## Running Present Value

The present value of a cash flow stream is easily calculated in the term structure framework. One simply multiplies each cash flow by the discount factor associated with the period of the flow and then sums these discounted values; that is, present value is obtained by appropriately discounting all future cash flows.

There is a special, alternative way to arrange the calculations of present value, which is sometimes quite convenient and which has a useful interpretation This different way is termed running present value. It calculates present value in a recursive manner starting with the final cash flow and working backward to the present This method uses the concepts of expectations dynamics from the previous section, although it is not necessary to assume that interest rates actually follow the expectations dynamics pattern to use the method Although this method is presented, at this point, as just an alternative to the standard method of calculation, it will be the preferred—indeed standard—method of calculation in later chapters.

To work out the process, suppose (.to, a-] , at, , , a„) is a cash flow stream, We denote the present value of this stream PV(0), meaning the present value at time zero. Now imagine that k time periods have passed and we are anticipating the remainder of the cash flow stream, which is (a*, a^.+i , , , a"„) We could calculate the present value (as viewed at time k) using the discount factors that would be applicable then, We denote this present value by PV(/c). In general, then, we can imagine the present value running along in time—each period's value being the present value of the remaining stream, but calculated using that period's discount factors These running values are related to each other in a simple way, which is the basis for the method we describe The original present value can be expressed explicitly as

PV(0) = ao + (i | a"! + (¡2 a'2 + + d„Xn where the ¿4's are the discount factors at time zero. This formula can be written in the alternative form

PV(0) - a-0 + d,[a-, + (d2/di)a-2 + ■ + {d„/d\)x„]. (4,2)

The values dk/d\, k — 2, 3, . . , n, are the discount factors 1 year front now under an assumption of expectations dynamics (as shown later) Hence,

To show how this works in general, for arbitrary time points, we employ the doubie-indexing system for discount factors introduced in the previous section, The present values at time k is

PV(A-) - xk + dk k+[xk+i ~f dk k+2Xk+2 + + dk llx„

Using the discount compounding formula, it follows that clkk+j — dk k+\dk+\ k+j Hence we may write this equation as

PV(Ar) = xk + dk k+l(xk+] + dk+i jc-i-zXk+i H-----h dk+inx„)

We can therefore write

This equation states that the present value at time k is the sum of the current cash flow and a one-period discount of the next present value. Note that dk k+\ = 1/(1 + fk where fk k+\ is the short rate at time k Hence in this method discounting always uses short rates to determine the discount factors

Present value updating The running present values satisfy the recursion

PV(k)=xk+dkMPV(k+l) where dk k+i — 1/(1 + fk k+i) the discount factor for the short rate at k

To carry out the computation in a recursive manner, the process is initiated by starting at the fined time. One first calculates PV(/i) as PV(/i) = .v„ and then PV(/i - 1) =..r(I_i +rf„_uPV(/i), and so forth until PV(0) is found.

You can visualize the process in terms of n people standing strung out, on a time line You are at the head of the line, at time zero, Each person can observe only the cash flow that occurs at that person's time point. Hence you can observe only the current, time zero, cash flow. How can you compute the present value? Use the running method.

The last person, person //, computes the present value seen then and passes that value to the first person behind. That person, using the short rate at that time, discounts the value announced by person n, then adds the observed cash flow at n — 1 and passes this new present value back to person n — 2 This process continues, each person discounting according to their short rate, until the running present value is passed to you. Once you hear what the person in front of you announces, you discount it using the initial short rate and add the current cash flow That is the overall present value.

The running present value PV(£) is, of course, somewhat of a fiction It will be the actual present value of the remaining stream at time k only if interest rates follow expectations dynamics. Otherwise, entirely different discount rates will apply at that

TABLE 43

Example of Running Present Value

TABLE 43

Example of Running Present Value

Year k | ||||||||

0 |
1 |
2 |
3 |
4 |
5 |
6 |
7 | |

Cash flow |
20 |
25 |
30 |
35 |
40 |
30 |
20 |
10 |

Discount |
943 |
9.35 |
93 |
926 |
923 |
921 |
917 | |

PV(fe) |
168 95 |
157 96 |
142.20 |
120 64 92 49 |
5687 |
29.17 |
10,00 |

The present value is found by •¡tinting at the final time and working backward, discounting one period at a time

The present value is found by •¡tinting at the final time and working backward, discounting one period at a time time However, when computing a present value at time zero, that is, when computing PV(0), the running present value method can be used since it is a mathematical identity,

Example 4.6 (Constant running rate) Suppose that the spot rate curve is flat, with = t for ali k = 1, 2, . , n Let (.vo, x\, .v2, . , x„) be a cash flow stream. In the flat case, ali forward rates are also equal to r (See Exercise 9.) Hence the present value can be calculated as

This recursion is run from the terminal time backward to k — 0

Example 4.7 (General running) A sample present value calculation is shown in Table 4.3. The basic cash flow stream is the first row of the table, We assume that the current term structure is that of Table 4.2, and the appropriate one-period discount rates (found in the first column of the discount factor table in Table 4,2) are listed in the second row of Table 4.3

The present value at any year k is computed by multiplying the discount factor listed under that year times the present value of the next year, and then adding the cash flow for year k This is done by beginning with the final year and working backward to time zero Thus we first find PV(7) = 10 00. Then PV(6) = 20+ 917 x 10 00 ~ 29,17, PV(5) = .30 + 921 x 29.17 = 56.87, and so forth. The present value of the entire stream is PV(0) = 168,95.

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