Compound and Discount Factor Tables There are different ways to translate values forward and backward in
Compound and Discount Factor Tables There are different ways to translate values forward and backward in
time. The basic way is through equations (7-1) and (7-5), using which- ever values of PV, FV, N, or i given, and solving for the present or future value required by the problem.
FOUNDATIONS
Another way is to use tables of discount factors and compound fac- tors. A table of compound factors for periods ranging from 1 to 20 and for rates of interest from 1% to 15% is provided in Exhibit 7.4. Simi- larly, a table of discount factors for the same range of periods and inter- est rates is provided in Exhibit 7.5. The compound factor to use for a problem is determined by choosing the table value corresponding to the row for the number of periods and the column for the interest rate per period given in the problem. A discount factor is determined in a like manner.
To see how to use a table of factors, let’s find the compound factors for several combinations of periods and interest rates. The compound factor for 10 periods and an interest rate of 5% per period is 1.6289. The compound factor for five periods and an interest rate of 10% per period is 1.6105. The compound factor for three periods and an interest rate of 6% per period is 1.1910.
The table of compound factors can also be used for situations where you need to determine the number of periods or the interest rate. For example, suppose that you are asked to find out how long it takes to double your money if the interest rate per period is 8%. Doubling your money would mean that the future value is twice the present value. Using the equation:
FV = PV (1 + i) N
and inserting the known values: 2.0000 = 1.0000 1 ( + 0.08 ) N or 2.0000 = ( 1 + 0.08 ) N the compound factor is 2.0000.
The compound factor for 8% per period over some unknown num- ber of periods is 2.0000. Looking at the top panel, going down the 8% interest rate column, we see that the factor closest to 2.0000 is nine periods (compound factor = 1.9990). Therefore, it takes nine periods to double your money if interest is compounded at 8% per period.
Consider another example. If you want to invest $1,000 for six peri- ods, at what interest rate must the account pay compounded interest in order for you to have $1,500 after six periods? We know
FV = PV 1 ( + i N ) $1,500 = $1,000 1 ( + i 6 ) 1.5000 = ( 1 + i 6 )
EXHIBIT 7.4 Table of Compound Factors
Number of
Compounding Rate
EXHIBIT 7.5 Table of Discount Factors
Number of
Discount Rate
Mathematics of Finance
Therefore, the compound factor is 1.5. Using Exhibit 7.4, we see going across the row corresponding to six periods that the compound factor is 1.5000 at (approximately) a 7% interest rate. Therefore, if you save $1,000 in an account that provides 7% per period compounded interest for 6 periods, you will have a balance of approximately $1,500 after six periods.
To see how to use Exhibit 7.5, let’s find the discount factors for sev- eral combinations of periods and interest rates. The discount factor for ten periods and an interest rate of 5% per period is 0.6139. The dis- count factor for five periods and an interest rate of 10% per period is 0.6209. The discount factor for three periods and an interest rate of 6% per period is 0.8396. Just as we did for the compound factors, these dis- count factors can be used to solve for N, given a value of the discount factor and an interest rate, or to solve for the interest rate, given the value for the discount factor and the number of discounting periods.
If we look at equations (7-1) and (7-5) and think about them for a moment, it becomes apparent that inverting the values in one table pro- duces the values in the other. For example, using the corresponding fac- tors for N = 10 and r = 5%, we see this inverse relation:
Compound factor = ⁄ 1 Discount factor 1.6289 = ⁄ 1 0.6139
Likewise,
Discount factor = ⁄ 1 Compound factor 0.6139 = ⁄ 1 1.6289
The compound and discount factors are inversely related to one another for any pair of N and i values.
Parts
» Financial Management and Analysis
» SECURITIES MARKETS The primary function of a securities market—whether or not it has a
» Stock Exchanges Stock exchanges are formal organizations, approved and regulated by
» Stock Market Indicators Stock market indicators have come to perform a variety of functions,
» Efficient Markets Investors do not like risk and they must be compensated for taking on
» THE FEDERAL RESERVE SYSTEM The United States has a central monetary authority known as the Fed-
» The Fed and the Money Supply Financial managers and investors are interested in the supply and
» Deposit Institutions Traditionally, the United States has had several types of deposit institu-
» Investment Banking The primary market involves the distribution to investors of newly
» Interest Rates and Yields Because bonds are traded in the secondary market, the price of the bond
» The Risk Premium Market participants talk of interest rates on non-Treasury securities as
» OPTIONS An option is a contract in which the writer of the option grants the
» Buying Call Options The purchase of a call option creates a position referred to as a long call
» Buying Put Options The buying of a put option creates a financial position referred to as a
» CAP AND FLOOR AGREEMENTS There are agreements available in the financial market whereby one
» I n assessing a company’s current and future cash flows, the financial
» Depreciation for Tax Purposes For accounting purposes, a firm can select a method of depreciation
» Capital Gains We tend to use the term “capital gain” loosely to mean an increase in the
» Current assets (also referred to as circulating capital and working
» Noncurrent Assets Noncurrent assets are assets that are not current assets; that is, it is not
» Deferred Taxes Along with long-term liabilities, the analyst may encounter another
» THE INCOME STATEMENT An income statement is a summary of the revenues and expenses of a
» THE STATEMENT OF CASH FLOWS The statement of cash flows is a summary over a period of time of a
» T he notion that money has a time value is one of the most basic con-
» DETERMINING THE PRESENT VALUE Now that we understand how to compute future values, let’s work the
» Shortcuts: Annuities There are valuation problems that require us to evaluate a series of level
» THE CALCULATION OF INTEREST RATES
» T here are a number of factors that affect a stock’s price and its value to
» Dividend Valuation Model If dividends are constant forever, the value of a share of stock is the
» Returns on Common Stock As we saw in the preceding section, the value of a stock is the present
» Straight Coupon Bond Suppose you are considering investing in a straight coupon bond that:
» Returns on Bonds If you invest in a bond, you realize a return from the interest it pays (if
» Coupon Bonds The present value of a bond is its current market price, which is the dis-
» Callable Bonds Some bonds have a feature, referred to as a call feature, that allows the
» RISK Whenever you make a financing or investment decision, there is some
» Financial Risk When we refer to the cash flow risk of a security, we expand our con-
» Reinvestment Rate Risk Another type of risk is the uncertainty associated with reinvesting cash
» Interest Rate Risk Interest rate risk is the sensitivity of the change in an asset’s value to
» Currency Risk In assessing the attractiveness of an investment, we estimated future cash
» 5 (Continued) Portfolio of Investment C and Investment D
» Portfolio Size and Risk What we have seen for a portfolio with two assets can be extended to
» I n Chapters 8 through 10, we discussed and practiced techniques for
» The Cost of Debt Because Congress allows you to deduct from your taxable income the
» The Cost of Common Stock The cost of common stock is the cost of raising one more dollar of com-
» INTEGRATIVE EXAMPLE: ESTIMATING THE COST OF CAPITAL FOR DUPONT
» CAPITAL BUDGETING Because a firm must continually evaluate possible investments, capital
» Investment Cash Flows When we consider the cash flows of an investment we must also consider
» Asset Disposition At the end of the useful life of an asset, the firm may be able to sell it or
» Change in Expenses When a firm takes on a new project, the costs associated with it will
» Putting It All Together Here’s what we need to put together to calculate the change in the firm’s
» The Analysis To determine the relevant cash flows to evaluate this expansion, let’s
» The Problem The new equipment costs $300,000 and is expected to have a useful life of
» T he value of a firm today is the present value of all its future cash
» Payback Period The payback period for a project is the length of time it takes to get your
» Discounted Payback Period The discounted payback period is the time needed to pay back the origi-
» Net Present Value If offered an investment that costs $5,000 today and promises to pay
» Net Present Value Decision Rule
» Profitability Index The profitability index (PI) is the ratio of the present value of change in
» Stand-Alone versus Market Risk If we have some idea of the uncertainty associated with a project’s
» Sensitivity Analysis Estimates of cash flows are based on assumptions about the economy,
» Simulation Analysis Sensitivity analysis becomes unmanageable if we change several factors
» Options on Real Assets The valuation of stock options is rather complex, but with the assis-
» OVERVIEW OF DEBT OBLIGATIONS In a debt obligation, the borrower receives money in exchange for a
» Repayment Schedule Term loans are usually repaid in installments either monthly, quarterly,
» Interest In the United States, interest is typically paid twice a year at six month
» Debt Retirement By the maturity date of the bond, the issuer must pay off the entire par
» Rating Systems In all systems the term high grade means low default risk, or conversely,
» S uppose you buy a new car that costs $20,000 and you pay cash for it.
» Limited Liability The corporate form of doing business is attractive to owners of a busi-
» Stock Ownership We can classify a corporation according to whether its shares of stock
» Voting Rights Common shareholders are generally granted rights to
» Corporate Democracy Corporate democracy gives owners of the corporation a say in how to
» Methods of Repurchasing Stock
» Dividends Although a firm’s board of directors declares a dividend on its preferred
» Sinking Funds Because there is no legal obligation to pay the preferred dividend and
» DEBT VERSUS EQUITY The combination of debt and equity used to finance a firm’s projects is
» CAPITAL STRUCTURE AND TAXES We’ve seen how the use of debt financing increases the risk to owners;
» Interest Tax Shield An interesting element introduced into the capital structure decision is
» Unused Tax Shields The value of a tax shield depends on whether the firm can use an interest
» PUTTING IT ALL TOGETHER As a firm increases the relative use of debt in the capital structure, its
» A s we saw in Part Three, managers base decisions about investing in
» CASH MANAGEMENT Cash flows out of a firm as it pays for the goods and services it pur-
» The Baumol Model The Baumol Model is based on the Economic Order Quantity (EOQ)
» The Miller-Orr Model The Baumol Model assumes that cash is used uniformly throughout the
» The Check Clearing Process The process of receiving cash from customers involves several time-
» RECEIVABLES MANAGEMENT When a firm allows customers to pay for goods and services at a later
» Captive Finance Subsidiaries Some firms choose to form a wholly-owned subsidiary—a corporation
» The Economic Order Quantity Model The Economic Order Quantity (EOQ) model helps us determine what
» Just-in-Time Inventory The goal of the just-in-time (JIT) inventory model is to cut down on the
» Monitoring Inventory Management We can monitor inventory by looking at financial ratios in much the
» Add-on-interest Another way of stating interest is with add-on interest, where the total
» Trade Credit Trade credit is granted by a supplier to a customer purchasing goods or
» Commercial Paper Commercial paper is an unsecured promissory note with a fixed matu-
» Types of Inventory Financing There are several different types of loan arrangements that involve
» SPECIALIZED COLLATERALIZED BORROWING ARRANGEMENT FOR FINANCIAL INSTITUTIONS
» RATIOS AND THEIR CLASSIFICATION
» RETURN-ON-INVESTMENT RATIOS Return-on-investment ratios compare measures of benefits, such as earn-
» The Du Pont System The returns on investment ratios give us a “bottom line” on the perfor-
» LIQUIDITY Liquidity reflects the ability of a firm to meet its short-term obligations
» PROFITABILITY RATIOS We have seen that liquidity ratios tell us about a firm’s ability to meet its
» Using a Benchmark To interpret a firm’s financial ratios we need to compare them with the
» INTEGRATIVE EXAMPLE: FINANCIAL ANALYSIS OF WAL-MART STORES 6
» Dilutive Securities For a company having securities that are dilutive—meaning they could
» ANALYSTS’ FORECASTS There are many financial services firms offering projections on different
» PRICE-EARNINGS RATIO Many investors are interested in how the earnings are valued by the mar-
» FREE CASH FLOW Cash flows without any adjustment may be misleading because they do
» NET FREE CASH FLOW There are many variations in the calculation of cash flows that are used
» Using Cash Flow Information The analysis of cash flows provides information that can be used along
» THE GLOBAL ECONOMY Many countries export a substantial portion of the goods and services
» FOREIGN CURRENCY Doing business outside of one’s own country requires dealing with the cur-
» The Euro The European Union consists of 15 European member countries that
» Global Equity Market In 1985, Euromoney surveyed several firms that either listed stock on a
» Currency Swaps When issuing bonds in another country where the bonds are not denom-
» Currency Option Contracts In contrast to a forward or futures contract, an option gives the option
» A s an alternative to the issuance of a corporate bond, a corporation
» WHAT RATING AGENCIES LOOK AT IN RATING ASSET-BACKED SECURITIES
» Third-Party Guarantees Perhaps the easiest form of credit enhancement to understand is insur-
» EXAMPLE OF AN ACTUAL STRUCTURED FINANCE TRANSACTION
» Accounting for Capital Leases
» FEDERAL INCOME TAX REQUIREMENTS FOR TRUE LEASE TRANSACTIONS
» Direct Cash Flow from Leasing When a firm elects to lease an asset rather than borrow money to pur-
» S tructured financing is a debt obligation that is backed by the value of
» CREDIT IMPACT OBJECTIVE While the sponsor or sponsors of a project financing ideally would pre-
» A business that maximizes its owners’ wealth allocates its resources
» Budgeting In budgeting, we bring together analyses of cash flows, projected income
» Taxes and Transaction Costs The Black-Scholes option pricing model ignores taxes and transaction
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