Decision tree with EMV

1. Alka is considering paying \(\pounds 5\) to play a game. The game involves rolling two fair six-sided dice. If the sum of the numbers on the two dice is at least 8 , she receives \(\pounds 10\), otherwise she loses and receives nothing.

If Alka loses, she can pay a further \(\pounds 5\) to roll the dice again. If both dice show the same number then she receives \(\pounds 35\), otherwise she loses and receives nothing.

1. Draw a decision tree to model Alka's possible decisions and the possible outcomes.
2. Determine Alka's optimal EMV and state the optimal strategy indicated by the decision tree.

2. An outdoor theatre is holding a summer gala performance. The theatre owner must decide whether to take out insurance against rain for this performance. The theatre owner estimates that

• on a fine day, the total profit will be \(\pounds 15000\)
• on a wet day, the total loss will be \(\pounds 20000\)

Insurance against rain costs \(\pounds 2000\). If the performance must be cancelled due to rain, then the theatre owner will receive \(\pounds 16000\) from the insurer. If the performance is not cancelled due to rain, then the theatre owner will receive nothing from the insurer. The probability of rain on the day of the gala performance is 0.2
Draw a decision tree and hence determine whether the theatre owner should take out the insurance against rain for this performance.

5. Sebastien needs to make a journey. He can choose between travelling by plane, by train or by coach. Sebastien knows the exact costs of all three travel options, but he also wants to account for his travel time, including any possible delays. The cost of Sebastien's time is \(\pounds 50\) per hour.
The table below shows the costs, the journey times (without delays), and the corresponding probabilities of delays, for each travel option.

1. By drawing a decision tree, evaluate the EMV of the total cost of Sebastien's journey for each node of your tree.
2. Hence state the travel option that minimises the EMV of the total cost of Sebastien's journey.
3. A cube root utility function is applied to the total costs of each option. Determine the travel option with the best expected utility and state the corresponding value.

4. A game uses a standard pack of 52 playing cards. A player gives 5 tokens to play and then picks a card. If they pick a \(2,3,4,5\) or 6 then they gain 15 tokens. If any other card is picked they lose. If they lose, the card is replaced and they can choose to pick again for another 5 tokens. This time if they pick either an ace or a king they gain 40 tokens. If any other card is picked they lose. Daniel is deciding whether to play this game.

1. Draw a decision tree to model Daniel's possible decisions and the possible outcomes.
2. Calculate Daniel's optimal EMV and state the optimal strategy indicated by the decision tree.

3. The table below shows the transport options, usual travel times, possible delay times and corresponding probabilities of delay for a journey. All times are in minutes.

1. Draw a decision tree to model the transport options and the possible outcomes.
2. State the minimum expected travel time and the corresponding transport option indicated by the decision tree.

Karl is considering investing in a villa in Greece. It will cost him 56000 euros (€ 56000). His alternative is to invest his money, £35000, in the United Kingdom. He is concerned with what will happen over the next 5 years. He estimates that there is a 60% chance that a house currently worth € 56000 will appreciate to be worth € 75000 in that time, but that there is a 40% chance that it will be worth only € 55000. If he invests in the United Kingdom then there is a 50% chance that there will be 20% growth over the 5 years, and a 50% chance that there will be 10% growth.

1. Given that £1 is worth € 1.60, draw a decision tree for Karl, and advise him what to do, using the EMV of his investment (in thousands of euros) as his criterion. [4]

In fact the £/€ exchange rate is not fixed. It is estimated that at the end of 5 years, if there has been 20% growth in the UK then there is a 70% chance that the exchange rate will stand at 1.70 euros per pound, and a 30% chance that it will be 1.50. If growth has been 10% then there is a 40% chance that the exchange rate will stand at 1.70 and a 60% chance that it will be 1.50.

1. Produce a revised decision tree incorporating this information, and give appropriate advice. [3]

A financial analyst asks Karl a number of questions to determine his utility function. He estimates that for x in cash (in thousands of euros) Karl's utility is \(x^{0.5}\), and that for y in property (in thousands of euros), Karl's utility is \(y^{0.75}\).

1. Repeat your computations from part (ii) using utility instead of the EMV of his investment. Does this change your advice? [3]
2. Using EMVs, find the exchange rate (number of euros per pound) which will make Karl indifferent between investing in the UK and investing in a villa in Greece. [2]
3. Show that, using Karl's utility function, the exchange rate would have to drop to 1.277 euros per pound to make Karl indifferent between investing in the UK and investing in a villa in Greece. [4]