An investor wishes to set up a retirement annuity by making semiannual deposits into an account...

Question:

An investor wishes to set up a retirement annuity by making semiannual deposits into an account which offers an annual stated rate of 12% per year compounded semiannually. The first of these semiannual deposits will be made on his 25th birthday and the last on his 45th birthday. His goal is to be able to make annual withdrawals of $50,000 when he retires starting on his 65th birthday and continuing to his 85th birthday inclusive. Assume the account continues to offer 12% per year compounded semiannually throughout the entire period of deposits and withdrawals.

a. What must be the amount of the semiannual deposits if this goal is to be achieved?

Value of an Annuity:

An annuity is a series of equal cash flows received at the start or at the end of a specified period. An annuity can be categorised into an annuity due when payments are made at the start or an ordinary annuity where payments is done at the end of the period. The present value and future value of annuities are calculated based on the time value of money concept with the applicable formulas.

Answer and Explanation:

a. What must be the amount of the semiannual deposits if this goal is to be achieved?

For withdrawals from 65th birthday to 85th is a total of 21

Determine the present value of these withdrawals at year 65

  • {eq}PV \ of \ annuity \ due = Annuity* (1 + r )* \dfrac{(1 - (1 + r )^{-n}) }{ r} {/eq}

Adjust r and n for semi annual compounding:

r =0.12/2 =0.06

n = 21*2=42

  • {eq}PV \ of \ annuity \ due = 50,000* (1 + 0.06 )* \dfrac{(1 - (1 + 0.06 )^{-42}) }{ 0.06} {/eq}
  • {eq}PV \ of \ annuity \ due = $806,900.80 {/eq}
  • On the 65th birthday as he makes the first withdrawal the account should have $806,900.80

The investor will make a total of 21 deposits from his 25th birthday to the 45th birthday

Discount the $806,900.80 to determine its worth on the 45th birthday

  • {eq}PV =\dfrac{FV}{(1+r)^{n}} {/eq}
  • {eq}PV =\dfrac{806,900.80}{(1+0.06)^{40}} {/eq}
  • {eq}PV =$78,448.66 {/eq}

Determine the value of the annuities

FV at end of 45th year = $78,448.66

  • {eq}FV \ of \ annuity \ due = annuity * (1 + r )* \dfrac{( (1 + r )^{n} - 1) }{r} {/eq}
  • {eq}78,448.66 = annuity * (1 + 0.06 )* \dfrac{( (1 + 0.06 )^{42} - 1) }{ 0.06} {/eq}
  • {eq}Annuity= $420.62 {/eq}
  • Semiannual deposits = $420.62

Learn more about this topic:

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How to Find the Value of an Annuity

from Algebra II Textbook

Chapter 21 / Lesson 15
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