Special Financial Concepts: Perpetuity and Sinking Funds

Perpetuity: Definition and Calculation of Present Value

Definition of a Perpetuity

A perpetuity is a unique type of annuity characterized by a stream of equal payments that are expected to continue indefinitely, or forever. Conceptually, it is an annuity where the term ($n$) extends infinitely ($n \to \infty$).

Perpetuities are financial constructs used in theory and as approximations in valuing certain real-world assets and income streams:

• Consols: Historically, these were perpetual bonds issued by the British government, promising fixed annual interest payments without a maturity date.

• Certain Preferred Stocks: Some preferred shares offer fixed dividend payments that are expected to continue indefinitely, provided the issuing company remains solvent and decides to pay dividends.

• Endowments: Institutions (like universities) often have endowments where a large sum is invested, and a fixed amount of the investment income is used annually (e.g., for scholarships) with the goal of never spending the principal, thus generating income effectively forever.

• Perpetual Lease or Ground Rent: Agreements to pay a fixed amount periodically for the use of land or property, with no specified end date.

The defining feature is the promise or expectation of receiving (or paying) the same amount at regular intervals forever.

Concept and Future Value

Given that the payments of a perpetuity continue without end, considering the total accumulated value of these payments at any point in the future leads to a theoretically infinite sum. As time goes to infinity, the number of payments goes to infinity, and the total interest earned also goes to infinity.

Therefore, the concept of a Future Value (FV) for a perpetuity is generally not meaningful or calculable in practice. The focus of perpetuity analysis is exclusively on determining its Present Value (PV) – the finite lump sum amount today that is equivalent to receiving (or paying) the infinite stream of future payments.

Present Value (PV) of an Ordinary Perpetuity

The Present Value of an ordinary perpetuity is the value today of receiving a series of equal payments ($R$) at the end of each period, starting one period from now, for an infinite number of periods. The calculation requires a discount rate ($i$) per period.

We can derive the formula for the Present Value of a perpetuity by taking the limit of the Present Value formula for an ordinary annuity as the number of periods ($n$) approaches infinity.

Recall the formula for the Present Value of an Ordinary Annuity (payments $R$ at the end of each period, discounted at rate $i$ per period, for $n$ periods):

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

A perpetuity is an ordinary annuity with $n \to \infty$. So, the Present Value of a perpetuity is:

$PV_{perpetuity} = \lim\limits_{n \to \infty} PV_{annuity} = \lim\limits_{n \to \infty} R \left[ \frac{1 - (1+i)^{-n}}{i} \right]$

Since $R$ and $i$ are constants with respect to $n$, we can move them outside the limit:

$PV_{perpetuity} = \frac{R}{i} \times \lim\limits_{n \to \infty} \left[ 1 - (1+i)^{-n} \right]$

$PV_{perpetuity} = \frac{R}{i} \times \left[ 1 - \lim\limits_{n \to \infty} (1+i)^{-n} \right]$

Now, we need to evaluate the limit of $(1+i)^{-n}$ as $n$ approaches infinity. This term can be written as $\frac{1}{(1+i)^n}$.

For the Present Value of the perpetuity to be a finite, positive number (which is usually the case in practical financial contexts, otherwise the PV would be infinite), the discount rate $i$ must be positive ($i > 0$). If $i > 0$, then $(1+i) > 1$.

When a number greater than 1 is raised to an increasingly large positive power, the result grows without bound:

If $i > 0$, then $\lim\limits_{n \to \infty} (1+i)^n = \infty$

Consequently, the reciprocal of this term approaches zero:

If $i > 0$, then $\lim\limits_{n \to \infty} (1+i)^{-n} = \lim\limits_{n \to \infty} \frac{1}{(1+i)^n} = 0$

Substituting this limit value back into the PV perpetuity equation:

$PV_{perpetuity} = \frac{R}{i} \times \left[ 1 - 0 \right]$

$PV_{perpetuity} = \frac{R}{i} \times 1$

$\mathbf{PV = \frac{R}{i}}$

Formula for Present Value of an Ordinary Perpetuity

The Present Value (PV) of an ordinary perpetuity, where equal payments $R$ are received (or paid) at the end of each period, starting one period from today, is given by the formula:

$\mathbf{PV = \frac{R}{i}}$

Where:

• PV = The Present Value of the ordinary perpetuity (the value today of the infinite stream of future payments).
• R = The amount of the single, regular periodic payment.
• i = The interest rate or discount rate per period, expressed as a decimal. It is essential that the discount rate ($i$) is positive ($i > 0$) and that its period matches the frequency of the payments (e.g., if payments are quarterly, $i$ must be the quarterly discount rate).

Perpetuity Due (Payments at the Beginning of the Period)

If the first payment of the perpetuity occurs immediately at time 0 (at the beginning of the first period), it is called a perpetuity due. The subsequent payments occur at the beginning of each following period forever.

The Present Value of a perpetuity due is the sum of the initial payment (which is already at time 0, so its PV is simply $R$) and the present value of all the remaining payments. The remaining payments form an ordinary perpetuity starting from Time 1.

$PV_{Perpetuity\_Due} = (\text{Payment at Time 0}) + (\text{PV of Ordinary Perpetuity starting from Time 1})$

$PV_{Perpetuity\_Due} = R + \frac{R}{i}$

This can also be written as:

$PV_{Perpetuity\_Due} = R \left(1 + \frac{1}{i}\right)$

$PV_{Perpetuity\_Due} = R \left(\frac{i+1}{i}\right)$

So, the present value of a perpetuity due is simply the present value of an ordinary perpetuity multiplied by $(1+i)$.

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Special Financial Concepts: Sinking Funds: Definition and Purpose

Definition of a Sinking Fund

A sinking fund is a dedicated fund created by an individual or entity (most commonly corporations or governments) through a series of regular, equal contributions over a specific period. The primary goal is to accumulate a predetermined target amount (a Future Value) by a particular date. The funds contributed are usually invested to earn interest, which helps the fund grow towards its objective.

In essence, a sinking fund involves a commitment to make scheduled payments into an investment account that compounds over time, with the aim of meeting a known future financial requirement.

Purpose of Sinking Funds

Sinking funds are established as a proactive financial planning tool to ensure that large future financial obligations or significant expenditures can be met without strain. By spreading the required savings or repayment amounts over time through regular contributions, the burden of a large lump-sum payment in the future is mitigated. Common purposes for establishing a sinking fund include:

1. Debt Repayment: This is perhaps the most classic use. Companies or governments that issue bonds or debentures often use sinking funds to set aside money systematically to repay the principal amount to bondholders when the debt matures. This prevents a liquidity crisis at maturity.

2. Asset Replacement: Businesses frequently use sinking funds to accumulate funds for the eventual replacement of major capital assets like machinery, buildings, or fleet vehicles. Regular contributions throughout the asset's useful life ensure that capital is available for replacement when the asset reaches the end of its economic life.

3. Future Capital Expenditures: A sinking fund can be used to save money for planned future investments in expansion projects, research and development, or acquisitions, allowing the entity to self-finance or reduce reliance on external borrowing.

4. Meeting Specific Future Financial Goals (Individual Context): While not always formally termed "sinking funds," individuals commonly adopt this principle to save for significant future expenses such as funding a child's college education, making a down payment on a house, or saving for retirement (though formal retirement plans have specific structures).

5. Environmental or Restoration Obligations: Companies in certain industries might establish sinking funds to cover future costs associated with environmental clean-up or site restoration required at the end of a project's life.

The fundamental principle of a sinking fund is converting a large future financial need into a series of smaller, manageable periodic savings, utilizing the power of compounding to reach the target amount.

Special Financial Concepts: Calculating Contributions to a Sinking Fund

Concept

When establishing a sinking fund, a key calculation is determining the amount of the regular, equal contribution ($R$) that must be made each period to ensure that the fund reaches the desired target Future Value ($FV$) by the end of the specified term ($n$), given the interest rate ($i$) that the fund's investments are expected to earn per period.

This calculation is essentially finding the payment amount ($R$) for an annuity where the Future Value ($FV$), the number of periods ($n$), and the interest rate per period ($i$) are known. Sinking fund contributions are almost always assumed to be made at the end of each period, meaning we are dealing with an Ordinary Annuity.

Formula Derivation

We begin with the formula for the Future Value (FV) of an Ordinary Annuity, which relates the future value, the periodic payment, the interest rate per period, and the number of periods:

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

In a sinking fund problem, we know the target FV, the rate $i$, and the number of periods $n$, and we need to find the regular payment $R$. To solve for $R$, we rearrange the formula by dividing both sides by the term in the square brackets:

Multiplying by the reciprocal of the fraction in the denominator gives us the formula for $R$:

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

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

Formula for Sinking Fund Contributions

The regular periodic contribution (R) required to be deposited into a sinking fund at the end of each period to accumulate a target Future Value (FV) over $n$ periods, earning interest at rate $i$ per period, is:

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

Where:

• R = The required regular periodic payment (contribution) into the sinking fund.
• FV = The target Future Value – the specific amount of money that the sinking fund needs to contain at the end of the term.
• i = The interest rate per period, expressed as a decimal. This rate must match the frequency of the contributions and the compounding frequency of the fund (e.g., if contributions are monthly and the nominal annual rate is 6% compounded monthly, $i=0.06/12=0.005$).
• n = The total number of periods over which contributions will be made. This must be consistent with the frequency of contributions and the total term of the sinking fund (e.g., if contributions are made quarterly for 10 years, $n=4 \times 10 = 40$).

The term $\left[ \frac{i}{(1+i)^n - 1} \right]$ is known as the Sinking Fund Factor or the "amortization factor" for future value. It represents the periodic payment required to accumulate $\textsf{₹}\$ 1$ by the end of $n$ periods at rate $i$.

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Problems involving Perpetuities and Sinking Funds

This section focuses on solving problems that apply the specific formulas derived for perpetuities and sinking funds. These problems typically involve finding the Present Value (PV) of a perpetuity or determining the required regular contribution (R) for a sinking fund to reach a target Future Value (FV).

Perpetuity Problems

Perpetuity problems usually involve finding the lump sum needed today (PV) to generate a perpetual stream of payments (R) or determining the rate (i) or payment (R) given the PV.

The core formula for an ordinary perpetuity (payments at the end of each period) is $\mathbf{PV = \frac{R}{i}}$. This formula can be rearranged to find R ($R = PV \times i$) or i ($i = \frac{R}{PV}$).

Worked Examples (Perpetuity)

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Sinking Fund Problems

Sinking fund problems typically involve calculating the required periodic payment (R) that must be deposited into a fund to accumulate a specific Future Value (FV) by a certain date. They use the formula derived from the Future Value of an Ordinary Annuity, solved for R: $\mathbf{R = FV \left[ \frac{i}{(1+i)^n - 1} \right]}$.

Worked Examples (Sinking Fund)

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