Annuities: Introduction and Valuation

Annuities: Definition and Concept

Definition of Annuity

An annuity is a finite sequence of periodic payments of equal amounts. The key characteristics are that the payments are uniform in size and occur at regular intervals over a defined period of time. The term "annuity" historically implied annual payments, originating from the Latin word "annus" meaning year, but in modern financial practice, the payment interval can be any fixed period, such as monthly, quarterly, semi-annually, or annually.

Key Characteristics of an Annuity

For a series of payments to be considered an annuity, it must meet the following criteria:

• Equal Payments: Each payment in the sequence must be of the exact same amount. This constant payment amount is often denoted by $R$ (Regular Payment), $C$ (Cash Flow), or $PMT$ (Payment).

• Regular Intervals: The time period between the start of one payment and the start of the next payment must be fixed and consistent throughout the series. This period is called the payment interval (e.g., one month, one quarter, one year).

• Specified Period (Term): The payments continue for a predetermined or calculable length of time. This total duration for which the annuity payments are made is called the term of the annuity.

• Interest Accumulation/Discounting: The calculations involving annuities inherently account for the time value of money, using a specific interest rate (for accumulation of savings) or discount rate (for present value of future payments) that is applied per payment interval.

Concept and Applications

The concept of an annuity allows us to deal with a stream of identical, periodic cash flows as a single financial unit. This is extremely useful in many real-world financial scenarios where payments or receipts are structured in this manner. Understanding how to calculate the value of an annuity (either its value today or its value in the future) is crucial in these contexts.

Annuities are fundamental components in the valuation of many financial products and obligations, including:

• Loan Repayments: Many loans, such as home loans, car loans, and personal loans, are repaid through a series of fixed, equal payments called Equated Monthly Installments (EMIs). Each EMI is an annuity payment.

• Savings and Investment Plans: Regular, equal contributions to savings accounts, recurring deposits, mutual fund Systematic Investment Plans (SIPs), or retirement savings plans (like PPF or NPS) form an annuity. Calculating the total accumulated value of these savings at a future point requires calculating the Future Value of an Annuity.

• Insurance and Pension Payouts: Many pension plans provide a series of regular, equal payments after retirement, which is a form of annuity payout. Similarly, certain life insurance policies may offer payouts as a stream of equal payments.

• Lease and Rental Agreements: Fixed, periodic rental payments for property or lease payments for equipment constitute an annuity.

• Valuation of Financial Instruments: Calculating the value of instruments like bonds often involves valuing a stream of equal coupon payments, which is an annuity.

The core problems involving annuities are calculating their Present Value (the value of the entire series of payments at the beginning of the term) and their Future Value (the value of the entire series of payments at the end of the term), considering the time value of money using an appropriate interest or discount rate.

Types of Annuities (Ordinary Annuity or Regular Annuity, Annuity Due)

Annuities are primarily classified based on the timing of the payment within each payment interval. This timing significantly impacts the calculation of their present and future values.

1. Ordinary Annuity (or Annuity Immediate or Regular Annuity)

• Definition: In an Ordinary Annuity, each payment is made at the end of the corresponding payment interval.

• Timing of Payments: If the first interval starts at Time 0 and ends at Time 1, the first payment occurs at Time 1. The second payment occurs at Time 2 (end of the second interval), and so on, until the last payment which occurs at Time $n$ (at the end of the $n$-th and final period).

• Timeline Representation:

  Time: 0 ----- | ----- 1 ----- | ----- 2 ----- | ----- 3 ----- | ... ----- n-1 ----- | ----- n Payment: R R R R (End of Period 1) (End of Period 2) ... (End of Period n)

  The term of the annuity runs from Time 0 to Time $n$. There are $n$ payments in total.

• Common Examples:

  • Equated Monthly Installments (EMIs) for home loans, car loans, personal loans (payments are typically due at the end of the month after the loan is disbursed).
  • Interest payments received on most conventional bonds (coupon payments are made at the end of coupon periods).
  • Regular contributions to a savings plan where the deposit is made at the end of a period (e.g., end of month savings).

2. Annuity Due (or Annuity Anticipated)

• Definition: In an Annuity Due, each payment is made at the beginning of the corresponding payment interval.

• Timing of Payments: If the first interval starts at Time 0 and ends at Time 1, the first payment occurs at Time 0 (the beginning of the first interval). The second payment occurs at Time 1 (the beginning of the second interval), and so on, until the last payment which occurs at Time $n-1$ (at the beginning of the $n$-th and final period).

• Timeline Representation:

  Time: P----- 0 ----- P----- 1 ----- P----- 2 ----- | ... P----- n-1 ----- | ----- n R R R R (Start of Pd 1 / Time 0) (Start of Pd 2 / Time 1) ... (Start of Pd n / Time n-1) (End of Pd n / Time n)

  The term of the annuity runs from Time 0 to Time $n$. There are $n$ payments in total, with the last payment occurring one period *before* the end of the term at time $n$.

• Common Examples:

  • Rental payments (landlords often require rent at the beginning of the month).
  • Insurance premiums (often paid at the start of the coverage period).
  • Lease payments.
  • Subscriptions (e.g., magazine subscriptions paid at the start of the subscription period).

Other Types of Annuities

While Ordinary Annuities and Annuities Due are the most common types encountered in basic financial mathematics, other classifications exist:

• Deferred Annuity: An annuity whose first payment is scheduled to occur at a date later than the end of the first period. There is a period of deferral during which no payments are made.

• Perpetuity: A special type of annuity where the payments continue indefinitely (forever). It is essentially an annuity with an infinite term.

• Certain Annuity: An annuity where the payments are guaranteed to be made for a fixed, predetermined number of periods. The term is certain.

• Contingent Annuity: An annuity where the payments depend on the occurrence of a specific event, usually the survival of a person. Life annuities, which pay out as long as the annuitant is alive, are common examples.

In standard financial problems, unless explicitly stated as an annuity due or another specific type, the term "annuity" usually implies an Ordinary Annuity (payments at the end of the period).

Calculating Future Value of a Regular Annuity

Concept of Future Value of an Ordinary Annuity

The Future Value (FV) of an Ordinary Annuity is the total value that a series of equal payments, made at the end of each period, will accumulate to at the time the *last* payment is made. It represents the total accumulated sum of all the payments plus the compound interest earned on each payment from the time it is made until the end of the annuity term.

This calculation is particularly relevant for savings plans or investments where regular contributions are made (like SIPs, recurring deposits, retirement funds). It helps answer questions such as: "How much will I have in my account after making regular deposits for a certain number of years?"

Derivation of the Formula

Let:

• $R$ = The amount of the regular payment made at the end of each period.
• $i$ = The interest rate per period (expressed as a decimal). This rate must match the payment interval and compounding frequency (e.g., if payments are monthly, $i$ is the monthly interest rate).
• $n$ = The total number of payment periods (which is also the total number of payments).
• $FV$ = The Future Value of the ordinary annuity, calculated at the end of period $n$.

Consider a timeline representing the payments and the point at which we want to find the future value (Time $n$):

We need to find the future value of each individual payment as of Time $n$ and sum them up. Using the future value formula for a single sum, $FV = PV(1+i)^{\text{periods}}$, where $PV=R$ for each payment:

• The 1st payment is made at the end of Period 1 (Time 1). It earns interest for the remaining $n-1$ periods until Time $n$. Its future value at Time $n$ is $R(1+i)^{n-1}$.
• The 2nd payment is made at the end of Period 2 (Time 2). It earns interest for the remaining $n-2$ periods until Time $n$. Its future value at Time $n$ is $R(1+i)^{n-2}$.
• The 3rd payment is made at the end of Period 3 (Time 3). It earns interest for the remaining $n-3$ periods until Time $n$. Its future value at Time $n$ is $R(1+i)^{n-3}$.
• ...
• The second-to-last payment is made at the end of Period $n-1$ (Time $n-1$). It earns interest for the remaining 1 period until Time $n$. Its future value at Time $n$ is $R(1+i)^1$.
• The last payment is made at the end of Period $n$ (Time $n$). It has no time to earn interest before Time $n$. Its future value at Time $n$ is $R(1+i)^0 = R$.

The total Future Value (FV) of the annuity is the sum of the future values of all these individual payments:

$FV = R(1+i)^{n-1} + R(1+i)^{n-2} + \dots + R(1+i)^1 + R$

It's often easier to write this sum in reverse order:

$FV = R + R(1+i)^1 + R(1+i)^2 + \dots + R(1+i)^{n-1}$

We can factor out the regular payment amount $R$:

$FV = R [ 1 + (1+i)^1 + (1+i)^2 + \dots + (1+i)^{n-1} ]$

The expression inside the square brackets is a finite geometric series. A geometric series is a series of the form $a, ar, ar^2, ar^3, \dots$.

In our series:

• The first term ($a$) is 1.
• The common ratio ($r'$) is $(1+i)$ (each term is the previous term multiplied by $1+i$).
• The number of terms is $n$ (corresponding to the $n$ payments).

The formula for the sum ($S_n$) of the first $n$ terms of a geometric series is $S_n = a \frac{(r'^n - 1)}{r' - 1}$, provided $r' \neq 1$.

Substituting the values from our annuity problem into the geometric series sum formula:

Sum $= 1 \times \frac{((1+i)^n - 1)}{(1+i) - 1}$

Simplify the denominator: $(1+i) - 1 = i$.

Sum $= \frac{(1+i)^n - 1}{i}$

Now, substitute this sum back into the FV equation for the annuity:

$FV = R \left[ \frac{(1+i)^n - 1}{i} \right]$

Formula for Future Value of an Ordinary Annuity

The formula for the Future Value (FV) of an Ordinary Annuity (where payments R are made at the end of each period) is:

$\mathbf{FV = R \left[ \frac{(1+i)^n - 1}{i} \right]}$

Where:

• FV = Future Value of the ordinary annuity at the end of the term.
• R = The amount of the regular payment made at the end of each period.
• i = The interest rate per period (as a decimal). It must be consistent with the payment interval (e.g., if payments are monthly, $i$ is the monthly rate).
• n = The total number of payment periods (total number of payments).

The term inside the square brackets, $\left[ \frac{(1+i)^n - 1}{i} \right]$, is a standard factor known as the Future Value Interest Factor for an Annuity (FVIFA) or the "s-angle-n" factor, often denoted as $s_{\overline{n}|i}$. This factor can often be found in financial tables for common values of $i$ and $n$.

Worked Example

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Time Value of Money: Net Present Value (NPV): Definition, Calculation, and Decision Rule

Definition

The Net Present Value (NPV) is a widely used capital budgeting technique for evaluating the financial attractiveness of an investment project or a series of cash flows. It quantifies the value that an investment is expected to add to the firm or individual, expressed in terms of today's money.

Specifically, NPV is defined as the difference between the present value of all expected future cash inflows from an investment and the present value of all expected cash outflows, including the initial investment outlay.

The underlying principle is the Time Value of Money (TVM) – future cash flows are discounted back to their present value using a discount rate that reflects the risk of the project and the required rate of return (opportunity cost of capital).

A positive NPV indicates that the project is expected to yield returns greater than the required rate, thereby creating wealth. A negative NPV suggests the project is expected to yield less than the required rate, destroying wealth.

Calculation Formula

The calculation of NPV involves summing the present values of all cash flows, both positive (inflows) and negative (outflows), associated with a project over its expected life. The initial investment is typically an outflow occurring at the beginning of the project (Time $t=0$). Subsequent cash flows occur at the end of each period (e.g., annually, quarterly).

Let:

• $C_t$ = The net cash flow expected at the end of period $t$. This can be a cash inflow (positive $C_t$) or a cash outflow (negative $C_t$). For the initial investment at time 0, it's common to denote it as $C_0$ (which is negative).
• $i$ = The discount rate per period, expressed as a decimal. This rate should match the period length of the cash flows (e.g., if cash flows are annual, $i$ is the annual rate; if monthly, $i$ is the monthly rate). It represents the required rate of return or the cost of capital.
• $n$ = The total number of periods over the life of the investment or project.

The formula for calculating NPV is:

$NPV = \text{Present Value of Future Cash Flows} - \text{Present Value of Initial Investment}$

The present value of future cash flows is the sum of the present values of each individual cash flow occurring from period 1 to period $n$. The present value of the initial investment at time 0 is simply the initial investment amount itself, as it occurs today.

$PV\$ of\$ Future\$ Cash\$ Flows = \frac{C_1}{(1+i)^1} + \frac{C_2}{(1+i)^2} + \dots + \frac{C_n}{(1+i)^n}$

Initial Investment (at Time 0) = $C_0$ (usually treated as a positive value for the initial cost, which is then subtracted)

So, the formula for NPV is:

$\mathbf{NPV = \frac{C_1}{(1+i)^1} + \frac{C_2}{(1+i)^2} + \dots + \frac{C_n}{(1+i)^n} - C_0}$

Using summation notation, this formula can be written concisely as:

Alternatively, if we consider the initial investment $C_0$ as a negative cash flow at time $t=0$, we can express NPV as the sum of the present values of *all* cash flows from time 0 to $n$:

$NPV = \frac{C_0}{(1+i)^0} + \frac{C_1}{(1+i)^1} + \frac{C_2}{(1+i)^2} + \dots + \frac{C_n}{(1+i)^n}$

Since $(1+i)^0 = 1$, this simplifies to:

Both formulas are equivalent and commonly used. Formula (1) explicitly shows the initial cost being subtracted, while Formula (2) treats the initial cost as the cash flow at $t=0$ and sums the present values of all cash flows, including the initial one.

Each term $\frac{C_t}{(1+i)^t}$ in the summation is the present value of the cash flow occurring at time $t$. It is calculated by multiplying the cash flow $C_t$ by the discount factor for period $t$, which is $(1+i)^{-t}$.

Decision Rule for NPV

The Net Present Value method provides a clear and intuitive criterion for making investment decisions. The decision rule is based on whether the project's expected returns, in present value terms, exceed its cost.

• If NPV > 0 (Net Present Value is Positive): This means the present value of the expected future cash inflows is greater than the present value of the costs, including the initial investment. The project is expected to generate a return greater than the required rate of return ($i$). Accepting such a project will increase the wealth of the investors or the value of the firm. Therefore, projects with a positive NPV should be accepted.

• If NPV < 0 (Net Present Value is Negative): This means the present value of the expected future cash inflows is less than the present value of the costs. The project is expected to generate a return less than the required rate of return ($i$). Accepting such a project would decrease the wealth of the investors or the value of the firm. Therefore, projects with a negative NPV should be rejected.

• If NPV = 0 (Net Present Value is Zero): This means the present value of the expected future cash inflows exactly equals the present value of the costs. The project is expected to generate a return exactly equal to the required rate of return ($i$). The project just breaks even in terms of value creation at the required rate. Financially, the company is indifferent between accepting or rejecting the project. The decision might then be based on qualitative factors or strategic considerations.

When faced with choosing among several mutually exclusive projects (where only one can be undertaken), the project with the highest positive NPV should generally be selected, as it promises the largest increase in wealth in present value terms.

Worked Example

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Time Value of Money: Applications of Present Value and Future Value in Financial Decisions

The Pervasiveness of PV and FV

The concepts of Present Value (PV) and Future Value (FV), underpinned by the Time Value of Money (TVM), are arguably the most fundamental tools in finance. Almost every significant financial decision, whether made by individuals, businesses, or financial institutions, relies implicitly or explicitly on comparing monetary values across different points in time. PV and FV provide the mathematical framework to make these comparisons accurately.

Key Areas of Application

Here are some of the major areas where PV and FV concepts are applied:

1. Evaluating Investment Opportunities (Capital Budgeting)

Businesses use PV and FV extensively to decide which long-term investment projects to undertake. Techniques like:

• Net Present Value (NPV): Directly calculates the value added by a project in today's terms (as discussed in I4). Projects with positive NPV are accepted.

• Internal Rate of Return (IRR): Determines the effective rate of return a project is expected to yield. It is the discount rate at which the NPV of the project becomes zero. Accepted if IRR > required rate of return.

• Profitability Index (PI): Measures the value created per unit of investment. Calculated as PV of future cash inflows divided by the initial investment. Accepted if PI > 1 (which is equivalent to NPV > 0).

2. Valuing Financial Assets and Businesses

Determining the fair price or intrinsic value of assets that promise future cash flows is a core application.

• Bond Valuation: The market price of a bond is the sum of the present values of all its future interest payments (coupons, which form an annuity) and the present value of its face value received at maturity. The discount rate used is the current market yield for similar bonds.

• Stock Valuation (Discounted Cash Flow - DCF Analysis): A widely used method where the estimated intrinsic value of a company's stock is the present value of its expected future cash flows, such as future dividends or free cash flows available to the company. The discount rate is typically the cost of equity or the weighted average cost of capital.

• Real Estate Valuation: For income-generating properties, their value can be estimated as the present value of the expected future net rental income plus the present value of the expected sale price at the end of the holding period.

3. Loan Structuring and Management

PV is fundamental to loan calculations.

• Calculating Loan Payments (EMIs): The amount of a fixed loan payment (like an EMI) is determined such that the present value of the entire stream of future payments, discounted at the loan's interest rate, equals the initial principal amount of the loan. (This involves the PV of an annuity formula).

• Determining Outstanding Loan Balance: The remaining balance on a loan at any point in time is the present value of all the remaining future loan payments, discounted back to that point.

4. Personal Financial Planning

Individuals use PV and FV concepts extensively for planning and achieving personal financial goals.

• Retirement Planning: Calculating how much a person needs to save regularly (using FV of annuity) to accumulate a target retirement fund (FV) by a certain age. Or, calculating the present value of the required income stream during retirement to determine how large the retirement corpus needs to be today.

• Saving for Specific Goals: Determining how much needs to be invested today (PV) to have a specific sum (FV) in the future for expenses like a child's education, a down payment for a house, or a major purchase. Alternatively, calculating how much regular savings (using FV of annuity) will accumulate over time for these goals.

• Comparing Financial Products: Using the Effective Annual Rate (EAR), which is derived from PV/FV relationships, to compare savings accounts, fixed deposits, or loan offers that have different nominal rates and compounding frequencies.

5. Insurance and Pension Products

The pricing and payout structures of insurance and annuity products rely heavily on actuarial calculations that utilize PV and FV. For instance, the premium charged for an insurance policy is the present value of the expected future payouts by the insurer, adjusted for factors like mortality and expenses.

6. Comparing Cash Flows at Different Times

Any time a decision involves comparing money received or paid at different points in time, PV or FV calculations are necessary to bring those amounts to a common point in time for a meaningful comparison. For example, would you prefer $\textsf{₹}\$ 10,000$ today or $\textsf{₹}\$ 11,500$ in 2 years? The answer depends on your required rate of return; you would compare the FV of $\textsf{₹}\$ 10,000$ or the PV of $\textsf{₹}\$ 11,500$ using that rate.

In essence, PV and FV are the essential tools that allow financial decision-makers to correctly account for the opportunity cost of capital and the impact of time on monetary value, leading to more rational and wealth-maximizing choices.

Annuities: Simple Applications of Regular Annuities (Calculations for limited periods) and Problems based on Annuities

Simple Applications of Regular Annuities (Calculations for limited periods)

The formulas for the Present Value (PV) and Future Value (FV) of ordinary annuities are widely applied in various financial scenarios involving a finite series of equal, regular payments. These applications often involve calculating one of the key components of the annuity (PV, FV, payment amount R, rate i, or number of periods n) when the others are known.

1. Calculating the Accumulation of Periodic Savings (Finding FV)

If you make regular, equal deposits into an interest-bearing account, the total accumulated amount at a future date is the Future Value of that series of deposits (an ordinary annuity, assuming deposits are at the end of the period). This is used for planning savings goals like down payments, education funds, or general wealth accumulation.

Scenario: Rina decides to deposit $\textsf{₹}\$ 4,000$ at the end of every quarter into a mutual fund SIP. The fund is expected to earn an average return of 12% per annum compounded quarterly. How much money will Rina have in the fund after 5 years?

• Regular Payment (R) = $\textsf{₹}\$ 4,000$ (quarterly).
• Nominal Annual Rate (r) = 12% = 0.12. Compounding frequency $m=4$ (quarterly).
• Periodic Rate (i) = $\frac{r}{m} = \frac{0.12}{4} = 0.03$. (3% per quarter).
• Time (t) = 5 years. Number of periods (n) = $m \times t = 4 \times 5 = 20$ quarters.
• Find Future Value (FV).
• Formula: $FV = R \left[ \frac{(1+i)^n - 1}{i} \right]$.
• $FV = 4000 \left[ \frac{(1+0.03)^{20} - 1}{0.03} \right] = 4000 \left[ \frac{(1.03)^{20} - 1}{0.03} \right]$.
• Using a financial calculator or table, $(1.03)^{20} \approx 1.80611$.
• $FV \approx 4000 \left[ \frac{1.80611 - 1}{0.03} \right] = 4000 \left[ \frac{0.80611}{0.03} \right] \approx 4000 \times 26.87033 = \textsf{₹}\$ 1,07,481.32$.

Rina will have approximately $\textsf{₹}\$ 1,07,481.32$ after 5 years.

2. Calculating the Maximum Loan Amount (Finding PV)

The maximum amount of loan you can borrow today, given that you can afford a fixed payment each period, is the Present Value of that stream of affordable payments (an ordinary annuity, assuming payments are at the end of the period). This is used to determine borrowing capacity for home loans, car loans, etc.

Scenario: You can afford to pay an EMI of $\textsf{₹}\$ 20,000$ at the end of each month for a 10-year housing loan. If the interest rate is 9% per annum compounded monthly, what is the maximum loan amount you can take?

• Regular Payment (R) = $\textsf{₹}\$ 20,000$ (monthly).
• Nominal Annual Rate (r) = 9% = 0.09. Compounding frequency $m=12$ (monthly).
• Periodic Rate (i) = $\frac{r}{m} = \frac{0.09}{12} = 0.0075$. (0.75% per month).
• Time (t) = 10 years. Number of periods (n) = $m \times t = 12 \times 10 = 120$ months.
• Find Present Value (PV).
• Formula: $PV = R \left[ \frac{1 - (1+i)^{-n}}{i} \right]$.
• $PV = 20000 \left[ \frac{1 - (1+0.0075)^{-120}}{0.0075} \right] = 20000 \left[ \frac{1 - (1.0075)^{-120}}{0.0075} \right]$.
• Using a financial calculator or table, $(1.0075)^{-120} \approx 0.40655$.
• $PV \approx 20000 \left[ \frac{1 - 0.40655}{0.0075} \right] = 20000 \left[ \frac{0.59345}{0.0075} \right] \approx 20000 \times 79.1266 = \textsf{₹}\$ 15,82,532$.

The maximum loan amount you can take is approximately $\textsf{₹}\$ 15,82,532$.

3. Calculating Required Periodic Savings/Payment (Finding R)

Often, the goal is to achieve a specific future sum (FV) or finance a specific present sum (PV) through regular payments. In these cases, the regular payment amount (R) is the unknown.

• Sinking Fund Deposits (Finding R given FV): This involves determining the equal, periodic amount that must be set aside to accumulate a specific future sum by a target date. Used by businesses to save for future debt repayment or asset replacement.
• Loan EMI Calculation (Finding R given PV): This involves determining the equal, periodic payment required to amortize (pay off) a loan over a specified term. This is a very common application.

Scenario (Loan EMI): You take a loan of $\textsf{₹}\$ 5,00,000$ for 5 years at 10% per annum compounded monthly. What is the Equated Monthly Installment (EMI)?

• Present Value (PV) = $\textsf{₹}\$ 5,00,000$.
• Nominal Annual Rate (r) = 10% = 0.10. Compounding frequency $m=12$ (monthly).
• Periodic Rate (i) = $\frac{r}{m} = \frac{0.10}{12} \approx 0.008333$.
• Time (t) = 5 years. Number of periods (n) = $m \times t = 12 \times 5 = 60$ months.
• Find Regular Payment (R or EMI).
• Formula: $PV = R \left[ \frac{1 - (1+i)^{-n}}{i} \right]$. Rearrange to solve for R: $\mathbf{R = PV \left[ \frac{i}{1 - (1+i)^{-n}} \right]}$.
• $R = 500000 \left[ \frac{0.10/12}{1 - (1+0.10/12)^{-60}} \right]$.
• Using a financial calculator, $\frac{0.10}{12} \approx 0.00833333$. $(1 + 0.10/12)^{-60} \approx (1.00833333)^{-60} \approx 0.607513$.
• $R \approx 500000 \left[ \frac{0.00833333}{1 - 0.607513} \right] = 500000 \left[ \frac{0.00833333}{0.392487} \right] \approx 500000 \times 0.021247 \approx \textsf{₹}\$ 10,623.50$.

The EMI for this loan would be approximately $\textsf{₹}\$ 10,623.50$.

These examples illustrate how the PV and FV annuity formulas serve as powerful tools for financial planning and analysis, linking lump sums today with streams of payments in the future.

Problems based on Annuities

Annuity problems can involve calculating PV, FV, the payment amount (R), the number of periods (n), or the interest rate (i), given the other parameters. The key is to correctly identify the type of annuity (ordinary vs. due), the known variables, and the variable you need to find, and then apply the appropriate formula or its rearrangement.

For Ordinary Annuities:

• $FV = R \left[ \frac{(1+i)^n - 1}{i} \right]$
• $PV = R \left[ \frac{1 - (1+i)^{-n}}{i} \right]$

For Annuities Due, the formulas are simply the Ordinary Annuity formulas multiplied by $(1+i)$ because each payment occurs one period earlier and thus earns or is discounted one extra period's interest.

• $FV_{Due} = FV_{Ordinary} \times (1+i) = R \left[ \frac{(1+i)^n - 1}{i} \right](1+i)$
• $PV_{Due} = PV_{Ordinary} \times (1+i) = R \left[ \frac{1 - (1+i)^{-n}}{i} \right](1+i)$

Unless specified as Annuity Due, assume it's an Ordinary Annuity.

Finding $n$ or $i$ (or $r$) in annuity problems often requires financial calculators, spreadsheets, or trial and error, as the formulas involve exponents or require solving polynomial equations. However, exam problems are sometimes designed such that $n$ or $i$ can be found by inspection or simplified calculations.

Worked Examples

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