# You are planning to save for retirement over the next 20 years. To do this, you will invest $700... ## Question: You are planning to save for retirement over the next 20 years. To do this, you will invest$700 a month in a stock account and $400 a month in a bond account. The return of the stock account is expected to be 9 percent, and the bond account will pay 4 percent. When you retire, you will combine your money into an account with a 7 percent return. How much can you withdraw each month from your account assuming a 25-year withdrawal period? ## Saving for Retirement: Most individuals saving for retirement need to invest at least a portion of their savings in risky assets, such as stocks. The reason is that safe investments, like bank deposits, offer almost no return as of 2019. ## Answer and Explanation: 1 Assuming a 25-year withdrawal period, you can withdraw$4,339 each month from your account.

The monthly returns of the various accounts are:

• Stock account monthly return = 9% / 12 = 0.750%
• Bond account monthly return = 4% / 12 = 0.333%
• Combined account monthly return = 7% / 12 = 0.583%

Let:

• FV = future value
• PV = present value
• rS = monthly interest rate of stocks
• rB = monthly interest rate of bonds
• rC = monthly interest rate of combined
• PMTS = monthly deposit in stock account
• PMTB = monthly deposit in bond account
• PMTC = monthly withdrawal from combined account
• n = number of months of saving = 20 * 12 = 240
• k = number of months of spending = 25 * 12 = 300

We know that at the time of retirement the value of savings must be equal to the value of withdrawals. So,

{eq}FV(Savings)=PV(Withdrawals)\\ FV(Stock \ savings)+FV(Bond \ savings)=PV(Withdrawals)\\ PMT_{S}*\frac{(1+r_{S})^{n}-1}{r_{S}}+PMT_{B}*\frac{(1+r_{B})^{n}-1}{r_{B}}=PMT_{C}*\frac{1-(1+r_{C})^{-k}}{r_{C}}\\ 700*\frac{(1+0.0075)^{240}-1}{0.0075}+400*\frac{(1+0.00333)^{240}-1}{0.00333}=PMT_{C}*\frac{1-(1+0.00583)^{-300}}{0.00583}\\ PMT_{C}=\$4,339\\ {/eq} The monthly withdrawals are$4,339.