In investment science, notions of investment are based on the concept of flows of expenditures and receipts over time. The goal is to tailor the pattern of these flows over time so that they satisfy some underly- ing well defined objective such as the maximization of the returns on investment. In this course we will think of the undelying expenditures and receipts as denominated in cash (currency). In this case, the net of expenditures and receipts in a given time period is called the cash flow in that period. The series of cash flows over a chosen number of time periods (or time horizon) is called the cash flow stream for that series of time periods. The units of time used will vary with the problem. under consideration. In some instances it will be months, others quar- ters, and others years. In each setting the time period and the time horizon to be considered must be precisely specified. We will describe cash flow streams as vectors of real numbers. Abstractly, such a vector can be written as I=(10, 11, 12. In) where n is the number of time periods. If the time period is months and the stream horizon is 10 years, then n = 12 x 10 = 120. Knowing the basic properties of geometric series is extremely helpful in the analysis of cash flow streams. We now take a moment to review these properties. Geometric Series: Consider a series of the form where p is any real number. Such a series is called a geometric series. It is said to be an infinite geometric series if we take no∞, in which case the collection of symbols Σκ 100 is to interpreted as the limit To sum this series simply observe that Sn = 1+p+p² + p² + + p = 1+p [1+p+p² + p² + + -1] = 1+p [1+p+p² + p² ++p" | - p+1 = 1 + pSn-p+1, Therefore, or equivalently (1-p)Sn=1p+1 Sn = 1-p+1 1-p If p1, then limp" = 0, and so 71 S = lim ∑ p = lim 00 k=1 100 1-p+1 1-p 1 1-p It is also usefull to consider the related series TL S = Σρ = S - 1. k=1 The formulas for Sn given above imply that Sn = 1-pn+1 1-p 1 = p-pn+1 1-p 1 - ρ P1-PS-1 1.1. Compound interest. Consider a simple savings account with. initial investment P dollars invested at an annual interest rate of r compounded monthly and closed after n years. After the first month the account will contain the original principal Pplus the interest earned. on that principal over the month P dollars, That is, the account will contain (1+) P dollars. At the end of the second month the account will earn interest on (1+) P dollars since this is the principal for that month. Hence, as before, the content of the account at the end of the second month is the sum of the principal at the begining of the month and the interest earned on this principal during the month which is (1+P+(1+P = (1+2P. 12 12 12 Continuing in this way we see that the content of the account after n years is (1+) (12) P We now describe a cash flow stream associated with this investment. The time period for the stream is one month, and the horizon for the stream is N years or 12N periods. In the first period we make an investment of P dollars which is the initial principal. Hence we set TP. Of the remaining periods, the only period during which money is received from the account is in the last period when the account is closed, Hence 212N (1+) (12) P. Hence the cash flow stream for this investment is a vector of length 12N+1 with 10-P. Tk=0, k=1,212N-1 and 12N=(1+) (12) P. There is another equally valuable way to view this cash flow stream. In this view we consider the initial principal Pas the purchase price for the stream and so do not included it in the stream itself. In this view the cash flow stream is again of length 12N+1 but now all entries in the stream are zero except for the last which is 112N = (1+) (12) P. In general, there are many ways to model the cash flow stream for the same investment opportunity. The choice of model depends on the questions being asked, Example 1.1. Home Mortgages) A home mortgage is a loan from a bank or broker to facilitate the purchase of a home. Some of the key features of a standard home mortgage are (a) the principal borrowed, Po, (b) the annual interest rate at which it is borrowed, r, (c) and the monthly payment, y, and (d) the number of years to pay off the loan, N. We assume that the payment and compounding periods coincide (this is almost always the case). Clearly, the values of the variables Por, y, and N are related in a special way so that the loan can be payed off in n years. If we denots Pk as the principal remaining after k periods, then we must have P12N0. This equation can be used to determine the relationship between the variables Po, r. y, and N if we can write Pi2n as a function of Pory and N. To begin with note that for each period k we have Calculus tells us that lim (1+) = e. 100 If we let m→∞ we say that the account is compounding contiuously. Such an account yields Pert from an initial investment of P dollars invested at an annual rater after years (heret need not be given in whole years, e.g. t = √2 works just fine). Continuous compounding is often used to simplify the analysis in very complicated settings. Interest rates are typically quoted at an annual rate regardless on the compounding schedule. This interest rate is called the nominal interest rate. The effective interest rate is the equivalent annually compounded rate. That is, it is the rate of return of the investment after one year. If an investment returns Q dollars after one year from an initial in- vestment of P dollars, then the effective interest rate of the investment is Q-P Te= P Therefore, an investment at an annual interest rate of 4 percent com- pounded continuously has an effective interest rate of approximately 081077419239 percent since e.041.04081077419239 1.3. Present Value. There are a wide variety of investment oppor- tunities available in the market place. All of these can be modeled as having an associated cash flow stream (perhaps unknowable), but the period and time to maturity of each such stream can vary considerably. In order to compare any two investment opportunities, or equivalently, their cash flow streams, we need a measure that is independent of the period and time to maturity of the stream. One such measure is present value, In effect the present value of a cash flow stream is its total value in todays dollars. One way to compute present value is to assume a constant prevailing rate of return r. This prevailing rate of return is sometimes taken to be a riskless rate of return (for example a suitable treasury bill rate), or it can be a historical average rate of return. The choice can depend on the time to maturity of the underlying invest- ments to be compared. Once a rater has been established, one can then derive the corresponding per period rate of return of a given cash flow stream rp. For example, if the rate is 04 (or 4 percent) and the period is months, then the per period rate is r, = (.04)/12. Let us now suppose we are given a cash flow stream = (10, 11, 12 In) with a prevailing per period rate of return r. The present value of this stream can be computed by considering each flow element separately. Consider flow element 70. This is the initial cash flow element today, so its present value today is 20. Next consider 1. This is the cash flow one period from now. Take the present value of z₁ to be the value of the principal after one period if it were invested at a per period rate of Tp. That is, we need to solve the equation 21 = (1+r) Pi for P, where P₁ is the principal this yields a present value for value of the cash flow element at the end of the first period. Obviously equal to In general, the present (1+rp) equals the principal required to obtain I after k periods if it were invested at a per period rate of r. That is, the present value of is the value of Pr, the value of the principal at the end of the kth period, that satisfies the equation k Th = (1+r) Pk which is (1+r) Definition 1.2. (Present Value) Given a cash flow stream z = (10.11..... In) and a per period interest rate r, the present value of this stream relative to this rate is 12 In PV = 10 + (1+rp) (1+r,)² (1+r) is called the discount factor for this stream. The factor d, = (1+rp) Another way to describe present value is in terms of discount fac- tors. In the definition given above, the factor d, (1+r) is called the discount factor for this stream. Thus, we compute present value of a cash flow stream relative to a given per period discount factor. Example 1.3. (Savings Accounts) In the discussion of compound interest given above, we described two different cash cash flow streams for a savings account with initial prin- cipal invested at annual rater compounded monthly and closed after N years. These are = (-Р.0..... 0. (1 + гр) 12NP), (0.....0. (1+) 12NP), and = where r=r/12 and the second stream has the underlying assumption that the initial principal is to be interpreted as the purchase price of the stream. The present value of these streams relative to r, is 0 and P, respectively. Thus, present value can be used to value the purchase price of a cash flow stream today. Why is the purchase price of x zeroо while that of y is P?