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.


Learn more about this topic:

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How to Find the Value of an Annuity

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Chapter 21 / Lesson 15
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An annuity is a type of savings account that pays back the investor in the future. Learn the formula used to calculate an annuity's value, and understand the importance of labeling specific numbers to calculate an output over time.


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