MICRO-Company X would like to hire Ricky to manage

Company X would like to hire Ricky to manage the Happy operations. The profits from the operations depend partially on how hard Ricky works, as follows: Probabilities Profit =

10,000 Profit =

50,000 Lazy Worker 60% 40% Hard Worker 20% 80% If Ricky is lazy, he will surf the Internet all day, and he views this as a zero-cost opportunity. However, Ricky would view working hard as a “personal cost” valued at $1,000. Assume Ricky only cares about his expected payment less any “personal cost.” (a) What fixed percentage of the profits should Company X offer Ricky? (Show your steps!) (8 marks) (b) What type of information problem the above situation will refer t

A

Key Concept

Expected Utility Maximization

Explanation

Ricky will choose the effort level that maximizes his expected utility, which is his expected payment minus any personal cost.

To calculate the fixed percentage of the profits Company X should offer Ricky, we need to ensure that Ricky's expected utility from working hard is at least as great as the utility he gets from being lazy. When Ricky is lazy, his expected profit is:

0.6 \times \$10,000 + 0.4 \times \$50,000 = \$6,000 + \$20,000 = \$26,000

Since he incurs no personal cost when lazy, his utility is the full expected profit of \

26,000. When Ricky works hard, the expected profit is:

0.2 \times \$10,000 + 0.8 \times \$50,000 = \$2,000 + \$40,000 = \$42,000

However, Ricky incurs a personal cost of \$1,000 when he works hard, so his utility from working hard is the expected profit minus the personal cost:

\$42,000 - \$1,000 = \$41,000

Let $ p $ be the fixed percentage of the profits that Company X offers Ricky. To make Ricky indifferent between working hard and being lazy, we set the expected utility from both options equal to each other:

p \times \$26,000 = p \times \$41,000 - \$1,000

Solving for $ p $, we get:

p \times \$26,000 + \$1,000 = p \times \$41,000

\$1,000 = p \times \$41,000 - p \times \$26,000

\$1,000 = p \times (\$41,000 - \$26,000)

\$1,000 = p \times \$15,000

p = \frac{\$1,000}{\$15,000}

p = \frac{1}{15}

p = 0.0667 \text{ or } 6.67\%

$ So, Company X should offer Ricky 6.67% of the profits to make him indifferent between working hard and being lazy.

B

Key Concept

Moral Hazard

Explanation

The information problem refers to moral hazard, where Company X cannot directly observe Ricky's effort level and must design a compensation scheme to incentivize him to work hard.

The ask discount rate on a particular money market instrument is 3.75%. The face value is

200,000 and it matures in 51 days. What is its current price? What would be the current price if it had 71 days to maturity? (5 marks) (b) What would be your ask discount rate % and bond equivalent rate % on the purchase of a 162-day Treasury bill for

4,875 that pays $5,000 at maturity? (

Answer

The current price for the money market instrument with 51 days to maturity is

198,537.50, and with 71 days to maturity, it is

198,212.50. The ask discount rate on the purchase of a 162-day Treasury bill is 5.13%, and the bond equivalent rate is 5.24%.

Solution

a

Calculation of Current Price with 51 Days to Maturity:

The formula for the current price using the bank discount method is: $ P = FV \times \left(1 - \frac{r_d \times t}{360} \right) $, where $ P $ is the current price, $ FV $ is the face value, $ r_d $ is the discount rate, and $ t $ is the time to maturity in days.

b

Calculation of Current Price with 71 Days to Maturity:

Using the same formula as in step a, we substitute 71 for $ t $ to find the current price for the instrument with 71 days to maturity.

c

Calculation of Ask Discount Rate and Bond Equivalent Rate:

The ask discount rate is calculated using the formula $ r_d = \frac{FV - P}{FV} \times \frac{360}{t} $, and the bond equivalent rate is calculated using the formula $ r_{BEY} = \frac{FV - P}{P} \times \frac{365}{t} $, where $ P $ is the purchase price of the Treasury bill.

Key Concept

Discounting and Pricing of Money Market Instruments

Explanation

The current price of a money market instrument is calculated by discounting its face value by the discount rate over the period until maturity. The ask discount rate and bond equivalent rate provide different annualized interest rate measures for the investment, reflecting the investor's return on a Treasury bill.

Answer

The current price for the money market instrument with 51 days to maturity is

198,537.50, and with 71 days to maturity, it is

198,212.50. The ask discount rate on the purchase of a 162-day Treasury bill is 5.13%, and the bond equivalent rate is 5.24%.

Solution

a

Calculation of Current Price with 51 Days to Maturity:

Using the formula $ P = FV \times \left(1 - \frac{r_d \times t}{360} \right) $, where $ FV = \$200,000 $, $ r_d = 3.75\% $, and $ t = 51 $, we get $ P = \$200,000 \times \left(1 - \frac{0.0375 \times 51}{360} \right) = \$198,537.50 $.

b

Calculation of Current Price with 71 Days to Maturity:

Substituting $ t = 71 $ into the same formula, we get $ P = \$200,000 \times \left(1 - \frac{0.0375 \times 71}{360} \right) = \$198,212.50 $.

c

Calculation of Ask Discount Rate and Bond Equivalent Rate:

For the Treasury bill, $ FV = \$5,000 $, $ P = \$4,875 $, and $ t = 162 $. The ask discount rate is $ r_d = \frac{\$5,000 - \$4,875}{\$5,000} \times \frac{360}{162} = 5.13\% $. The bond equivalent rate is $ r_{BEY} = \frac{\$5,000 - \$4,875}{\$4,875} \times \frac{365}{162} = 5.24\% $.

Key Concept

Discounting and Pricing of Money Market Instruments

Explanation

The current price of a money market instrument is calculated by discounting its face value by the discount rate over the period until maturity. The ask discount rate and bond equivalent rate provide different annualized interest rate measures for the investment, reflecting the investor's return on a Treasury bill.

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