (1 + i)−n OR PV = 𝐅𝐕 (𝟏 + 𝐢)𝐧. ANNUITIES Classifying rationale Type of Annuity formula An ordinary annuity is a stream of N equal cash flows paid at regular intervals. … The mathematical derivation of the PV formula The present value of an N-period annuity A with payment C and interest r is given by: + = 1+ + 1+ + 1+ +⋯+ 1+ , + =∗ 1 1+ , You may recognize this, from Calculus classes, as a finite geometric

    • Future Value Interest Factor for an Annuity 97-98 • Present Value Interest Factor 99-100 • Present Value Interest Factor for an Annuity 101-102 2. Standard Normal Probability Distribution Table 103 3. t Distribution Table 104 4. Area in the Right Tail of a Chi-Square (χ2) Distribution Table 105-106 5. F Distribution Table 107-108 6. Control Chart Factors Table 109 7. Table for Value … Future value of a lump sum investment is explained on the future value of a single sum page. In this article future value or sum of an annuity is determined. Formula: The following formula is used to calculate future value of an annuity:

    The Present Value of an Ordinary Annuity •The present value of an ordinary annuity measures the value today of a stream of cash flows occurring in the future. •For example, we will compute the PV of ordinary annuity if we wish to answer the question: what is the value today equivalent of receiving every year for the The formula for calculating the future value of an annuity must take into account the fact that cash received today is more valuable than cash in the future.

    Future Value of a Growing Annuity. The future value of a growing annuity can be calculated by working out each individual cash flow by (a) growing the initial cash flow at g; (b) finding future value of each cash flow at the interest rate r and (c) then summing up all the component future … Future value of a lump sum investment is explained on the future value of a single sum page. In this article future value or sum of an annuity is determined. Formula: The following formula is used to calculate future value of an annuity:

    At the end of period 9 what is the value of these future payments? Here the answer is 3a 8j = 4a 8j = 11s 8j = 12 s nj Adeferred annuity is one that begins payments at some time in the future. Using the setting above, we could describe this stream of payments from the time t = 0 as 12ja 8j = (8 payment annuity immediate deferred 12 periods.) Calculating a life annuity The calculation an annuity payable for the remaining lifetime of an annuitant needs to take account of four primary factors: The Pattern of Income The present value of all future income payments must always equal the investment lump sum. The graph below illustrates the resulting change in

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