The Advanced Fixed Income and Derivatives Management Guide. Saied Simozar

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      Since long term cash flows are worth less than near term ones, the value of cash flows is different from the sum of cash flows. Using a continuous compounding method, we can calculate the present value of cash flows discounted by a yield of y as

2.2

Equations (2.1) and (2.2) are examples of discrete processes. We can also write equivalent equations in a continuous form. If the pedestrian walks at a speed of v per unit of time or a fixed income instrument has a cash flow of v per unit of time, we can write (2.1) as either

2.3

      or

2.4

      Likewise, (2.2) can be written as

2.5

Equation (2.4) is the integral form of (2.3); for our example (2.3) and (2.4) are identical. The most important difference between (2.4) and (2.5) is that (2.4) is linear and (2.5) is non-linear (exponentially decaying). The marginal value of a step in the linear case is the same whether it is at the beginning of the walk process or at the end of it. On the other hand, the marginal value of a cash flow in the future is less than the present cash flow.

      Let us define the function u as

      2.6

      so that

2.7

By substituting the integrand in (2.5) using (2.7), we have