Lesson 09 - Probability Theory and Contingent Payments.pptx

Probability

1. 1 LESSON 09 PROBABILITY THEORY AND CONTINGENT PAYMENTS LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS FIN 3230: Financial Mathematics Bachelor of Business Administration in Finance, Semester V, 2024 Department of Finance Faculty of Management and Finance University of Colombo

2. Learning Outcomes Understanding basic concepts of probability. Differentiate between mutually exclusive and independent events. Understanding calculation of expected value with different situations with different probabilities. Understand dealings related to contingent payments that affects time value of money. 2 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

3. Content Introduction Probability Mutually exclusive events and independent events Mathematical expectations Contingent payments with time value 3 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

4. Introduction Any happening whose all-possible results are known, however uncertain as to which result will occur is known as an, Random E xperiment. Examples : Tossing a coin, Rolling a die, selecting a defective item in storage The possible results of the experiment are O utcomes . Examples: Rolling a die and getting 4 The set of all possible outcomes of an experiment is the Sample Space (S) of the experiment. Example: Rolling a die, then the sample space is the set, S = {1, 2, 3, 4, 5, 6} Any sub collection of a sample space is an E vent (E) . Example: Rolling a die and getting an even number 4 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

5. Probability To a given event E , we assign a number that measures the likelihood that E occurs, P(E) , out of whole collection, called its Probability. This number is between 0 and 1 ( ) If event E never occurs, P(E)=0 If event E inevitably occurs (Certain), P(E)=1 If an event E can happen in h ways, and fail to happen in f ways, all of these ways being equally likely, then the probability p=P(E) of the occurrence of the event is given by, The probability that the event will fail to occur is;   5 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

6. But for some events we cannot assign a probability a priori . Examples: Dying a person Borrower’s defaulting In such occasions, we must instead make a prediction based on past experience. Specifically, we observe a (large) number, N , of trials of the event E in the question, and let   6 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

7. Mutually Exclusive Events Events are said to be mutually exclusive if the occurrence of one of the events precludes (prevents) the occurrence of any of the other events. Example: In a single toss of a coin, getting head rules out getting tail. Given n mutually exclusive, E 1 , E 2 , E 3 ,….., E n , the probability of occurrence of any one of the events is the sum of the respective probabilities of the individual events . P ( E 1 or E 2 or E 3 or….. or E n ) = P (E 1 ) + P (E 2 ) + P (E 3 ) +…… + P ( E n ) 7 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

8. Independent Events The events are said to be independent if the occurrence of one of the events has no effect on the occurrence of any of the other events. Example: Successive flips of a fair coin. The outcome of one toss has no effect on the outcome of any other toss. The probability of the occurrence of all n independent events E 1 , E 2 , E 3 ,….., E n is the product of the respective probabilities of the individual events. P ( E 1 and E 2 and E 3 and….. and E n ) = P (E 1 ) * P (E 2 ) * P (E 3 ) *…… * P (E n ) 8 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

9. Find the probability that in throwing a die you would get either five or six. Find the probability that in drawing two cards from a standard deck of 52 cards, you get an ace and a king. Assume no replacements. 3. A random number generator on a computer selects three integers from 1 to 20 (assume all the numbers are positive). What is the probability that all three numbers are less than or equal to 5? 9 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

10. A box contains six green, three blue and five yellow marbles. Determine the probabilities of drawing a green marble (on one draw) drawing two yellow marbles (no replacement) drawing a blue marble twice (with replacement) drawing a green marble followed by a yellow marble (with replacement) drawing a blue marble and a green marble (no replacement) 10 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

11. Mathematical Expectation Let X denote the numerical outcome of an experiment, whose possible outcomes are the numerical values x 1 , x 2 ,… . Then X will take on the values of x 1 , x 2 ,… with given probabilities f ( x 1 ), f ( x 2 ),… . The mathematical expectation or expected value of X is defined by, E (X) = x 1 f ( x 1 ) + x 2 f ( x 2 ) +.... + x n f ( x n ) 11 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

12. A box c ontains four $10 bills, six $5 bills, and three $1 bills. You are allowed to pull one bill from the box and keep it. What is the expected value of your winnings? Mr. A pays $ 10 to enter a betting game. If he can get 3 tails in a row by tossing a fair coin, he wins $ 50; otherwise, he loses $ 10. What is his expected gain? A and B play a dice game. A wins if, when the dice are thrown, the two faces showing are identical; otherwise, B wins. If A wagers $ 15, how much should B wager, to make the game fair? 12 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

13. Contingent Payments with Time Value We can now mathematically analyse situations where investors discount the value of an investment to account for the probability of receipt of payment. If p is the probability of receiving a sum S and 1-p is the corresponding probability of receiving nothing, then the amount S actually received has expected value, E (X) = Sp + 0 (1-p) = pS The discounted value of the expectation pS to be received n periods from today, assuming interest at rate i per period is, pS (1 + i ) -n If sums of money S 1 , S 2 ,…, S n are received with probabilities p 1 , p 2 , …, p n at times t 1 , t 2 ,…, t n then the discounted value of the expectation is, p 1 S 1 (1 + i ) -t 1 + p 2 S 2 (1 + i ) -t 2 + … + p n S n (1 + i ) - t n 13 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

14. Mr. Jones wants to borrow some money. He can repay the loan with a single payment of Rs. 8000 in a one-years time. The lending institution determines that there is a 5% chance that Mr. Jones will not repay the loan. The normal lending rate at that time is j 1 = 10%. How much will they lend Mr. Jones? If he repays the loan in full, what rate of interest was realized? Roberts’ Aunt dies and leaves him an inheritance of Rs. 50 000 in a bank account earning interest at 8% per annum. He gains title to the money if he lives to age 21, he is now aged 8. If he dies before age 21 (probability 0.05), the money goes to his sister Sandra, aged 5, if she survives to age 21 (probability 0.97). Determine Sandra’s expectation. 14 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

15. Edna is in very poor health, is to receive Rs. 5000 at the end of each year as long as she is alive. Her survival probabilities are as follows. If what is the expected value of these payments?   n Probability of surviving n years 1 0.75 2 0.40 3 15 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS

16. ADDITIONAL QUESTIONS LESSON 6 - LOAN AMORTIZATION (1)

17. According to statistics gathered in New York city, in 1995 there were 200,000 births, of which 2087 were multiple births ( ie ., twins, triplets, etc.). If there were 24,000 in Atlanta in 1995, estimate the number of multiple births there. Consider a lottery with three possible outcomes: $100 will be received with probability .1, $50 with probability .2, and $10 with probability .What is the expected value of the lottery? Find a fair price, ignoring expenses, for a term insurance policy that will pay Rs.1,000,000 at the end of first year if the policy holder dies during the year. The policy is issued to a female, aged 30, whose probability of death is 0.00135, assume   LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS 17

18. Thank you ! 18 LESSON 7 - PROBABILITY THEORY AND CONTINGENT PAYMENTS