an annuity has an initial balance of $5000. annual withdrawals are made in the amount of $800 for 9 years, at which point the account balance is zero. what annual rate of interest, compounded annually, was earned over the duration of this annuity?
PV = $5000
R = $800
n = 9
i = ?
PV = (R[1-(1+i)^-n])/i
5000 = (800[1-(1+i)^-9])/i
IDK how to isolate for i and solve for it. please help me!
I think you will have to try various interest rates and graph that function
if i = .1 or 10%
6.25 = 5.76
if i = .08
6.25 = 6.24 close to 8%
i know the answer is 0.08 but how can i isolate for i instead of using the method guess and check
5000 = (800[1-(1+i)^-9])/i
5000 = (800[1-(1+i)^-9])/i
or
(1-(1+i)^-9)/i = 6.25
Unfortunaltely, there is no easy way to algebraically isolate the i in this type of equation.
The method that Damon has used is about the best way to go about it.
notice that if we use .08 exactly,
800(1 - 1.08^-9)/.08 = 4997.51 so we were everso-slightly off by 2.50
we could try something like .0801
800(1 - 1.0801^-9)/.0801 = 4995.43 off by 4.50 and more than before, so I actually "guessed" too high.
rate result
.08 -- 4997.51
r ----5000
.0801 -4995.43
so now could set up a ratio:
(r-.08)/(5000-4997.51) = (.0801-.08)/(4995.43-4997.51)
(r-.08)/2.49 = .0001/-2.07715
r-.08 = -.0001199
r = .07988
let's try that:
800(1 - 1.07988^-9)/.07988 = 5000.0025
This method is called interpolation, one finds two values very close to the unknown one, and then use a ratio similar to the one I used.
(Actually for my example it would be extrapolation)
There are other methods, such as Newton's Method, but it requires Calculus.
We could alway use the Magic of Wolfram:
http://www.wolframalpha.com/input/?i=solve+%281-%281%2Bx%29%5E-9%29%2Fx+%3D+6.25
Notice how close my answer of .07988 was to .0798802..
Also notice that I had to change the i to x, Wolfram knows that i = √-1
okay thank you ! :)
To solve for the interest rate (i), we need to isolate it in the formula:
PV = (R[1-(1+i)^-n])/i
Step 1: Simplify the equation:
Multiply both sides of the equation by i to remove the denominator:
5000i = (800[1-(1+i)^-9])
Step 2: Expand and simplify further:
Distribute the 800 and simplify the expression inside the brackets:
5000i = 800 - 800(1+i)^-9
Step 3: Move all terms to one side of the equation:
Subtract 800 from both sides:
5000i - 800 = -800(1+i)^-9
Step 4: Divide both sides by -800:
This will make the right side of the equation positive:
(5000i - 800)/-800 = (1+i)^-9
Step 5: Simplify further:
Divide the numerator by the denominator on the left side of the equation:
-((5000i - 800)/800) = (1+i)^-9
Step 6: Take the reciprocal of both sides:
This will invert the exponent, making it positive:
-800/(5000i - 800) = (1+i)^9
Step 7: Raise both sides to the power of 1/9:
This will cancel out the exponent on the right side of the equation:
[-800/(5000i - 800)]^(1/9) = 1+i
Step 8: Subtract 1 from both sides:
This will isolate i:
[-800/(5000i - 800)]^(1/9) - 1 = i
Now, you can use a calculator to substitute the values inside the brackets on the left side of the equation and calculate the value of i.