Question

An individual faces an uncertain prospect, where wealth could be Rs. 10 Lakh with probability 0.75 and Rs. 7 Lakh with probability 0.25.
Let the utility function be U(w) = w³. Then the individual will buy full insurance by paying a premium of Rs. _____ Lakh (round off to 2 decimal places).

Answer 1

Correct Answer: 0.56

To determine the premium for full insurance, we must first calculate the expected utility without insurance and with insurance, then equate these to find the premium.

Without insurance, the expected utility is calculated as:
1. Wealth if outcome 1 occurs: Rs. 10 Lakh with probability 0.75, hence utility is \(U(10^3) = (10)^3 = 1000\).
2. Wealth if outcome 2 occurs: Rs. 7 Lakh with probability 0.25, hence utility is \(U(7^3) = (7)^3 = 343\).
Thus, the expected utility without insurance is:
\( E[U] = 0.75 \times 1000 + 0.25 \times 343 = 750 + 85.75 = 835.75 \).

With full insurance, wealth becomes constant, requiring calculation of a premium P. Post-premium wealth is Rs. (10 - P) Lakh in all scenarios. The utility with insurance becomes:
\( U(\text{insured}) = ((10 - P)^3) \).
For the individual to be indifferent between insuring and not insuring, expected utility with insurance equals expected utility without insurance:
\( (10 - P)^3 = 835.75 \).

Solving for P:
\( 10 - P = 835.75^{1/3} \approx 9.445 \)
\( P = 10 - 9.445 = 0.555 \).
Hence, the full insurance premium is Rs. 0.56 Lakh, rounded to two decimal places.