THE TIME VALUE OF A SERIES OF CASH FLOWS Managers regularly need to determine the present or future value of a
THE TIME VALUE OF A SERIES OF CASH FLOWS Managers regularly need to determine the present or future value of a
series of cash flows rather than simply a single cash flow. The principles of determining the future value or present value of a series of cash flows are the same as for a single cash flow, yet the math becomes a bit more cumbersome.
Suppose that the following deposits are made in a Thrifty Savings and Loan account paying 5% interest, compounded annually:
1 e is approximately equal to 2.718. The natural logarithm function can be found on most calculators, usually indicated by “ln”.
Mathematics of Finance
Time when Deposit is Made
Amount of Deposit
Today
At the end of the first year
At the end of the second year
What is the balance in the savings account at the end of the second year if no withdrawals are made and interest is paid annually?
Let’s simplify any problem like this by referring to today as the end of period 0, and identifying the end of the first and each successive period as 1, 2, 3, and so on. Represent each end-of-period cash flow as “ CF” with a subscript specifying the period to which it corresponds.
Thus, CF 0 is a cash flow today, CF 10 is a cash flow at the end of period
10, and CF 25 is a cash flow at the end of period 25, and so on. Representing the information in our example using cash flow and period notation:
Period Cash Flow End of Period Cash Flow
0 CF 0 $1,000 1 CF 1 $2,000 2 CF 2 $1,500
The future value of the series of cash flows at the end of the second period is calculated as follows:
End of Period Number of Periods Compounding Future Period
Cash Flow
Interest is Earned
Factor Value
The last cash flow, $1,500, was deposited at the very end of the second period—the point of time at which we wish to know the future value of the series. Therefore, this deposit earns no interest. In more formal terms, its future value is precisely equal to its present value.
Today, the end of period 0, the balance in the account is $1,000 since the first deposit is made but no interest has been earned. At the end of period 1, the balance in the account is $3,050, made up of three parts:
FOUNDATIONS
1. the first deposit, $1,000
2. $50 interest on the first deposit
3. the second deposit, $2,000 The balance in the account at the end of period 2 is $4,702.50,
made up of five parts:
1. the first deposit, $1,000
2. the second deposit, $2,000
3. the third deposit, $1,500
4. $102.50 interest on the first deposit, $50 earned at the end of the first period, $52.50 more earned at the end of the second period
5. $100 interest earned on the second deposit at the end of the second period
These cash flows can also be represented in a time line. A time line is used to help graphically depict and sort out each cash flow in a series. The time line for this example is shown in Exhibit 7.6. From this exam- ple, you can see that the future value of the entire series is the sum of each of the compounded cash flows comprising the series. In much the same way, we can determine the future value of a series comprising any number of cash flows. And if we need to, we can determine the future value of a number of cash flows before the end of the series.
For example, suppose you are planning to deposit $1,000 today and at the end of each year for the next ten years in a savings account paying 5% interest annually. If you want to know the future value of this series after four years, you compound each cash flow for the number of years it takes to reach four years. That is, you compound the first cash flow over four years, the second cash flow over three years, the third over two years, the fourth over one year, and the fifth you don’t compound at all because you will have just deposited it in the bank at the end of the fourth year.
EXHIBIT 7.6 Time Line for the Future Value of a Series of Uneven Cash Flows
Deposited to Earn 5% Compounded Interest Per Period
End of period 0 1 2 Time
Cash flows CF 0 =$1,000.00
CF 2 =$1,500.00 ➥ $2,000.00(1.05)=
CF 1 =$2,000.00
1,102.50 FV = $4,702.50
Mathematics of Finance
To determine the present value of a series of future cash flows, each cash flow is discounted back to the present, where the beginning of the first period, today, is designated as 0. As an example, consider the Thrifty Savings & Loan problem from a different angle. Instead of cal- culating what the deposits and the interest on these deposits will be worth in the future, let’s calculate the present value of the deposits. The present value is what these future deposits are worth today.
In the series of cash flows of $1,000 today, $2,000 at the end of period 1, and $1,500 at the end of period 2, each are discounted to the present, 0, as follows:
End of Period Number of Periods Discount Present Period
Cash Flow
of Discounting
The present value of the series is the sum of the present value of these three cash flows, $4,265.30. For example, the $1,500 cash flow at the end of period 2 is worth $1,428.57 at the end of the first period and is worth $1,360.54 today.
The present value of a series of cash flows can be represented in notation form as:
0 1 2 N 1 1 1 1 PV = CF 0 ----------- + CF 1 ----------- + CF 2 ----------- + … CF +
N -----------
1 + i For example, if there are cash flows today and at the end of periods 1
and 2, today’s cash flow is not discounted, the first period cash flow is discounted one period, and the second period cash flow is discounted two periods.
We can represent the present value of a series using summation notation as shown below:
t ----------- 1 PV = ∑ CF t
(7-7)
This equation tells us that the present value of a series of cash flows is the sum of the products of each cash flow and its corresponding dis- count factor.
FOUNDATIONS
We can also use the cash flow program in a financial calculator to solve for the present value of an uneven series of cash flows:
Hewlett-Packard Hewlett-Packard Hewlett-Packard Texas Instruments 10B
12C
17B
BA-II Plus
1000 CFj
CF 2000 CFj
1000 CF 0 FIN CFLO
1000 ENTER 1500 CFj
2000 CFj
1000 INPUT
↑ 1 ENTER 5 I/YR
1500 CFj
1 INPUT
↑ 2000 ENTER ■ NPV
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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