Sloan MIT School of Management
J. Wang E52-456 15.407 Fall 2003
Recitation Notes 9/18/2003
Things to cover today: Basic Concepts: 1. No arbitrage 2. Utility - Consumption choice 3. Risk Aversion PV: 1. Timing 2. Compounding 3. Annuity
Definition of an Arbitrage:
An arbitrage opportunity is a trade that gives you either (i) positive income today and nonnegative payoff in the future; or (ii) a zero income today and a non-negative payoff in the future, with a positive probability of getting a strictly positive payoff. Examples of Arbitrage: (i) The 1 year (simple) interest rate is 5%, 2-year (simple) interest rate is 10%, but the 1-year forward rate 1-year from now is 5% (ii)You can buy 1-kg pack of rice for $10 from the supermarket, but someone is offering to buy rice from you for $1.2 per 100g. Examples of things that are NOT arbitrage: (i) A lottery that cost you $1, but have a 10% chance of paying you $15 (ii) GE is offering a 10-year bond with annual coupon of 10%, while government bond of the same life and is only paying a 5% annual coupon.
Consumption Choice:
Recap on utility What is utility function? Measures people's "satifaction" Assumptions about utility function: (i) It is strictly increasing in all goods (ii) It is concave (people are risk averse) (iii) Time discount (the same consumption in the future is less valuable than consumption today) Example: 2-period riskless consumption choice:
max U (c0 , c1 ) = u(c0 ) + u(c1 ) s.t.c0 = e - s c1 = s(1 + rf ) The problem is equivalent to:
(1)
max u(e - s) + u(s(1 + rf )) F OC : -u (e - s) + (1 + rf )u (s(1 + rf )) = 0 u (c0 ) = (1 + rf ) u (c1 ) Therefore, if rf goes up, then we can see that c1 goes up.
(2)
Risk Aversion
Basic idea: People do not like risk: Given a choice of a sure payment of $100 and a 50-50 chance of $0 or $200, people will choose the $100. This is characterized by a concave utility function. Is this always true in the real world, though? Think gambling.
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Present Value
Timing: How to compare (sure) cashflows that occurs at different points of time? Ans: Discount/grow it to the same point of time using the riskless rate Example: How to compare the (i) $105 today and (ii) $50 today and $60 tomorrow if interest rate is 10%? Compounding: Example: Annual vs Semi-annual APR of 6% Deposit $1: Using annual compounding, you get $1.06 at the end of 1 year Using semi-annual compounding, you get $1.032 = 1.0609 at the end of the year Difference? Interest on the 0.03 you earned at time 1 2 To compare rates with different compounding, use the same idea as above. Grow payments the to the same point of time. Continuous compounding: suppose you deposit in rate r with k compoundings per year: r you get paid (1 + k )k at the end of the year, and this converges to er . Annuity: Example: Show that the present value of any growing annuity is: PV = Answer: Case r = g: PV 1+g ) 1+r 1+g PV - PV ( ) 1+r 1+g P V (1 - ) 1+r r-g PV ( ) 1+r PV ( = A[ = = = = 1 1+g (1 + g)T -1 + + ... + ] 1 + r (1 + r)2 (1 + r)T (1 + g)2 (1 + g)T 1+g A[ + + ... + ] (1 + r)2 (1 + r)3 (1 + r)T +1 1 (1 + g)T A[ - ] 1 + r (1 + r)T +1 A 1+g T [1 - ( ) ] 1+r 1+r A 1+g T [1 - ( ) ] 1+r 1+r 3
A [1 r-g
- ( 1+g )T ] : r = g 1+r T : r=g 1+r
Therefore, PV = Case r = g: P V = A[ Since r = g, P V = A[ Therefore, PV = T 1+r 1 1 1 + + ... + ] 1+r 1+r 1+r 1 1+g (1 + g)T -1 + + ... + ] 1 + r (1 + r)2 (1 + r)T A 1+g T (1 - ( ) ) r-g 1+r
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John is 30 years old at the beginning of the new millenium and is thinking about getting an MBA. John is currently making $40000 per year and expects the same for the remainder of his working years (until age 65). If he goes to a business school, he gives up his income for two years and, in addition, pays $20000 per year for tuition. In return, John expects an increase in his salary after his MBA is completed. Suppose that the post-graduation salary increases at a 5% per year and that the discount rate is 8%. What is minimum expected starting salary after graduation that makes going to a business school a positiveNPV investment for John? For simplicity, assume that all cash flows occur at the end of each year. Answer: Assumptions: John starts MBA immediately John gets his salary in the end of the year John pays his tuition in the end of the year Let x be John's minimum salary after the MBA education If John does not go to MBA, P VN oM BA = 40000 40000 + ... + 1 + 0.08 (1 + 0.08)35 1 40000 (1 - ) = 0.08 (1 + 0.08)35 = 466183
If John goes to MBA, P VM BA = -20000 -20000 x x(1 + 0.05) x(1 + 0.05)32 + + + + ... + 1 + 0.08 (1 + 0.08)2 (1 + 0.08)3 (1 + 0.08)4 (1 + 0.08)35
To justify the cost of MBA, P VN oM BA P VM BA Solving for x, we have x 29000.
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Suppose, for the next 20 years you will receive a perpetuity each year. The first perpetuity has a payment of $1 per year, with each subsequent perpetuity increasing the annual payment by 5%. What is the value of all your payments? The discount rate is 10%. Answer: Consider the problem as receiving the PV of each perpetuity. The PV of the first perpetuity is $10, and then grows at 5% for 20 years. Therefore, the total PV =
10 [1 10%-5% 1+5% - ( 1+10% )20 ] = $121.12
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