Rates and Rational Models

pe_quotient_ruleMany situations require us to take the derivative of a quotient. One situation like this is a model of a stocks price to earning ratio. In this situation the price is modeled by a function P(t) and the earnings by E(t).To find the rate at which the PE ratio is changing, you need to need to take the derivative of the model for the PE ratio.

The ratio is modeled by a quotient, P(t)/E(t). Since this is a quotient, we need to apply the quotient rule for derivative, pe_quo_01

but apply this to our functions. This means u will be replaced with P and v will be replaced with E to give


Let’s assume that the PE ratio R(t) is modeled by


To complete the quotient rule we need to find the derivative of the numerator and denominator:


Using the quotient rule we get


Where the numerator has been simplified by multiplying the trinomials by the binomials. There is one set to multiply in blue and another set to multiply in green. In doing this, be careful with the subtraction! Remember to subtract each term from the green multiplication.

The rate at t = 2 is found by substituting 2 into the derivative,


Along the tangent line at t = 2, the ratio would be 11.26567797 + 4.87405 or 16.13972797 at t = 3.

pe_quotient_rule_02Following the red tangent line yields a conservative estimate since it is below the prediction of the model in black. If we were to use the tangent line closer to t = 2, we would still get conservative estimates. But those estimates would be closer to what is predicted by the model. An analyst might say that the value of the ratio at t = 3 will be anywhere from 16.14 to 30.82. This range indicates the variability that results from applying different types of strategies.

In this example, the price and earnings were modeled by quadratic functions. If other combinations of functions are used, we simply change the functions for P(t) and/or E(t) as well as the corresponding derivatives and apply the quotient rule.