Question

In: Finance

(Solve manually) Lucy, has just turned 27 years old today and wants to retire at age...

(Solve manually) Lucy, has just turned 27 years old today and wants to retire at age 63. When she retires she wants to have enough money to purchase a 27-year annuity that will pay $4.000 per month. Monthly payments should start on her birthday when she reaches age 63 years old. The annuity earns 3.25% rate of return (compounded annually).

To achieve her financial goals, Lucy needs to invest $X every month. She has identified 2 types of investments she weill invest her money into:

option 1:

a portfolio of technology stocks managed by Lucy's bank. this portfolio offers an annual rate of return of 7.50% (compounded annually).

option 2:

A mutual fund comprising of fixed-income instruments. This fund offers and annual rate of return of 4.25% (compounded semi-annually).

Lucy will put 50% of $X into option 1) and 50% of $X into opotion 2). How much will she have to save each month (I.e solve for $X) starting one month from now to age 63 for her to reach her retirement goal?

Solutions

Expert Solution

First we calculate the present value of 27year annuity paying $4000 every month at 3.25% annual rate - this is the value she needs to have to purchase the annuity on retirement. Moreover since the annuity should start from her birthday itself, it is annuity due.

PV of annuity due = Periodic Cash flow * [1-(1+r)-t]/r * (1+r) ; where r is the applicable interest rate and t is the time period.

In this case, the periodic cash flow = $4000, r = (3.85%/12) = 0.32% and t = (27*12) = 324 months

PV = 4000 * (1+0.32%) * [1-(1+0.32%)-324]/0.32% = $ 807,710.96

Now the two investment annuities that Lucy will invest into should total to 807710.96 on her 63 birthday.

The formula for future of annuity (ordinary in this case) = Periodic cash flow * [(1+r)t - 1]/r

t = 36*12 = 432 months

rtechnology = (7.50%/12) = 0.625% and rfixed income = 4.25%/12 = 0.35%

Now we have : x/2 * [(1+0.625%)432 - 1]/0.625% + x/2 * [(1+0.35%)432 - 1]/0.35% = 807710.96

solving for x, we get x = 501.85


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