Find the amount of periodic payment necessary for the deposit to a sinking fund. (Round your answer to the nearest cent.)
Amount Needed A $50,000
Frequency n semiannually
Rate r 11%
Time t 15 years
Thank you for your time, it is appreciated.
Future value = 50000
period = semi-annual
number of periods, n = 15*2=30
Annual interest rate, = 11%
i = 0.11/2=0.055
Periodic payment (6 months) = P
Let r=1+i=1.055
Future value, 50000
= P(1+r+r²+r³+...+rn-1)
= P(rn-1)/(r-1) by factorizing
therefore, using (r-1)=i,
P=50000*i/(rn-1)
=50000*0.055/(1.05530-1)
=690.27 per period of 6 months.
thank you for your time it is appreciated.
To find the amount of periodic payment necessary for the deposit to a sinking fund, you can use the formula for the future value of an annuity:
PMT = A * ((r/n) / (1 - (1 + r/n)^(-n*t)))
Where:
PMT = Periodic payment
A = Amount needed
r = Rate (in decimal)
n = Frequency
t = Time
Plugging in the given values:
A = $50,000
r = 11% = 0.11 (as a decimal)
n = semiannually = 2
t = 15 years
PMT = $50,000 * ((0.11/2) / (1 - (1 + 0.11/2)^(-2*15)))
Calculating:
PMT = $50,000 * (0.055 / (1 - (1 + 0.055)^(-30)))
PMT = $50,000 * (0.055 / (1 - 0.2746))
PMT = $50,000 * (0.055 / 0.7254)
PMT = $50,000 * 0.0758
PMT = $3,790
Therefore, the amount of periodic payment necessary for the deposit to a sinking fund is approximately $3,790.
To find the amount of periodic payment necessary for the deposit to a sinking fund, we can use the formula for the future value of an ordinary annuity.
The future value of an ordinary annuity formula is given as:
A = P * [(1 + r)^n - 1] / r
Where:
A = Amount Needed
P = Periodic payment
r = Rate per period
n = Number of periods
In this case, we have:
A = $50,000
r = 11% (convert to decimal by dividing by 100: 11% / 100 = 0.11)
n = 15 years (since the frequency is semiannually, we need to multiply it by 2: 15 * 2 = 30)
Now let's substitute these values into the formula and solve for P:
A = P * [(1 + r)^n - 1] / r
$50,000 = P * [(1 + 0.11)^30 - 1] / 0.11
Next, we can solve this equation to find the value of P:
$50,000 * 0.11 = P * [(1 + 0.11)^30 - 1]
$5,500 = P * (1.11^30 - 1)
Now, let's calculate the value of (1.11^30 - 1):
(1.11^30 - 1) ≈ 12.858373
Now, we can substitute this value back into our equation:
$5,500 = P * 12.858373
Dividing both sides by 12.858373, we get:
P ≈ $5,500 / 12.858373
P ≈ $427.50
Therefore, the amount of periodic payment necessary for the deposit to a sinking fund is approximately $427.50 per period.