Battery warrant valuations

Actuarial modeling is helpful to value a car manufacturer's warrant liabilities, on newly developed batteries.  Recently we have seen a case where Tesla Motors' earnings were driven off both the level and accounting treatment of battery warrant liabilities. 

There are two basic significant considerations for this sort of mathematical analysis.  The first is the battery "mortality" model to be used, and the second is the survival probability relationships against both the vehicle age and the mileage limits.  For the first consideration, popular frameworks have too large of an initial mortality to be applied in practice here.  These models include the exponential, uniform, and Makeham type models.  A customized and more rigorous mortality function, grounded initially at zero and increasing at a 4% annual rate fits the parameters of this problem.

For the second consideration, we must model an in-between, positive relationship between a vehicle’s age and its mileage.  In the current case of one of the popular Tesla models, insurance payment is dependent upon the new battery's failure prior to 6 years and prior to 125,000 miles.  A small number of vehicles for example would have a battery failure at 6 years and with less than 20,000 miles.  Or a small number would fail in the first year and with 125,000 miles.  But a more substantial case would see some directional relationship between these survival probabilities against the two variables here.  In this note we will model a 50% co-relationship.

We begin our model proof with the whole annuity coverage cost designed to be:

Here we will use a net δ of 10%, to incorporate both the underlying interest rate and the future price discount rate on the replacement value of the battery.  The central exponent to the first integral above is the survival probability, which we take to the ¾ power to incorporate the joint survival probability on both age and mileage.  The erfc(x) function is the error function noted by:

If one wants another approach to erfc(x) versus the integral above, they can use the incomplete gamma function of half and x^2, and divided by the square root of π.  Moving on, this comes to a whole annuity of 8.5*0.6=5.2.  The whole coverage would therefore be 1-δ*(whole annuity).  Or 1-0.1(5.2)=0.45.  In English, this value shows that the automotive company would be able to offer a warrant liability for 0.45 per dollar of current replacement value.

In Tesla’s case, we have a warrant liability for a limited period of the first failure, prior to either 6-years or 125,000 miles.  So we must take our closed form definite integral and calculate it using numerical methods to be a fraction of 0.45, or in this case with Tesla’s limitations it is 0.26.

So the average warrant liability of 0.26 per dollar of replacement value has a standard deviation of 0.2.  To get there, we see that the variance of the coverage estimate is computed using the following formula:

The Coverage@2δ is equal to 1-2δ*(term-annuity@2δ).  And the standard deviation of the sample average would be equal to the square root of this variance divided by the square root of the sample size, or in Tesla’s case we have 10,000 vehicles.  Again this comes to 0.2 per dollar of replacement value.

As a final estimation we show that the 90% confidence interval (i.e., +1.65 standard deviations) for the warrant liability valuation, given these term limitations, would be $20 million to $30 million.  And note for the case of Tesla, there is an approximately $21 million cumulative provisions.  A company can actuarially sit at the low end of the valuation estimate if they are assuming (right or wrong) that future replacement costs are driven down at a substantially faster rate.  Else we see a downward bias on future earnings to compensate for the provisions estimates now.