Find the amount of periodic payment necessary for the deposit to a sinking fund:
A=50,000 n=semiannually , r=13%, t=15 yrs.
Let P be the semiannual payment
P(1.065^30 - 1)/.065 = 50000
P = 578.87
R = A (r/n)/(1+ r/n)^nt -1))
R = 50000(.13/2)/(1+ .13/2)^30 -1))
50000(0.065)/((1.065)^30 -1))
$578.85
To find the amount of periodic payment necessary for the deposit to a sinking fund, you can use the formula for the present value of an annuity. The formula is given as:
A = P * [(1 - (1 + r)^(-n*t)) / r]
Where:
A = the desired amount in the sinking fund
P = the periodic payment
r = interest rate per compounding period
n = number of compounding periods per year
t = number of years
Given the values:
A = $50,000
r = 13% (or 0.13 as a decimal)
n = semiannually (which means twice a year, so n = 2)
t = 15 years
Substituting the values into the formula:
$50,000 = P * [(1 - (1 + 0.13/2)^(-2*15)) / (0.13/2)]
Now, let's simplify the equation:
$50,000 = P * [(1 - (1.065)^(-30)) / (0.065)]
To solve for P, we need to isolate it on one side of the equation:
$50,000 * (0.065) = P * (1 - (1.065)^(-30))
$50,000 * (0.065) = P * (1 - 0.220100651)
$3,250 = P * 0.779899349
To solve for P, divide both sides of the equation by 0.779899349:
P = $3,250 / 0.779899349
P ≈ $4,167.67
Therefore, the amount of periodic payment necessary for the deposit to a sinking fund is approximately $4,167.67.