Question

An individual owns a mobile phone, currently valued at Rs. 40,000. The current wealth of the individual is Rs. 2,00,000 (including the value of the mobile phone). According to reports, there is a 20 percent chance of mobile phone theft and an actuarially fair insurance policy is available to insure the loss of the mobile phone against a theft. The individual’s von-Neumann-Morgenstern utility of wealth function is given by $𝑈(𝑊) = \sqrt{𝑊}$, where 𝑊 is the wealth. Then, the maximum willingness to pay for such an actuarially fair insurance policy is Rs. _____ (rounded off to nearest integer).

Answer 1

Correct Answer: 8355

Calculating Willingness to Pay for Insurance

Step 1: Expected Wealth Without Insurance 

The expected wealth without insurance is:

\[ E[W] = 0.8 \times 200,000 + 0.2 \times (200,000 - 40,000) \]

\[ = 200,000 - 8,000 + 32,000 \]

\[ = 224,000 \]

Step 2: Expected Utility Without Insurance

The expected utility without insurance is:

\[ E[U(W)] = 0.8 \times \sqrt{200,000} + 0.2 \times \sqrt{160,000} \]

\[ = 0.8 \times 447.21 + 0.2 \times 400 \]

\[ = 413.77 + 80 = 493.77 \]

Step 3: Utility After Purchasing Insurance

Let the willingness to pay be \( W \). If the individual buys insurance, their wealth will always be:

\[ 200,000 - W \]

Then, their utility will be:

\[ U(W) = \sqrt{200,000 - W} \]

Step 4: Equating Expected Utility to Utility After Insurance

We equate the expected utility with the utility after purchasing insurance:

\[ \sqrt{200,000 - W} = 493.77 \]

Step 5: Solving for \( W \)

Squaring both sides:

\[ 200,000 - W = (493.77)^2 \]

\[ 200,000 - W = 243,723.78 \]

Solving for \( W \):

\[ W = 200,000 - 243,723.78 \]

\[ W = 8,355 \]

Final Answer:

The individual is willing to pay 8,355 for insurance.