Annuities

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Chapter 2

• Annuities
• (sample extract)

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Learning Outcomes

• Annuity-immediate and annuity-due
• Present and future values of annuities
• Perpetuity and deferred annuity
• Payment periods and compounding periods
• Variable annuities

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• Examples
• Car loan repaid with equal monthly installments
• Annuity purchased upon retirement
• Insurance policy with monthly premiums
• Bond with semi-annual coupon payments
• Remark
• In this chapter we consider annuities with
  payments which are certain and made at equal
  intervals.

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2.1 Annuity-Immediate

• Payments are made at the end of every year for n
  years, called annuity-immediate (or ordinary
  annuity or annuity in arrears)
• The present value (PV) of an annuity is the sum
  of the PV of each payment.
• Example 2.1
• Calculate the PV of an annuity-immediate of
  amount 100
• paid annually for 5 years at the rate of interest
  of 9.

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• Solution
• Table 2.1 Present value of annuity

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• Time Diagram of an annuity immediate

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• Suppose the rate of interest per period is i, let
  denote the PV of the annuity
• Sometimes denoted as when the rate of
  interest is understood.
• The PV of the annuity is
• (2.1)

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• Accumulated value at time n will be denoted by
• , where
• (2.2)

• is the future value (FV) of the
  annuity.
• If annuity is of regular payments of P, the PV
  and FV of the annuity are respectively.

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• Example 2.2
• Calculate the PV of an annuity-immediate of
  amount 100 paid annually for 5 years at the rate
  of interest of 9 using (2.1). Also, calculate
  its FV at time n.
• PV of the annuity is
• FV can be calculated as

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• Example 2.3
• Calculate the PV of an annuity-immediate of
  amount 100 payable quarterly for 10 years at the
  annual rate of interest of 8 convertible
  quarterly. Also, calculate its FV at the end of
  10 years.
• Rate of interest per payment period (Qtr) is
  (8/4) 2, and there are 40 payments.
• PV of annuity-immediate is
• FV of annuity-immediate is

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• Example 2.4
• A loan of 20,000 to purchase a car at annual
  rate of interest of 6 will be paid back through
  monthly installments over 5 years, with 1st
  installment to be made 1 month after the release
  of the loan. What is the monthly installment?
• Rate of interest per payment period (6/12)
  0.5. Let P be the monthly installment.
• so that

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2.2 Annuity-Due

• An annuity-due is an annuity for which the
  payments are made at the beginning of the payment
  periods.
• The PV of the annuity-due at time 0 will be
  denoted by
• The FV of the annuity at time n is

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• Note that
• (2.3)
• Also,
• (2.4)

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• We conclude
• (2.5)
• (2.6)
• Also, we have
• (2.7)
• (2.8)