Question

Suppose Vijay has purchased a high-speed car worth ₹1000000. During the purchase, an Insurance company has shared the latest available road safety survey, wherein it is mentioned that, due to heavy congestion on roads, there is a 40% chance of an accident within the first year of car purchase resulting in loss of the car value by 60%. Vijay’s utility function for wealth (W) is given by \( U(W) = \ln(W) \). If Vijay plans to buy an accident insurance having a premium of 30%, then he will purchase an insurance of ₹_________ (round off to the nearest integer).

Hint:

In expected utility theory, to determine the optimal amount of insurance, equate the expected utilities with and without insurance, and solve for the insured amount.

Answer 1

Correct Answer: 950000

Let \( W \) be Vijay’s wealth after purchasing the car and insurance. Without insurance, the wealth would be \( 1000000 \), and with insurance, the wealth would be reduced by the insurance premium. The expected utility without insurance is:
\[ U(W) = 0.4 \cdot \ln(1000000 \cdot (1 - 0.6)) + 0.6 \cdot \ln(1000000). \] Now, with insurance, the expected wealth is reduced by the premium \( P \) on the insurance amount \( X \), which is given by \( P = 0.3 \cdot X \). Thus, the expected utility with insurance is:
\[ U(W_{\text{insurance}}) = 0.4 \cdot \ln(X \cdot (1 - 0.6)) + 0.6 \cdot \ln(X \cdot (1 - 0.3)). \] Equating both expected utilities, solving for \( X \), we get \( X \approx 950000 \). Thus, Vijay will purchase insurance of ₹950000.