Answer: i) We can use the present value of an annuity formula to determine how much Mr. Kyeremeh needs to deposit today:
PV = A * [(1 - (1 + r)^-n)/r]
Where PV is the present value, A is the annual payment, r is the effective annual interest rate, and n is the number of periods.
In this case, A = $1000, r = 3.5%, and n = 10. We can substitute these values into the formula to get:
PV = $1000 * [(1 - (1 + 0.035)^-10)/0.035] ≈ $8,578.32
Therefore, Mr. Kyeremeh needs to deposit approximately $8,578.32 today to provide his daughter with $1000 a year for ten years, assuming a 3.5% effective annual interest rate.
ii) If the daughter saves the income from her father in an account also paying 3.5% effective annual interest, then the future value of the ten payments would be:
FV = A * [(1 + r)^n - 1]/r
Where FV is the future value, A is the annual payment, r is the effective annual interest rate, and n is the number of periods.
In this case, A = $1000, r = 3.5%, and n = 10. We can substitute these values into the formula to get:
FV = $1000 * [(1 + 0.035)^10 - 1]/0.035 ≈ $12,365.86
Therefore, if the daughter saves the income from her father in an account also paying 3.5% effective annual interest, she will have approximately $12,365.86 when she receives the final $1000 payment.
b) After three years, Kofi will have:
FV = P * (1 + r/n)^(n*t)
Where FV is the future value, P is the principal, r is the annual interest rate, n is the number of times interest is compounded per year, and t is the number of years.
In this case, P = $1000, r = 4%, n = 4 (since interest is compounded quarterly), and t = 3. We can substitute these values into the formula to get:
FV = $1000 * (1 + 0.04/4)^(4*3) = $1,343.92
After six years, Kofi will have:
FV = $1,343.92 + $1000 = $2,343.92
Now, we need to find the amount that Ama needs to deposit into her account such that after five years, she will have the same amount as Kofi.
Let X be the amount that Ama deposits. Then after five years, Ama will have:
FV = X * e^(∫0^5 1/6+t dt)
Where FV is the future value and e is the exponential function.
Simplifying the integral, we have:
FV = X * e^(ln(2)) = 2X
Therefore, we need to solve the equation:
2X = $2,343.92
X = $1,171.96
Therefore, Ama needs to deposit approximately $1,171.96 into her account to have the same amount as Kofi after five years.