I'm trying to solve this problem: What sinking fund payment would be required at the end of each three-month period, at 8% interest compounded quarterly, in order to amount to 20,000. I just need to know how to work the problem correctly.
you didn't give a time limit for the $20,000, so I'll assume 5yrs
FV = PMT * (((1 + i)^n - 1)/i )
x = amt of PMT
i = rate
n = no. of payments
i = .08/4 quarters = .02
n = 4 * 5 yr = 20 payments
20,000 = x *((1 + .02)^20 - 1 )/.02
20,000 = x *(( 1.02 )^20 - 1)/ .02
20,000 = x *(1.48595 - 1)/.02
20,000 = x * (0.48595)/.02
20,000 = 0.48595x/.02
400 = 0.48595x
x = 823.13
PMT = x = $823.13
To solve this sinking fund problem, we can use the formula for the future value of an ordinary annuity. The formula is:
FV = P * [(1 + r)^n - 1] / r
Where:
FV is the future value of the sinking fund (20,000 in this case).
P is the sinking fund payment at the end of each period (what we need to find).
r is the interest rate per period (8% compounded quarterly).
n is the number of periods (in this case, each period is three months).
Let's plug in the values and solve for P:
20,000 = P * [(1 + 0.08/4)^(n*4) - 1] / (0.08/4)
Simplify the equation further:
20,000 = P * [(1 + 0.02)^n - 1] / 0.02
Next, let's solve for the value inside the square brackets:
(1 + 0.02)^n = (20,000 * 0.02) / P + 1
Now, take the logarithm of both sides to solve for n:
log((1 + 0.02)^n) = log((20,000 * 0.02) / P + 1)
n * log(1 + 0.02) = log((20,000 * 0.02) / P + 1)
Finally, we can solve for n:
n = log((20,000 * 0.02) / P + 1) / log(1 + 0.02)
Using this value of n, we can substitute it back into the original equation to solve for P.