Annuity Due in Terms of Ordinary Annuity

Annuity Due in Terms of Ordinary Annuity

Annuity:  Ordinary vs. Due

The distinction is determined when the first payment is made:

Ordinary Annuity:   the first payment is made at the end of the first period, with no payment made at the beginning (period 0) of the annuity. 

Annuity Due:  the first payment is made at the very beginning of the annuity, with no payment made at the end of the annuity.  

For today's exercise, assume all payments in annuity are all equal.  To simplify, I am going to set payments to 1 monetary unit (Dollars, Euros, Lira, Yen, etc).

Present Value

The present value of an annuity is today's value of the annuity, with future payments and cash flows are discounted back to today using the periodic interest rate.  On financial calculators, the present value is symbolized by the PV key.  

Let i be the interest rate of the annuity, in decimal format.  

Example, for 5% periodic rate, i=0.05.

Let v = 1/(1 + i).   

Present Value of an Ordinary Annuity

Given the payment of 1 monetary unit, an annuity of n periods, equally spaced out, and interest rate i, the present value of this ordinary annuity is the sum:

PVAF = v + v^2 + v^3 + ... + v^(n-1) + v^n = Σ(v^x for x=1 to n)

The above is known as the present value annuity factor, or PVAF.  

A closed formula for PVAF is:

PVAF = (1 - (1 + i)^-n) / i

Present Value of an Annuity Due

Likewise, the present value of the annuity due is the sum:

PVAFD = 1 + v + v^2 + v^3 + ... + v^(n-1) = Σ(v^x for x=0 to n-1)

A closed formula for PVAFD is:

PVAFD = (1 - (1 + i)^-n) / (1 - 1/(1 + i))

We can express PVAFD in terms of PVAF by:

PVAFD = 1 + v + v^2 + v^3 + ... + v^(n-1)

PVAFD + v^n = 1 + v + v^2 + v^3 + ... + v^(n-1) + v^n

PAVFD + v^n = 1 + PVAF

PAVFD = 1 + PVAF - v^n

PAVFD = 1 + (1 - (1 + i)^-n) / i - (1 + i)^-n

Example:  i = 0.04, n = 36

PAVFD = 1 + (1 - 1.04^-36) / 0.04 - 1.04^-36 ≈ 19.6646

This is the same as setting up TVM (time value of money) keys on most financial calculators or any calculator with a TVM solver as:

n = 36, I = 4, PMT = -1, FV = 0, (if needed, P/Y = 1),  BEGIN mode

Solve for PV:  19.6646...

Future Value

The future value of an annuity is the final value of the annuity, including the accumulated interest of payments and cash flows from the date the payment to the end of the annuity.  On financial calculators, the future value is symbolized by the FV key.  

Let w = 1 + i

Future Value of an Ordinary Annuity

Given the payment of 1 monetary unit, an annuity of n periods, equally spaced out, and interest rate i, the future value of this ordinary annuity is the sum:

FVAF = 1 + w + ... + w^(n-3) + w^(n-2) + w^(n-1) = Σ(w^x for x=0 to n-1)

The above is known as the future value annuity factor, or FVAR.  

A closed formula for FVAF is:

FVAF = ( (1 + i)^n - 1 ) / i

Future Value of an Annuity Due

Likewise, the future value of the annuity due is the sum:

FVAFD = w + ... + w^(n-3) + w^(n-2) + w^(n-1) + w^n = Σ(w^x for x=1 to n)

A closed formula for FVAFD is:

FVAFD = (1 + i) * ( (1 + i)^n - 1 ) / i 

We can express FVAFD in terms of FVAF by:

FAVFD = w + ... + w^(n-3) + w^(n-2) + w^(n-1) + w^n 

1 + FAVFD = 1 + w + ... + w^(n-3) + w^(n-2) + w^(n-1) + w^n 

1 + FAVFD = FAVD + w^n

FAVFD = FAVD + w^n - 1

FAVFD = ( (1 + i)^n - 1 ) / i + (1 + i)^n - 1

Example:  i = 0.04, n = 36

FAVFD = (1.04^36 - 1) / 0.04 + 1.04^36 - 1 ≈ 80.7022

Using the TVM keys or solver:

n = 36, I = 4, PMT = -1, PV = 0, (if needed, P/Y = 1),  BEGIN mode

Solve for FV:  80.7022...

On tomorrow's blog, January 24, I will cover two actuarial problems. 

Source:

Finan, Marcel B.  A Basic Course in the Theory of Interest and Derivatives Markets:  A Preparation for the Actuarial Exam FM/2   Arkansas Tech University, 2017.  

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author. 

Swiss Micros DM42 and HP71B: Present Value of a Growing Annuity

Swiss Micros DM42 and HP71B: Present Value of a Growing Annuity

Introduction

Today we are going to calculate the present value of a growing annuity.  Unlike most annuities where the payment is constant, in a growing annuity, the payment increases each period.  For this particular blog, we are working with annuities that payments increase by a growth percent (g%) each period.  The annuity has an different interest rate (r%) in which payments are discounted.  

Since the payments are not constant, the time value of money (TVM) keys on a financial calculator are not going to be used.  If your calculator has the the net present value (NPV) function, this can assist you in these calculations.

I am going to use a different approach.

Derivation

Variables:

P = base payment (the first payment)

g = growth rate per period 

r = interest rate per period

n = number of periods 

PV = present value

Ordinary Growing Annuity

In an ordinary growing annuity, the first payment will be received after one period (typically a year or a month) has passed.  Discounting all the payments to calculate present value:

PV 

=  P/(1+r) + P * (1+g)/(1+r)^2 + P * (1+g)^2/(1+r)^3 + ... + P * (1+g)^(n-1)/(1+r)^n

=  P/(1+r) * [ 1 + (1+g)/(1+r) + (1+g)^2/(1+r)^2 + ... + (1+g)^(n-1)/(1+r)^(n-1)

Let w = (1+g)/(1+r), then:

PV

= P/(1+r) * [ 1 + w + w^2 + ... + w^(n-1) ]

The result is a geometric series.  In a general geometric series:

a + a*r + a*r^2 + ... + a*r^(n-1) = Σ(a*r^k, k=0 to n-1) = a * (1 - r^n)/(1 - r)

Then:

PV

= P/(1+r) * [ 1 + w + w^2 + ... + w^(n-1) ]

= P/(1+r) * Σ(w^k, k=0 to n-1) 

= P/(1+r) * (1 - w^n)/(1 - w)

Alternatively, change w back to (1+g)/(1+r):

PV

= P/(1+r) * (1 - (1+g)^n/(1+r)^n) / (1 - (1+g)/(1+r))

= [ P/(1+r) * (1 - (1+g)^n/(1+r)^n) ] / [ 1 - (1+g)/(1+r) ]

= [ P/(1+r) * P/(1+r) * (1+g)^n/(1+r)^n ] / [ 1 - (1+g)/(1+r) ]

The article from finaceformulas.net (see source) suggests multiplying by (1+r) / (1+r):

= [ P/(1+r) - P/(1+r) * (1+g)^n/(1+r)^n ] / [ 1 - (1+g)/(1+r) ] * (1 + r) / (1 + r)

= [ P - P * (1+g)^n/(1+r)^n ] / [ 1 + r - (1 + g) ]

= [ P - P * (1+g)^n/(1+r)^n ] / [ r - g ]

= P / (r - g) * (1 - (1+g)^n/(1+r)^n )

Growing Annuity Due

On an annuity due, the first payment takes place immediately.  The present value is calculated as:

PV 

=  P +  P * (1+ g)/(1+r) + P * (1+g)^2/(1+r)^2 + P * (1+g)^3/(1+r)^3 + ... + P * (1+g)^n/(1+r)^n

=  P * [ 1 + (1+ g)/(1+r) + (1+g)^2/(1+r)^2 + (1+g)^3/(1+r)^3 + ... + (1+g)^n/(1+r)^n ]

Let w = (1+g)/(1+r), then:

PV 

= P * [1 + w + w^2 + w^3 + ... + w^n ]

We have another geometric progression:

PV 

= P  * (1 - w^(n+1))/(1 - w)

Summary:

Present Value of a Growing Annuity - Ordinary

PV = P/(1+r) * (1 - w^n)/(1 - w)

Present Value of a Growing Annuity - Due

PV = P  * (1 - w^(n+1))/(1 - w)

HP 42S/DM42 Program:  PVGROW

Both PVGROW and PVGDUE use only one register, R01.

00  {79-Byte Prgm}

01  LBL "PVGROW"

02  "BASE PMT?"

03  PROMPT

04  "INTEREST?"

05  PROMPT

06  1

07  X<>Y

08  %

09  +

10  STO 01

11  ÷

12  1

13  "GROWTH?"

14  PROMPT

15  %

16  +

17  RCL÷ 01

18  STO 01

19  "N?"

20  PROMPT

21  Y↑X

22  1

23  X<>Y

24  -

25  1

26  RCL- 01

27  ÷

28  ×

29  "PV="

30  ARCL ST X

31  AVIEW

32  END

HP 42S/DM42 Program:  PVGDUE

00  {79-Byte Prgm}

01  LBL "PVGDUE"

02  "BASE PMT?"

03  PROMPT

04  "INTEREST?"

05  PROMPT

06  1

07  X<>Y

08  %

09  +

10  1

11  "GROWTH?"

12  PROMPT

13  %

14  +

15  ÷

16  1/X

17  STO 01

18  "N?"

19  PROMPT

20  1

21  +

22  Y↑X

23  1

24  X<>Y

25  -

26  1

27  RCL- 01

28  ÷

29  ×

30  "PV="

31  ARCL ST X

32  AVIEW

33  END

HP 71B Program: PVGROW

Note:  This is for both ordinary and growing annuities due.

100  DESTROY P,G,R,W,A,C

105  INPUT "PAYMENT? "; P

110  INPUT "INTEREST? "; R

115  R=.01*R

120  INPUT "GROWTH? "; G

125  G=.01*G

130  W=(1+G)/(1+R)

135  INPUT "N? "; N

140  INPUT "DUE?(Y=1,N=0) ";C

145  IF C=1 THEN 200

150  IF C=0 THEN 300 ELSE 140 

200  A=P*(1-W^(N+1))/(1-W) @ ! DUE

205  GOTO 400

300  A=P*(1-W^N)/((1+R)*(1-W)) @ ! ORD

305 GOTO 400

400 PRINT "PV ="; A

  

Examples:

Base Payment:  P = 20.00

Interest Rate:  r = 4%

Growth Rate:  g = 5%

n = 5

Ordinary Growing Annuity

Result:  PV = 98.02

Timeline of Payments:

Period 0:  0.00

Period 1:  20.00

Period 2:  21.00

Period 3:  22.05

Period 4:  23.15

Period 5:  24.31

Growing Annuity Due

Result:  PV = 122.92

Period 0:  20.00

Period 1:  21.00

Period 2:  22.05

Period 3:  23.15

Period 4:  24.31

Period 5:  25.53

Source:

"Present Value of a Growing Annuity" financeformulas.net  https://financeformulas.net/Present_Value_of_Growing_Annuity.html   Retrieved December 13, 2020.  

Eddie

All original content copyright, © 2011-2021.  Edward Shore.   Unauthorized use and/or unauthorized distribution for commercial purposes without express and written permission from the author is strictly prohibited.  This blog entry may be distributed for noncommercial purposes, provided that full credit is given to the author.