Annuities are a key concept in financial accounting, representing a series of equal payments made at regular intervals. They're crucial for understanding loans, leases, and long-term obligations, impacting how businesses report their financial positions.

Present and future value calculations for annuities are essential for informed decision-making. Different types of annuities, such as ordinary and due, require specific formulas. Understanding these calculations helps in valuing financial instruments and planning for future obligations.

Annuity definition and types

• Annuities are a series of equal payments made at regular intervals over a specified period of time
• In financial accounting, annuities are commonly used for loans, leases, and other long-term financial obligations
• Understanding the different types of annuities is crucial for accurate financial reporting and decision-making

Ordinary vs due annuities

• Ordinary annuities have payments made at the end of each period (end of month, quarter, or year)
• Annuities due have payments made at the beginning of each period
• The timing of payments affects the present and future value calculations

Annuities certain vs contingent

• Annuities certain have a fixed number of payments and a predetermined end date (loan with fixed term)
• Contingent annuities have payments that depend on the occurrence of an event, such as the death of the annuitant (life insurance payouts)
• Contingent annuities require additional assumptions and calculations to determine their value

Present value of annuities

• The present value of an annuity is the sum of all future payments discounted back to the present at a given interest rate
• Calculating the present value is essential for making informed financial decisions and reporting the value of long-term obligations

Derivation of present value formula

• The present value formula for an annuity is derived by summing the present values of each individual payment
• The formula is: $PV = PMT \times \frac{1 - (1 + r)^{-n}}{r}$
  • $PV$ = Present Value
  • $PMT$ = Payment amount per period
  • $r$ = Interest rate per period
  • $n$ = Number of periods

Present value of ordinary annuities

• For ordinary annuities, the present value is calculated using the formula above
• The first payment is discounted for one full period, as it occurs at the end of the first period

Present value of annuities due

• For annuities due, the present value formula is adjusted to account for the first payment occurring at the beginning of the first period
• The adjusted formula is: $PV = PMT \times \frac{1 - (1 + r)^{-n}}{r} \times (1 + r)$
• The additional $(1 + r)$ term accounts for the fact that the first payment is not discounted

Future value of annuities

• The future value of an annuity is the sum of all payments accumulated with interest to a future date
• Calculating the future value helps in determining the total amount that will be received or paid over the life of the annuity

Derivation of future value formula

• The future value formula for an annuity is derived by summing the future values of each individual payment
• The formula is: $FV = PMT \times \frac{(1 + r)^n - 1}{r}$
  • $FV$ = Future Value
  • $PMT$ = Payment amount per period
  • $r$ = Interest rate per period
  • $n$ = Number of periods

Future value of ordinary annuities

• For ordinary annuities, the future value is calculated using the formula above
• The first payment earns interest for $(n - 1)$ periods, as it occurs at the end of the first period

Future value of annuities due

• For annuities due, the future value formula is adjusted to account for the first payment occurring at the beginning of the first period
• The adjusted formula is: $FV = PMT \times \frac{(1 + r)^n - 1}{r} \times (1 + r)$
• The additional $(1 + r)$ term accounts for the fact that the first payment earns interest for the entire $n$ periods

Annuity payment calculations

• In many cases, the payment amount, number of payments, or interest rate may be unknown and need to be calculated
• Understanding how to solve for these variables is essential for creating and analyzing annuity-based financial arrangements

Solving for annuity payments

• To calculate the payment amount, rearrange the present value or future value formula to solve for $PMT$
• For example, using the present value formula: $PMT = PV \times \frac{r}{1 - (1 + r)^{-n}}$

Determining number of payments

• To calculate the number of payments, rearrange the present value or future value formula to solve for $n$
• For example, using the future value formula: $n = \frac{\ln(1 + \frac{FV \times r}{PMT})}{\ln(1 + r)}$

Calculating interest rate in annuities

• To calculate the interest rate, rearrange the present value or future value formula to solve for $r$
• This typically requires the use of a financial calculator or spreadsheet, as the equation cannot be solved algebraically

Deferred annuities and values

• Deferred annuities are annuities where the payments begin at a future date, rather than immediately
• Calculating the present and future values of deferred annuities is important for long-term financial planning and reporting

Deferred annuity calculations

• To calculate the present or future value of a deferred annuity, first determine the value at the start of the payment period
• Then, discount or accumulate this value to the present or future date, using the appropriate single sum formula

Present and future values

• The present value of a deferred annuity is calculated by discounting the value at the start of the payment period to the present
• The future value of a deferred annuity is calculated by accumulating the value at the start of the payment period to the future date

Annuities with non-annual periods

• Many annuities have payment periods that are not annual, such as monthly or quarterly
• Adjusting the annuity formulas for non-annual periods is necessary for accurate financial calculations

Adjusting for monthly payments

• To adjust for monthly payments, use the monthly interest rate $(r / 12)$ and the total number of monthly periods $(n \times 12)$
• For example, the present value formula for a monthly ordinary annuity would be: $PV = PMT \times \frac{1 - (1 + \frac{r}{12})^{-n \times 12}}{\frac{r}{12}}$

Adjusting for quarterly payments

• To adjust for quarterly payments, use the quarterly interest rate $(r / 4)$ and the total number of quarterly periods $(n \times 4)$
• For example, the future value formula for a quarterly annuity due would be: $FV = PMT \times \frac{(1 + \frac{r}{4})^{n \times 4} - 1}{\frac{r}{4}} \times (1 + \frac{r}{4})$

Annuities with changing payments

• Some annuities have payments that change over time, either increasing or decreasing by a fixed amount or percentage
• Calculating the present and future values of these annuities requires additional steps and formulas

Increasing annuity payments

• For annuities with payments that increase by a fixed amount each period, use the graduated annuity formula
• The present value formula for an increasing annuity is: $PV = \frac{PMT}{r} \times [1 - \frac{1 - (1 + r)^{-n}}{r} - n \times (1 + r)^{-n}]$

Decreasing annuity payments

• For annuities with payments that decrease by a fixed amount each period, subtract the present value of the decreasing portion from the present value of a level annuity
• First, calculate the present value of a level annuity with the initial payment amount
• Then, calculate the present value of an increasing annuity with the payment decrease amount and subtract it from the level annuity present value

Perpetuities and calculations

• Perpetuities are annuities that continue indefinitely, with no fixed end date
• Calculating the present value of perpetuities is simpler than annuities, as there is no need to account for a fixed number of periods

Present value of perpetuities

• The present value formula for a perpetuity is: $PV = \frac{PMT}{r}$
• This formula assumes that payments occur at the end of each period (ordinary perpetuity)
• For a perpetuity due, multiply the ordinary perpetuity present value by $(1 + r)$

Distinguishing perpetuities from annuities

• The key difference between perpetuities and annuities is the lack of a fixed end date for perpetuities
• Annuities have a fixed number of periods $(n)$, while perpetuities continue indefinitely
• When calculating the present value, perpetuities do not include the $(1 + r)^{-n}$ term found in the annuity formula

Annuities in financial statements

• Annuities can have significant impacts on a company's financial statements, particularly the balance sheet and income statement
• Understanding how annuities are reported in financial statements is crucial for accurate financial analysis and decision-making

Annuity impacts on balance sheet

• The present value of an annuity obligation is reported as a liability on the balance sheet
• If the company is the recipient of an annuity, the present value is reported as an asset
• Changes in the present value of an annuity (due to interest rate changes or the passage of time) are reflected in the balance sheet

Annuity impacts on income statement

• The interest expense associated with an annuity obligation is reported on the income statement
• If the company is the recipient of an annuity, the interest income is reported on the income statement
• The difference between the interest expense (or income) and the actual cash payments is reported as an adjustment to the annuity balance on the balance sheet

Annuity applications and examples

• Annuities have numerous applications in both personal and business finance
• Understanding how annuities are used in various financial instruments and planning strategies is important for making informed decisions

Loan amortization schedules

• Amortization schedules break down the payments of a loan (an annuity) into principal and interest components
• Each payment consists of interest on the outstanding balance and a portion of the principal
• The interest portion decreases over time as the principal is paid down, while the principal portion increases

Retirement planning with annuities

• Annuities are often used in retirement planning to provide a steady stream of income
• Immediate annuities begin payments right away, while deferred annuities start payments at a later date
• Annuities can be fixed (payments remain constant) or variable (payments vary based on investment performance)

Valuing annuity-based financial instruments

• Many financial instruments, such as bonds with coupon payments, are based on annuities
• To value these instruments, calculate the present value of the annuity payments and any lump-sum payments (face value for bonds)
• The discount rate used in the present value calculation is typically the market interest rate for similar instruments