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Unformatted text preview: 2010 THE UNIVERSITY OF HONG KONG DEPARTMENT OF STATISTICS AND ACTUARIAL SCIENCE STAT1802 Financial Mathematics LN3: Discounted Cash Flow Analysis The materials presented here are related to Sections 7.1-7.6 of the reference book. Let the net cash ﬂow C t at time t be C t = cash inﬂow at time t – cash outﬂow at time t. Then, the yield equation is given as P ( i ) = ∑ t C t (1 + i )- t = ∑ t C t v t = 0 , where the summation extends over all t such that C t ̸ = 0 and T is the time that the project ends. The yield equation is also known as the net present value function. The yield rate (often called the internal rate of return) i is the solution of P ( i ) = 0, provided that a unique solution exists. Assume for the moment that the investor may borrow and lend money at a fixed rate of interest i per unit time. The investor could accumulate the net cash ﬂows connected with the project in a separate account in which interest is payable or credited at this fixed rate. By the time the project ends (say, at time T ), the balance in this account will be ∑ t C t (1 + i ) T- t . If the project continues indefinitely, this value is not defined. But the net present value may be defined with T = ∞ . The practical interpretation of the net present value function P ( i ) and the yield is as follows: Suppose that the investor may lend and borrow money at a fixed rate of interest i 1 . Since P ( i 1 ) is the present value at rate of interest i 1 of the net cash ﬂows associated with the project, we conclude that the project will be profitable if and only if P ( i 1 ) > 0. Let us assume that, as is usually the case in practice, the yield i exists and P ( i ) changes from positive to negative when i = i . Under these conditions, it is clear that the project is profitable if and only if i 1 < i . That is, the yield exceeds the rate of interest at which the investor may lend or borrow money. 1 Example. C = − 20 , 000, C 1 = − 10 , 000, C t = 3 , 000 for t = 3 , ··· , 12, and C 13 = 6 , 000. 3 , 000 3 , 000 3 , 000 6 , 000 ······ 1 2 3 4 12 13 ······ − 20 , 000 − 10 , 000 Soln: The yield equation is P ( i ) = − 20 , 000 − 10 , 000 v + 3 , 000 v 2 a 10 | + 6 , 000 v 13 . Since P (0 . 02) = 735 . 66 and P (0 . 025) = − 412 . 45, the yield rate is approximately equal to i = 0 . 02 + 0 . 005 735 . 66 735 . 66 + 412 . 45 = 0 . 0232 . The project is profitable if and only if P ( i 1 ) > 0. 2 Comparison of Two Investment Projects Suppose now that an investor is comparing the merits of two possible investments or business ventures (denoted by Projects A and B). We assume that the borrowing powers of the investor are not limited. Let P A ( i ) and P B ( i ) denote the respective net present value functions and let i A and i B be the yields (which are assumed to exist). It might be thought that the investor should always select the project with higher yield, but this is not invariably the best policy....
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This note was uploaded on 07/30/2011 for the course STAT 3801 taught by Professor Kc during the Fall '11 term at HKU.
- Fall '11