Notes on Symbolic Representation of Annuities and Perpetuities

Introduction

17 Payments: Monthly payments, valued three months before the first payment.
Intermediate Valuation Date: Used to represent the value of the annuity.
Symbolic Representation:
- Value at the dotted line: 4 * a̅(17)
- Discounted back to valuation date:
  - Value = 4 * a̅(17) * V²
Deferred Annuity Notation:
- Represents the expression as a̅(2|17), which means:
  - Present value of a 2-month deferred 17-month basic annuity immediate.

If Valuation Date is First Payment:
- Use 4 * a̅̅(17) * V³
- Symbolically represented as a̅(3|17), indicating a 3-month deferred annuity due.
Last Payment Valuation Date:
- Represented as 4 * S̅(17) * V¹⁹, where:
  - Discounting 19 periods back to original valuation date.

Comparing Expressions:
- 4 * a̅(17) * V² can be expressed as:
  - 4 * a̅̅(17) * V³
- Relationship between a̅(17) and S̅(17) emphasizes the 17-period discounting.

Monthly Payments: Quarterly payments of 7 forever, valued two months before the first payment.
Intermediate Valuation Date: Set at time of the first payment.
Symbolic Representation:
- Value at first payment: 7 * a̅̅(∞).
- Discount for 2 months:
  - Value = 7 * a̅̅(∞) * V²
Interest Rates:
- Q for quarterly and M for monthly.
- Adjusting V for the correct interest rate when discounting.

Confusion between quarterly and monthly payments when using 7 * a̅(∞).
- Correct Usage: 7 * a̅(∞) gives value 1 quarter before the first payment (not 1 month).
- Necessary to accumulate to valuation date properly using respective interest rates.

Various ways to symbolically represent annuities and perpetuities.
Importance of understanding and correctly applying symbols will aid in future numeric calculations.
Next video will continue with more symbolic examples.