Suppose you decide to invest in an annuity that pays 5% interest, compounded semiannually. How much money do you need to invest semiannually to reach a savings goal of $300,000 at the end of 25 years.

In order to reach the target amount in 25 years with 5% interest we need to invest 87,282.8838 $.

Step-by-step explanation:

To find out the amount to be invested we can use the compound interest formula wich is given by:

A = P (1 + r/n) ^ (n*t)

Where A is the final amount, in our case we want to reach a value of $ 300,000.

P is the initial amount, this is what we want to find out.

r is the interest rate, it was given to us in the question that is 5% or 0.05

n is the number of compoundings a year, since it is semiannually we know n = 2

t is the total time for wich the money will be applied, t = 25 years.

Applying these values on the formula, we have:

300,000 = P (1 + 0.05/2) ^ (2*25)

300,000 = P (1 + 0.025) ^ (50)

300,000 = P (1.025) ^ (50)

300,000 = P (3.4371)

P * (3.4371) = 300,000

P = 300,000/3.4371 = 87,282.8838 $

In order to reach the target amount in 25 years with 5% interest we need to invest 87,282.8838 $.

Given Information:

Interest rate = r = 5%

Compounding semi-annually = n = 2

Accumulated amount = A = $300,000

Number of years = t = 25

Required Information

Semi-annually payment = P = ?

Answer:

Semi-annually payment ≈ $87,283

Step-by-step explanation:

Accumulated and principle amounts in terms of compound interest is given by

P = A / (1 + i) ^N

Where

i = r/n = 0.05/2 = 0.025

N = n*t = 2*25 = 50

P = 300,000 / (1 + 0.025) ^50

P ≈ $87,283

Therefore, you need to invest $87,283 two times a year for 25 years to reach a savings goal of $300,000.