Lecture 11 - Mathematics

							Lecture 11: Arithmetically
   Varying Annuities

        Feb 14, 2011
Geometrically Varying Annuities
• Annuity-Immediate
• Present value



• Accumulated value
Geometrically Varying Annuities
• Annuity-Due
• Present value



• Accumulated value
Geometrically Varying Annuities
• Special cases:
• r = i: n*present/accumulated value of
  each payment
• Perpetuity, r < i: 1/(i - r)
• Payment period different from
  increasing period: convert to
  increasing period
Arithmetically Increasing Annuities

• Basic case:
• First payment: 1
• Subsequent payments: previous
  payment + 1
• Annuity-immediate, n payments,
  interest rate i
Present Value
Accumulated Value
            Annuity-Due
• Corresponding formula for annuity-
  immediate x (1+i)
             Example 1
• 100 is deposited into an account at
  the end of the 1st year, and
  subsequent deposits are 200, 300,
  400, 500 at the ends of year 2, 3, 4,
  5. If the account earns 8% a year,
  what is the account balance right after
  the 5th deposit?
Example 1
               Remarks
• Use these formulas when the increments
  are of equal amount, and the same as the
  amount of the smallest (first) payment.
             Example 2
• 600 is deposited into an account at
  the end of the 1st year, and
  subsequent deposits are 700, 800,
  900, 1000 at the ends of year 2, 3, 4,
  5. If the account earns 8% a year,
  what is the account balance right after
  the 5th deposit?
Example 2
     Increasing Perpetuities
• Perpetuity immediate with payments
  1, 2, 3, …
• Present value:
              Example 3
• A perpetuity pays 10, 10, 20, 20, 30,
  30, … at the end of successive years.
  Calculate the present value of this
  perpetuity at an annual interest rate of
  10%.
Example 3
Arithmetically Decreasing Annuities

• Base case:
• First payment: n
• Subsequent payment: previous
  payment – 1
• Last payment: 1, at time n
• n payments, interest rate i
Present Value
Accumulated Value
               Remarks
• Use these formulas when the increments
  are of equal amount, and the same as the
  amount of the smallest (last) payment.
              Example
• Karen bought an increasing
  perpetuity-immediate with annual
  payments starting at amount 5 and
  increasing by 5 each year until the
  payment amount reaches 100. The
  payments remain at 100 thereafter.
  The effective annual interest rate is
  7.5%. Determine the present value of
  this perpetuity.
Solution 1
Solution 2
        Financial Calculators
• Recommended but not required
• All calculators are allowed in this course
• Review of Calculator Functions For The
  Texas Instruments BA II PLUS
• Samuel Broverman, University of Toronto
• http://www.soa.org/files/pdf/FM-23-05.pdf
• Recommended: p1. – p15.