You are planning to save for retirement over the next 30 years. To do this, you will invest $750 per month in a stock account and $325 per month in a bond account. That's 360 deposits. The return of the stock account is expected to be an APR of 10.5 percent, and the bond account will earn an APR of 6.1 percent. When you retire in 30 years, you will combine your money into an account with an APR of 6.9 percent. All interest rates are compounded monthly. In 30 years, how much can you begin to withdraw each month from your account assuming a withdrawal period of 25 years? Hint: That's 25x12 withdrawals (or payments to your self at the end of each month)

Hi, first, we have to take to future value (30 years in the future) the invested capital (both the stock account and the bond account). From there, we will consider the sum of both future values as the present value of the annuity that you are about to receive for the next 25 years (300 months). But before we do all that, we need to convert the return rates (compounded monthly) into effective monthly rates, for that we just go ahead and divide each one by 12, as follows

r(Stock) = 0.105/12= 0.00875

r(Bond)= 0.061/12 = 0.00508

r(Combined Account)= 0.069/12=0.00575

Now we are ready, first, let´s find the future value of the stock account.

[tex]FV(stock)=\frac{750((1+0.00875)^{360}-1) }{0.00875} =1,887,300.74[/tex]}

Now, let´s find out how much will it be in 30 years, investing $325 per month, at the end of the month, at 0.508% effective monthly.

[tex]FV(Bond)=\frac{325((1+0.00508)^{360}-1) }{0.00508} =332,526.95[/tex]

And then we add them up and we get:

[tex]FV(stock)+FV(bond)=1,887,300.74+332,526.95=2,219,827.69[/tex]

Ok, now let´s find the annuity (monthly withdraw) taking into account that we are going to make 300 withdraws at a rate of 0.575% effective monthly,

[tex][tex]2,219,827.69=A(142.7729593)[/tex]

[tex]\frac{2,219,827.69}{142.7729593} =A[/tex]

[tex]A=15,547.96[/tex]\frac{A((1+0.00575)^{300}-1) }{0.00575(1+0.00575)^{300} }[/tex]

Best of luck.