Dr. Maddah ENMG 602 Intro. to Financial Eng g 11/03/08. Economic Concepts (Chapter 6, Antle)

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1 Dr. Maddah ENMG 602 Intro. to Financial Eng g 11/03/08 Economic Concepts (Chapter 6, Antle) Adustments and Valuation Recall that adustments are necessary changes to account values not resulting from transactions. (See Chapter 3.) We make adustments to bring the amounts in the accounts closer to their economic value. GAAP are increasingly framed on requiring value estimation for assets first and then deriving income implications. Valuation is a complex combination of art and science. The value of an asset is tied to cash flows that it generates. This chapter focuses on two important properties of cash flows that affect valuation: Timing and uncertainty. Cash flows and economic values The economic value of an asset, used in business, is determined by the cash flows it generates. E.g., the value of a car used for business is the financial value of the stream of services (benefits) it provides for business. Cash flows are influenced by o Timing. A dollar today is worth more than a dollar tomorrow (time value of money concept). o Uncertainty. When cash flows extend into the future, it s difficult to estimate their values with certainty. 1

2 Principal and interest Interest is the manifestation of time value of money. If you invests P dollars for one year at an interest rate of r per year, then one year later your fortune is F = P(1 + r) = P + Pr. The amount P is the principal and the amount Pr is the interest. Compound interest Under a compound interest rule, an investment earns interest on interest. Specifically, P dollars invested for n years (periods) at an interest rate of r per year will have a total value of F = P(1 + r)(1 + r) K (1 + r) = P(1 + r) n. Note that the investment value grows geometrically. Example: $100 invested at an interest rate of 10% Fcompund( n) n 2

3 Present value Consider two situations (i) having $110 a year from now; or (ii) receiving a $100 today and depositing it in a bank account at 10% annual interest rate for 1 year. These two situations are equivalent. The $100 today is the present value of the $110 future amount. In general, an amount F received n years from now is equivalent to having P = F/(1 + r) n now when the market interest rate is r per year. (P is the present value of F.) The process of evaluating future amount(s) F as an equivalent present value P is called discounting. We also say that P is the discounted value of F. The term d n = 1/(1+r) n is called the discount factor. Future value The future value, n years from now, of an amount P one has today is F = P(1 + r) n when the interest rate is r per year. Present value of a stream of cash flows A stream of cash flows is a series of payments or receipts. It is assumed that payment/receipts occur at end of periods. 3

4 For example, when the period is one year, the stream ($ 10, $5, $5, $5) indicates the payment of $10 now (end of year zero) and the receipt of $5 at end of years 1 to 3. The present value of a cash flow stream x = (x 0, x 1,, x n ) at an interest rate of r per period is n x1 x2 x x n PV = x K + = 2 n 1 + r (1 + r) (1 + r) (1 + r) = 0. Future value of a stream of cash flows The future value of a cash flow stream x = (x 0, x 1,, x n ) at an interest rate of r per period is n n n 1 0(1 ) 1(1 ) n K n (1 ). = 0 FV = x + r + x + r + + x = x + r Relation of PV concept to accounting GAAP require making an adustment if a company s discounted future cash flow (i.e. PV of cash flows) from certain fixed assets is less than its book value. The company must write down the value of the asset in the balance sheet to the economic value based on PV. This is called asset impairment adustment. Uncertain cash flows: Expected value Suppose that a future cash flow has an uncertain value F. E.g., amounts generated from sales or from accounts receivable, amounts paid for warranty or a health care plan. 4

5 Suppose that F takes on the value f with probability p, = 1,, m. How to estimate f and p? No straightforward approach: Function of information. Looking at history (if any) and current economic indicators usually helps. Subective estimates are often needed. The expected value of F is a weighted average : Values from different outcomes with weights = probabilities, m E[ F] = p f. Example of expected value and effect of information It s Dec 1, you own an asset that will pay an amount F on Dec 31 which is equally likely to be f 1 = $1,000 or f 2 = $1,500. (I.e., p 1 = p 2 = 0.5.) The expected cash flow from the asset is 2 = 1 = 1 EF [ ] = p f = = $1, 250. Suppose that throughout the month, you obtain new information. You revise your estimate of the probabilities, while not changing the estimates of the payoffs (f 1 = $1,000 and f 2 = $1,500). Specifically, suppose that the new estimates of the probabilities on different dates are as follows. 5

Date p 1 p 2 f 1 f 2 E[F] 1 Dec 0.5 0.5 $ 1,000 $ 1,500 $ 1,250 8 Dec 0.2 0.8 $ 1,000 $ 1,500 $ 1,400 15 Dec 0.7 0.3 $ 1,000 $ 1,500 $ 1,150 22 Dec 0.1 0.

6 Date p 1 p 2 f 1 f 2 E[F] 1 Dec $ 1,000 $ 1,500 $ 1,250 8 Dec $ 1,000 $ 1,500 $ 1, Dec $ 1,000 $ 1,500 $ 1, Dec $ 1,000 $ 1,500 $ 1, Dec 1 0 $ 1,000 $ 1,500 $ 1,000 As you can see, values affected by uncertainty do not follow a smooth traectory. This is perhaps what happens to stock prices. (E.g. Dow Jones index in 1988.) 6

7 Discounted expected cash flow If we combine the concepts of PV and EV we can determine a company s discounted expected cash flows from an investment. The change in discounted expected value is a function of two components: Normal economic earnings due to timing (changes due to time value of money; i.e., due to interest). Abnormal economic earnings due to uncertainty (changes in expected value from investment). Example of discounted expected cash flow Suppose the payoff from an asset will occur on Dec 31, On Dec 31, 2004, it is estimated that the asset is equally likely to pay $1,000 or $1,500 on Dec 31, This leads to an expected payoff of $1,250 on Dec 31, Assume the interest rate is 6%, then the discounted expected cash flow (DECF) from the asset is 1,250/ = $ On Dec 31, 2005, new information is available and it is estimated that the asset will pay $1,000 w.p. 0.2 and $1,500 w.p This leads to an expected payoff of $1,400 on Dec 31, Then, the DECF from the asset is 1,400 / = $1, The total earnings as a result of the new information in 2005 is 1, = $

These can be divided into normal and abnormal earnings as follows. Suppose the probability estimates did not change, then the DECF on Dec 31, 2005 is 1250 / 1.06 3 = $1,049.52.

8 These can be divided into normal and abnormal earnings as follows. Suppose the probability estimates did not change, then the DECF on Dec 31, 2005 is 1250 / = $1, Therefore, the normal economic earnings (due to time value of money) on Dec 31, 2005 are = $ This leaves = $ to abnormal economic earnings (due to uncertainty). The following figure illustrates what s happening. 8

Chapter Organization. The future value (FV) is the cash value of. an investment at some time in the future.

Chapter Organization. The future value (FV) is the cash value of. an investment at some time in the future. Chapter 5 The Time Value of Money Chapter Organization 5.2. Present Value and Discounting The future value (FV) is the cash value of an investment at some time in the future Suppose you invest 100 in a