0.4 The First and Second Derivatives Illustrated

We have just demonstrated that changes in longer-term discount rates affect present value (i.e., prices) more than in the case of short-term rates and cash flows. In our example, the rate of changes in present value factors increased in time, e.g., from 26%, to 101%, to 303%.

However, the rate of increase is decreasing (i.e., “deceleration” or second derivative), e.g., from 126%, to 71%, to 38%.

In Calculus, we would say here that the first derivative is positive while the second derivative is negative. If we think of the price of the cash flows as distance covered as a function of time (“speed”), then we can think of the “First Derivative.” We can think of the “Second Derivative” here as the change in the rate of speed (“deceleration”) Were the second derivative positive, we would be speaking of “acceleration.” We observe these characteristics in this diagram:

Imagine a payment due in the future; its present value would be determined by its rate – a short-term rate (and PVF!) for a short-term payment and so on. The rate of change in present value was noted in the first derivative column, while the “rate of change in the rate of change” in the present value was noted in the second derivative column.

In other words, as time increases, the differences in present values – for 5% versus 10% – increase, but the rate of increase decreases! Remember: the price of a financial asset today is its present value!

The Take-away (again):

If rates change, prices (i.e., present values) could change dramatically!