Susan Orman wants to pay $1,550 semiannually to her granddaughter for 10 years for helping her around the house. If Susan can invest money at 4% compounded semiannually, how much must she invest today to meet this goal

She needs to deposit $25,344.72

Explanation:

Giving the following information:

Semiannual payment= $1,550

Number of periods= 20 semesters

Interest rate= 0.04/2= 0.02

First, we need to calculate the future value of the payments:

FV= {A*[(1+i)^n-1]}/i

A= Semiannual payment

FV= {1,550*[(1.02^20) - 1]} / 0.02

FV= $37,660.92

Now, the present value (initial investment):

PV= FV/(1+i)^n

PV= 37,660.92/1.02^20

PV= $25,344.72

She needs to deposit $25,344.72

Prove:

Annual payment= (PV*i) / [1 - (1+i)^(-n)]

Annual payment= (25,344.72*0.02) / [1 - (1.02^-20)]

Annual payment= $1,550

b

explanation:

the investment was $14,174.196

explanation:

a lump sum means that the father of the example made one deposit and leave it be.

we need to know the value of a capital that generated at 4.05% per year during 28 years the sum of $48,613.24

now clearing the formula to solve for capital we got:

and we reach the answer investment = $14,174.20