S an Annuity With an Infinite Life Making Continual Annual Payments

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Terminology (AKA jargon) can be a major impediment to understanding the concepts of finance. Fortunately, the vocabulary of time value of money concepts is pretty straightforward. Here are the basic definitions that you will need to understand to get started (calculator key abbreviations are in parentheses where appropriate):

Types of Cash Flow Streams

Annuity

An annuity is a series of equal cash flows paid at equal time intervals for a finite number of periods. A lease that calls for payments of $1000 each month for a year would be referred to as a "12-period, $1000 annuity." Note that, strictly speaking, in order for a series of cash flows to be considered an annuity, each cash flow must be identical and the amount of time between each cash flow must be the same in all cases. There are two types of annuities that vary only in the timing of the first cash flow:

• Regular Annuity – The first payment is made one period in the future (at period 1).
• Annuity Due – The first payment is made immediately (at period 0).

Graduated Annuity

A graduated annuity (also called a growing annuity) is a series of cash flows that increases over time at a constant rate for a finite number of periods. A common example of a graduated annuity would be a lottery payout. A lottery winner (e.g., Powerball) may opt to receive their winnings as a series of 30 annual payments (the first payment is immediate, and there are 29 additional annual payments). In the case of Powerball, each payment will be 4% greater than the previous payment. Note that, strictly speaking, a graduated annuity requires that the growth rate of the payments be constant for the life of the annuity.

Lump Sum

A lump sum is a single cash flow. For example, an investment that is expected to pay $100 one year from now would have a "lump sum payment" of $100. Please note that all time value of money problems can be decomposed into a series of lump sum problems (see Principle of Value Additivity)

Perpetuity

A perpetuity is simply a type of annuity that has an infinite life. In other words, it is a "perpetual annuity."

Uneven Cash Flow Stream

Any series of cash flows that doesn't conform to the definition of an annuity is considered to be an uneven cash flow stream. For example, a series such as: $100, $100, $100, $200, $200, $200 would be considered an uneven cash flow stream. However, you might also note that it is also a series of two consecutive annuities (a 3-period $100 annuity followed by a 3-period $200 annuity). For help with valuing such cash flows, please choose your calculator from the navigation links on the left.

Other TVM Definitions

Amortization Schedule

An amortization schedule is a table that shows each loan payment over the life of a loan, and a breakdown of the amount of interest and principal paid. Typically, it will also show the remaining balance after each payment has been made. Please see my tutorial on how to create an amortization schedule in Excel for more information.

Cash Flow Sign Convention

This convention, used by financial calculators and spreadsheet functions, specifies that the sign (i.e., positive or negative numbers) indicates the direction of the cash flow. Cash inflows are entered as positive numbers, and cash outflows are entered as negative numbers. Failure to properly adhere to this convention will usually result in incorrect answers from your calculator or spreadsheet. Please note that whether a cash flow is an inflow (+) or outflow (-) depends on the part that you play in a transaction. For example, loan payments are an outflow (-) for the borrower, but an inflow (+) for the lender.

Principle of Value Additivity

This fundamental principle states that the present value (future value) of a series of cash flows is the sum of the present value (future value) of each of the individual cash flows. For example, we can calculate the present value of an annuity by using a single formula, or by calculating the present value of each individual cash flow and then adding them together. This principle is very often useful for simplifying the calculation of the present or future value of uneven cash flow streams, particularly if the cash flows follow some identifiable pattern (such as several consecutive annuities).

Rule of 72

A simple rule that can be used to approximate how long it will take a given amount of money to double at a particular interest rate. It can also be used to determine the interest rate that is required to double your money in a particular amount of time. To determine how long it will take to double your money, simply divide 72 by the interest rate (in decimal form). For example, we know that it will take about 7.2 years to double your money at a 10% interest rate (72/10 = 7.2 years). Alternatively, we can see that to double your money in 5 years you would have to earn about 14.40% per year (72/5 = 14.40).

Time Line

A time line is a graphical depiction of the cash flows in a time value of money problem. Drawing a time line can be very helpful in solving a problem as it will help you to keep track of each cash flow and the time that it occurs. The image below shows an example of a time line for an uneven cash flow stream:

Banker's Year

A banker's year is 12 months, each of which contains 30 days. Therefore, there are 360 (not 365) days in a banker's year. This is a convention that goes back to the days when "calculator" and "computer" were job descriptions instead of electronic devices. Using 360 days for a year made calculations easier to do. This convention is still used today in some calculations such as the Bank Discount Rate that is used for discount (money market) securities.

Compound Interest

This refers to the situation where, in future periods, interest is earned not only on the original principal amount, but also on the previously earned interest. This is a very powerful concept that means money can grow at an exponential rate.

Compounding Frequency

This refers to how often interest is credited to the account. Once interest is credited it becomes, in effect, principal. Note that the compounding frequency and the frequency of cash flows are not always the same. In that case, the interest rate is typically adjusted to an effective rate that is of the same periodicity as the cash flows. For example, if we have quarterly cash flows with monthly compounding, we would typically convert the monthly rate into an effective quarterly rate to solve the problem.

Discount Rate

This is the interest rate that is used to convert between future values and present values. Note that the process of calculating present values is often referred to as "discounting" because present values are generally less than future values.

Frequency of Cash Flows

When using the cash flow functions, many financial calculators prompt you for both the cash flow (CFx) and then the frequency (Fx or #Times). The frequency is simple a shortcut to save both time and memory. If a cash flow occurs more than one time in a row, then you would enter the number of times that it occurs (in most cases, you will leave it at 1). The next cash flow that is entered will be the next different cash flow.

Future Value

This term refers to the value of a cash flow (or series of them) at some specific future time. Any cash flow that is scheduled to occur sometime later than today is referred to as a "future value." Literally translated, future value means "what will it be worth at some future point in time?" For example, if an investment promises to pay $100 one year from now, then the $100 is the future value of the investment because that investment will be worth $100 at that point in time.

Internal Rate of Return

The compound average annual rate of return that is expected to be earned on an investment, assuming that the investment is held for its entire life and that the cash flows are reinvested at the same rate as the IRR. Investments that have an IRR that is greater than or equal to the cost of funds (WACC) should be accepted.

Modified Internal Rate of Return

The compound average annual rate of return that is expected to be earned on an investment, assuming that the investment is held for its entire life and that the cash flows are reinvested at a rate that is different from the IRR. Typically, the reinvestment rate is assumed to be the WACC. Investments that have an MIRR that is greater than or equal to the cost of funds (WACC) should be accepted. Note that the difference between MIRR and IRR is in the assumed reinvestment rate.

Net Present Value

The present value of the future cash flows less the cost of the investment. The NPV is a direct measure of "cost versus benefit." It represents the economic profit to be earned by making an investment. Rational investors will take all investment opportunities that have an expected NPV greater than or equal to zero. If you use Excel (or any other spreadsheet program) you should read my post about the misleading nature of the NPV function.

Number of Periods

The total number of periods is a key variable in all time value of money problems. It is important to distinguish between the number of periods and the number of years. For example, we may refer to a "30-year mortgage." However, unless the payments are made annually, the number of periods is not 30. Instead, the number of periods would be 360 (= 30 years x 12 months per year). Similarly, we may say that "this bond has 10 years to maturity." In this case, the number of periods would be 20 (= 10 years x 2 semiannual periods per year) because bonds typically pay interest semiannually.

Payment

The payment is the amount of a cash flow. Typically, payment refers to the amount of the cash flow in an annuity. This is especially true when using financial calculators or spreadsheet functions.

Period

A period is simply a unit of time. Note that, depending on the problem under consideration, the relevant definition of a period can vary. The length of a period is most often defined by the amount of time that passes between cash flows. Most commonly, a period will be a day, a week, a month, a quarter (three months), six months, or a year. In a problem that involves mortgage payments, a period would typically be one month since payments are usually made monthly. However, a problem that involves the valuation of a bond will usually have a period of six months since bond interest payments are typically made every six months (semiannually).

Present Value

This term refers to the current (today's) value of a series of future cash flows. In other words, it is the amount that you would be willing to pay today in order to receive a cash flow (or a series of them) in the future. Literally translated, present value means "what is it worth right now?"

Required Return

The required return is simply the return that an investor believes he/she needs to earn in order to make an investment in a particular security. It is based on the perceived riskiness of the security, the rate of return available on alternative investments, and the investor's degree of risk aversion. It is likely that two investors looking at the same investment will have different required returns because of their differing risk tolerance. The required return, along with the size and timing of the expected cash flows, determines the value of the investment to the investor. Note that the required return is different from the yield (or promised rate of return), which is a function of the cost of the investment and the cash flows, and not of investor preferences.

TVM

A common abbreviation for "time value of money." This concept is most succinctly described by saying that a dollar today is worth more than a dollar tomorrow. The concept underlies much of financial mathematics and is the main purpose of this site. But you knew that already, didn't you? If you are interested in learning more about TVM math, please take a look at my introduction to time value of money math.

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Source: http://www.tvmcalcs.com/terminology

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