Glossary

Perpetuities are financial instruments that provide endless streams of equal cash flows. They come in various forms, like government bonds, preferred stocks, and endowments, offering constant or growing payments indefinitely. Understanding perpetuities is crucial for valuing long-term investments and income streams.

Calculating the of perpetuities involves simple formulas that factor in payment amounts, discount rates, and growth rates. These calculations help investors determine the current worth of future cash flows, making perpetuities valuable tools in financial analysis and investment decision-making.

Perpetuities

Definition and examples of perpetuities

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  • Provides an infinite stream of equal periodic cash flows with no end date
  • Periodic payments continue indefinitely at equal time intervals
  • Cash flows remain constant in each period
  • Consols are government bonds that pay fixed coupon payments forever with no maturity date
  • Preferred stocks pay fixed dividends in , assuming the company remains solvent (able to pay debts)
  • Endowments provide a perpetual income stream for charitable organizations (Red Cross) or educational institutions (universities)

Present value of constant perpetuities

  • Calculated by summing the present values of all future cash flows
  • Formula: PV=CrPV = \frac{C}{r}
    • PVPV represents the present value of the
    • CC represents the constant periodic payment amount
    • rr represents the or (also known as the )
  • Divide the constant periodic payment by the to determine present value
  • If a perpetuity pays 1,000annuallyandthe[requiredrateofreturn](https://www.fiveableKeyTerm:RequiredRateofReturn)is51,000 annually and the [required rate of return](https://www.fiveableKeyTerm:Required_Rate_of_Return) is 5%, the present value would be 20,000 (PV=1,0000.05=PV = \frac{1,000}{0.05} = 20,000$)
  • This calculation relies on the concept of

Growing perpetuities and valuation

  • Periodic payments grow at a constant rate over time
  • impacts the present value of the perpetuity
  • Formula: PV=Cr−gPV = \frac{C}{r - g}
    • PVPV represents the present value of the
    • CC represents the initial periodic payment amount
    • rr represents the discount rate or required rate of return
    • gg represents the of the periodic payments
  • Subtract the growth rate from the discount rate in the denominator
  • Growth rate must be less than the discount rate for the formula to be valid (g<rg < r)
  • If a perpetuity pays 1,000initiallyandgrowsat21,000 initially and grows at 2% annually, with a required rate of return of 6%, the present value would be 25,000 (PV=1,0000.06−0.02=1,0000.04=PV = \frac{1,000}{0.06 - 0.02} = \frac{1,000}{0.04} = 25,000$)

Perpetuities in financial analysis

  • Perpetuities are a type of used in financial modeling
  • They are related to annuities, which have a finite number of payments
  • Perpetuities are useful for of certain financial instruments and streams

Key Terms to Review (26)

Annuity: An annuity is a series of equal payments made at regular intervals over a specified period. These payments can be either incoming (received) or outgoing (paid).
Annuity: An annuity is a series of equal payments made at regular intervals, such as monthly, quarterly, or annually, over a specified period of time. It is a financial instrument that provides a stream of income or payments, and it is commonly used in retirement planning, insurance, and investment strategies.
Capitalization Rate: The capitalization rate, or cap rate, is a measure used in real estate valuation to estimate the potential return on an investment property. It represents the ratio of the property's net operating income (NOI) to its market value or purchase price.
Cash flow: Cash flow is the net amount of cash being transferred into and out of a business. It represents the company's operating, investing, and financing activities over a specific period.
Cash Flow: Cash flow refers to the net amount of cash and cash-equivalents moving in and out of a business or an individual's possession over a given period of time. It is a crucial measure of financial health and performance, as it reflects the ability to generate and manage the inflow and outflow of cash necessary for operations, investments, and financing activities.
Consol: A consol, short for consolidated annuity, is a perpetual bond issued by a government that has no fixed maturity date. It represents a long-term debt obligation where the issuer is not required to repay the principal, but instead makes regular interest payments to the bondholder in perpetuity.
Constant perpetuity: A constant perpetuity is a financial instrument that pays a fixed amount of money at regular intervals indefinitely. It is valued by discounting the perpetual series of cash flows back to their present value.
Constant Perpetuity: A constant perpetuity is a type of perpetuity, which is a stream of equal cash flows that continue indefinitely. In a constant perpetuity, the cash flows remain the same amount in each period, unlike a growing perpetuity where the cash flows increase over time.
Discount rate: The discount rate is the interest rate used to determine the present value of future cash flows. It reflects the time value of money and risk associated with those future cash flows.
Discount Rate: The discount rate is a key concept in finance that represents the interest rate used to determine the present value of future cash flows. It is a crucial factor in various financial analyses and decision-making processes, as it reflects the time value of money and the risk associated with the cash flows being evaluated.
Endowment: An endowment is a permanent fund that is invested to provide a steady stream of income for an organization, such as a university or a non-profit. It is a long-term financial asset that is designed to generate revenue to support the organization's mission and operations in perpetuity. In the context of perpetuities, an endowment is a prime example of a financial instrument that generates a constant stream of income over an indefinite period of time. The principal of the endowment is typically not spent, but rather invested to produce a stable and predictable source of funding for the organization.
Growing perpetuity: A growing perpetuity is a series of cash flows that continue indefinitely with each payment growing at a constant rate. The payments occur at regular intervals and increase by a fixed percentage every period.
Growing Perpetuity: A growing perpetuity is a type of perpetuity where the cash flows increase at a constant growth rate over an infinite time horizon. It is an important concept in the valuation of assets that generate a stream of cash flows that are expected to grow indefinitely.
Growth rate: Growth rate is the measure of the increase in value of an investment or a company's earnings over a specific period. It is typically expressed as a percentage.
Growth Rate: The growth rate is a measure of the change in a variable over time, often expressed as a percentage. It is a critical concept in various finance topics, including time value of money, perpetuities, dividend discount models, discounted cash flow analysis, and forecasting cash flow to assess the value of growth.
Infinite Series: An infinite series is a mathematical expression that represents the sum of an infinite number of terms, where each term is a function of the index of the series. Infinite series are particularly relevant in the context of perpetuities, which are a type of annuity that continues indefinitely.
Perpetuity: A perpetuity is a financial instrument that provides infinite periodic payments with no end date. The value of a perpetuity is calculated by dividing the periodic payment by the discount rate.
Perpetuity: A perpetuity is an infinite stream of equal cash flows that continues forever. It is a financial concept that describes a situation where a series of payments or cash flows goes on indefinitely without end.
Preferred stock: Preferred stock is a type of equity security that gives shareholders preferential treatment regarding dividends and asset liquidation. It typically does not provide voting rights in corporate decisions.
Preferred Stock: Preferred stock is a type of equity security that provides shareholders with certain preferences over common stockholders. These preferences typically include priority in dividend payments and asset distribution in the event of liquidation.
Present Value: Present value is a fundamental concept in finance that refers to the current worth of a future sum of money or stream of cash flows, discounted at an appropriate rate of interest. It is a crucial tool for evaluating the time value of money and making informed financial decisions across various topics in finance.
Required rate of return: Required rate of return is the minimum annual percentage earned by an investment that will induce individuals or companies to put money into a particular security or project. It accounts for the risk-free rate plus a risk premium.
Required Rate of Return: The required rate of return is the minimum rate of return an investor demands in order to make an investment. It represents the opportunity cost of the capital being invested and is a crucial factor in various financial decisions and analyses.
Time Value of Money: The time value of money is a fundamental concept in finance that recognizes the difference in value between a sum of money available today and the same sum available at a future point in time. It is based on the principle that money available at the present time is worth more than the identical sum in the future due to its potential to earn interest or be invested to generate a return.
Time value of money (TVM): Time Value of Money (TVM) is the concept that money available now is worth more than the same amount in the future due to its potential earning capacity. This principle underlines why receiving money today is preferable to receiving it later.
Valuation: Valuation is the process of determining the economic value or worth of an asset, such as a company, property, or financial instrument. It is a crucial concept in finance that is used to assess the intrinsic value of an investment and make informed decisions about its acquisition, management, or sale.
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