We use cookies to improve your experience on our site and to show you relevant advertising. By browsing this website, you agree to our use of cookies. Learn more   

AtoZmath.com
Support us
(New) All problem can be solved using search box
I want to sell my website www.AtoZmath.com with complete code
 
 
Home What's new College Algebra Games Feedback About us
Algebra Matrix & Vector Numerical Methods Statistical Methods Operation Research Word Problems Calculus Geometry Pre-Algebra
Home > Algebra calculators > Present value of Annuity example
7. Present value of Annuity example ( Enter your problem )
( Enter your problem )
  1. Find Present value (PV) Example
  2. Find Regular Deposit (C) Example
  3. Find Interest Rate (i) Example
  4. Find Time (n) Example
 		Other related methods
  1. Future value using Simple Interest
  2. Future value using Compound Interest
  3. Future value of Annuity
  4. Future value of Annuity Due
  5. Present value using Simple Interest
  6. Present value using Compound Interest
  7. Present value of Annuity
  8. Present value of Annuity Due
  9. Contineous Compounding
2. Find Regular Deposit (C) Example
(Previous example)		4. Find Time (n) Example
(Next example)

3. Find Interest Rate (i) Example


1. Find Interest Rate i = ?
Regular Deposit
(PMT Amount) C = 1000, Time n = 5 Year, Present value PV = 3790.79,
Deposit Frequency = at the end (Ordinary Annuity) of every Year (1/year)
for Present value of Annuity method

Solution:
`C=1000` (Cash flow per year)

`n=5` years (Number of periods)

`PV=3790.79` (Present value)

Now, Present value (Ordinary Annuity) formula is
`PV_("Ordinary Annuity")=C*[(1-(1+i)^(-n))/(i)]`

`:.3790.79=1000*[(1-(1+i)^-5)/(i)]`

`:.(3790.79)/(1000)=[(1-(1+i)^-5)/(i)]`

`:.[(1-(1+i)^-5)/(i)]=3.79`

Now, find one solution using Newton Raphson method

Here `((1-(1+x)^-5)/x)=3.79`

`:.(1-1/((1+x)^5))/x-3.79=0`

Let `f(x) = (1-1/((1+x)^5))/x-3.79`

`d/(dx)((1-1/((1+x)^5))/x-3.79)=((5x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(30x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(5x^5)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-1+1/(1+5x+10x^2+10x^3+5x^4+x^5))/(x^2)`


`d/(dx)((1-1/((1+x)^5))/x-3.79)`

`=d/(dx)((1-1/((1+x)^5))/x)-d/(dx)(3.79)`

`d/(dx)((1-1/((1+x)^5))/x)=((5x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(30x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(5x^5)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-1+1/(1+5x+10x^2+10x^3+5x^4+x^5))/(x^2)`
`d/(dx)((1-1/((1+x)^5))/x)`

`=1-1/(1+5x+10x^2+10x^3+5x^4+x^5)`

`=((x) * d/(dx)(1-1/(1+5x+10x^2+10x^3+5x^4+x^5))-(1-1/(1+5x+10x^2+10x^3+5x^4+x^5)) * d/(dx)(x))/(x)^2`

`d/(dx)(1-1/(1+5x+10x^2+10x^3+5x^4+x^5))=5/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(30x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(5x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)`
`d/(dx)(1-1/(1+5x+10x^2+10x^3+5x^4+x^5))`

`=d/(dx)(1)-d/(dx)(1/(1+5x+10x^2+10x^3+5x^4+x^5))`

`d/(dx)(1/(1+5x+10x^2+10x^3+5x^4+x^5))=-5/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(20x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(30x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(20x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(5x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)`
`d/(dx)(1/(1+5x+10x^2+10x^3+5x^4+x^5))`

`=-1/((1+5x+10x^2+10x^3+5x^4+x^5)^2)*d/(dx)(1+5x+10x^2+10x^3+5x^4+x^5)`

`d/(dx)(1+5x+10x^2+10x^3+5x^4+x^5)=5+20x+30x^2+20x^3+5x^4`
`d/(dx)(1+5x+10x^2+10x^3+5x^4+x^5)`

`=d/(dx)(1)+d/(dx)(5x)+d/(dx)(10x^2)+d/(dx)(10x^3)+d/(dx)(5x^4)+d/(dx)(x^5)`

`=0+5+20x+30x^2+20x^3+5x^4`

`=5+20x+30x^2+20x^3+5x^4`


`=-1/((1+5x+10x^2+10x^3+5x^4+x^5)^2)*5+20x+30x^2+20x^3+5x^4`

`=-5/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(20x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(30x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(20x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(5x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)`


`=0-(-5/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(20x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(30x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(20x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-(5x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2))`

`=5/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(30x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(5x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)`


`=((x) * (5/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(30x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(5x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2))-(1-1/(1+5x+10x^2+10x^3+5x^4+x^5)) * (1))/(x^2)`

`=((5x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(30x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(5x^5)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-1+1/(1+5x+10x^2+10x^3+5x^4+x^5))/(x^2)`


`=(((5x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(30x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(5x^5)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-1+1/(1+5x+10x^2+10x^3+5x^4+x^5))/(x^2))-0`

`=((5x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(30x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(5x^5)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-1+1/(1+5x+10x^2+10x^3+5x^4+x^5))/(x^2)-0`

`=((5x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(30x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(5x^5)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-1+1/(1+5x+10x^2+10x^3+5x^4+x^5))/(x^2)`


`:. f'(x) = ((5x)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^2)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(30x^3)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(20x^4)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)+(5x^5)/((1+5x+10x^2+10x^3+5x^4+x^5)^2)-1+1/(1+5x+10x^2+10x^3+5x^4+x^5))/(x^2)`

`x_0 = 0.1`


`1^(st)` iteration :

`f(x_0)=f(0.1)=(1-1/((1+0.1)^5))/0.1-3.79=0.000787`

`f'(x_0)=f'(0.1)=((5*0.1)/((1+5*0.1+10*0.1^2+10*0.1^3+5*0.1^4+0.1^5)^2)+(20*0.1^2)/((1+5*0.1+10*0.1^2+10*0.1^3+5*0.1^4+0.1^5)^2)+(30*0.1^3)/((1+5*0.1+10*0.1^2+10*0.1^3+5*0.1^4+0.1^5)^2)+(20*0.1^4)/((1+5*0.1+10*0.1^2+10*0.1^3+5*0.1^4+0.1^5)^2)+(5*0.1^5)/((1+5*0.1+10*0.1^2+10*0.1^3+5*0.1^4+0.1^5)^2)-1+1/(1+5*0.1+10*0.1^2+10*0.1^3+5*0.1^4+0.1^5))/(0.1^2)=-9.684171`

`x_1 = x_0 - f(x_0)/(f'(x_0))`

`x_1=0.1 - (0.000787)/(-9.684171)`

`x_1=0.100081`


`2^(nd)` iteration :

`f(x_1)=f(0.100081)=(1-1/((1+0.100081)^5))/0.100081-3.79=0`

`f'(x_1)=f'(0.100081)=((5*0.100081)/((1+5*0.100081+10*0.100081^2+10*0.100081^3+5*0.100081^4+0.100081^5)^2)+(20*0.100081^2)/((1+5*0.100081+10*0.100081^2+10*0.100081^3+5*0.100081^4+0.100081^5)^2)+(30*0.100081^3)/((1+5*0.100081+10*0.100081^2+10*0.100081^3+5*0.100081^4+0.100081^5)^2)+(20*0.100081^4)/((1+5*0.100081+10*0.100081^2+10*0.100081^3+5*0.100081^4+0.100081^5)^2)+(5*0.100081^5)/((1+5*0.100081+10*0.100081^2+10*0.100081^3+5*0.100081^4+0.100081^5)^2)-1+1/(1+5*0.100081+10*0.100081^2+10*0.100081^3+5*0.100081^4+0.100081^5))/(0.100081^2)=-9.680944`

`x_2 = x_1 - f(x_1)/(f'(x_1))`

`x_2=0.100081 - (0)/(-9.680944)`

`x_2=0.100081`


Approximate root of the equation `(1-1/((1+x)^5))/x-3.79=0` using Newton Raphson method is `0.100081` (After 2 iterations)

`n``x_0``f(x_0)``f'(x_0)``x_1`Update
10.10.000787-9.6841710.100081`x_0 = x_1`
20.1000810-9.6809440.100081`x_0 = x_1`



`:.i=0.100081`

`:.i=10.01 %` per year
2. Find Interest Rate i = ?
Regular Deposit
(PMT Amount) C = 5000, Time n = 3 Year, Present value PV = 12434.26,
Deposit Frequency = at the end (Ordinary Annuity) of every Year (1/year)
for Present value of Annuity method

Solution:
`C=5000` (Cash flow per year)

`n=3` years (Number of periods)

`PV=12434.26` (Present value)

Now, Present value (Ordinary Annuity) formula is
`PV_("Ordinary Annuity")=C*[(1-(1+i)^(-n))/(i)]`

`:.12434.26=5000*[(1-(1+i)^-3)/(i)]`

`:.(12434.26)/(5000)=[(1-(1+i)^-3)/(i)]`

`:.[(1-(1+i)^-3)/(i)]=2.49`

Now, find one solution using Newton Raphson method

Here `((1-(1+x)^-3)/x)=2.49`

`:.(1-1/((1+x)^3))/x-2.49=0`

Let `f(x) = (1-1/((1+x)^3))/x-2.49`

`d/(dx)((1-1/((1+x)^3))/x-2.49)=((3x)/((1+3x+3x^2+x^3)^2)+(6x^2)/((1+3x+3x^2+x^3)^2)+(3x^3)/((1+3x+3x^2+x^3)^2)-1+1/(1+3x+3x^2+x^3))/(x^2)`


`d/(dx)((1-1/((1+x)^3))/x-2.49)`

`=d/(dx)((1-1/((1+x)^3))/x)-d/(dx)(2.49)`

`d/(dx)((1-1/((1+x)^3))/x)=((3x)/((1+3x+3x^2+x^3)^2)+(6x^2)/((1+3x+3x^2+x^3)^2)+(3x^3)/((1+3x+3x^2+x^3)^2)-1+1/(1+3x+3x^2+x^3))/(x^2)`
`d/(dx)((1-1/((1+x)^3))/x)`

`=1-1/(1+3x+3x^2+x^3)`

`=((x) * d/(dx)(1-1/(1+3x+3x^2+x^3))-(1-1/(1+3x+3x^2+x^3)) * d/(dx)(x))/(x)^2`

`d/(dx)(1-1/(1+3x+3x^2+x^3))=3/((1+3x+3x^2+x^3)^2)+(6x)/((1+3x+3x^2+x^3)^2)+(3x^2)/((1+3x+3x^2+x^3)^2)`
`d/(dx)(1-1/(1+3x+3x^2+x^3))`

`=d/(dx)(1)-d/(dx)(1/(1+3x+3x^2+x^3))`

`d/(dx)(1/(1+3x+3x^2+x^3))=-3/((1+3x+3x^2+x^3)^2)-(6x)/((1+3x+3x^2+x^3)^2)-(3x^2)/((1+3x+3x^2+x^3)^2)`
`d/(dx)(1/(1+3x+3x^2+x^3))`

`=-1/((1+3x+3x^2+x^3)^2)*d/(dx)(1+3x+3x^2+x^3)`

`d/(dx)(1+3x+3x^2+x^3)=3+6x+3x^2`
`d/(dx)(1+3x+3x^2+x^3)`

`=d/(dx)(1)+d/(dx)(3x)+d/(dx)(3x^2)+d/(dx)(x^3)`

`=0+3+6x+3x^2`

`=3+6x+3x^2`


`=-1/((1+3x+3x^2+x^3)^2)*3+6x+3x^2`

`=-3/((1+3x+3x^2+x^3)^2)-(6x)/((1+3x+3x^2+x^3)^2)-(3x^2)/((1+3x+3x^2+x^3)^2)`


`=0-(-3/((1+3x+3x^2+x^3)^2)-(6x)/((1+3x+3x^2+x^3)^2)-(3x^2)/((1+3x+3x^2+x^3)^2))`

`=3/((1+3x+3x^2+x^3)^2)+(6x)/((1+3x+3x^2+x^3)^2)+(3x^2)/((1+3x+3x^2+x^3)^2)`


`=((x) * (3/((1+3x+3x^2+x^3)^2)+(6x)/((1+3x+3x^2+x^3)^2)+(3x^2)/((1+3x+3x^2+x^3)^2))-(1-1/(1+3x+3x^2+x^3)) * (1))/(x^2)`

`=((3x)/((1+3x+3x^2+x^3)^2)+(6x^2)/((1+3x+3x^2+x^3)^2)+(3x^3)/((1+3x+3x^2+x^3)^2)-1+1/(1+3x+3x^2+x^3))/(x^2)`


`=(((3x)/((1+3x+3x^2+x^3)^2)+(6x^2)/((1+3x+3x^2+x^3)^2)+(3x^3)/((1+3x+3x^2+x^3)^2)-1+1/(1+3x+3x^2+x^3))/(x^2))-0`

`=((3x)/((1+3x+3x^2+x^3)^2)+(6x^2)/((1+3x+3x^2+x^3)^2)+(3x^3)/((1+3x+3x^2+x^3)^2)-1+1/(1+3x+3x^2+x^3))/(x^2)-0`

`=((3x)/((1+3x+3x^2+x^3)^2)+(6x^2)/((1+3x+3x^2+x^3)^2)+(3x^3)/((1+3x+3x^2+x^3)^2)-1+1/(1+3x+3x^2+x^3))/(x^2)`


`:. f'(x) = ((3x)/((1+3x+3x^2+x^3)^2)+(6x^2)/((1+3x+3x^2+x^3)^2)+(3x^3)/((1+3x+3x^2+x^3)^2)-1+1/(1+3x+3x^2+x^3))/(x^2)`

`x_0 = 0.1`


`1^(st)` iteration :

`f(x_0)=f(0.1)=(1-1/((1+0.1)^3))/0.1-2.49=-0.003148`

`f'(x_0)=f'(0.1)=((3*0.1)/((1+3*0.1+3*0.1^2+0.1^3)^2)+(6*0.1^2)/((1+3*0.1+3*0.1^2+0.1^3)^2)+(3*0.1^3)/((1+3*0.1+3*0.1^2+0.1^3)^2)-1+1/(1+3*0.1+3*0.1^2+0.1^3))/(0.1^2)=-4.378116`

`x_1 = x_0 - f(x_0)/(f'(x_0))`

`x_1=0.1 - (-0.003148)/(-4.378116)`

`x_1=0.099281`


`2^(nd)` iteration :

`f(x_1)=f(0.099281)=(1-1/((1+0.099281)^3))/0.099281-2.49=0.000003`

`f'(x_1)=f'(0.099281)=((3*0.099281)/((1+3*0.099281+3*0.099281^2+0.099281^3)^2)+(6*0.099281^2)/((1+3*0.099281+3*0.099281^2+0.099281^3)^2)+(3*0.099281^3)/((1+3*0.099281+3*0.099281^2+0.099281^3)^2)-1+1/(1+3*0.099281+3*0.099281^2+0.099281^3))/(0.099281^2)=-4.387515`

`x_2 = x_1 - f(x_1)/(f'(x_1))`

`x_2=0.099281 - (0.000003)/(-4.387515)`

`x_2=0.099282`


Approximate root of the equation `(1-1/((1+x)^3))/x-2.49=0` using Newton Raphson method is `0.099282` (After 2 iterations)

`n``x_0``f(x_0)``f'(x_0)``x_1`Update
10.1-0.003148-4.3781160.099281`x_0 = x_1`
20.0992810.000003-4.3875150.099282`x_0 = x_1`



`:.i=0.099282`

`:.i=9.93 %` per year

This material is intended as a summary. Use your textbook for detail explanation.
Any bug, improvement, feedback then Submit Here


2. Find Regular Deposit (C) Example
(Previous example)		4. Find Time (n) Example
(Next example)


Share this solution or page with your friends.
 
 
Home What's new College Algebra Games Feedback About us
 
Copyright © 2025. All rights reserved. Terms, Privacy
 
 

.