Overview:Time and Uncertainty • Intertemporal Prices and Present Value • Uncertainty • Irreversible Investments and Option Value Economics of Time: Some Issues • Cash now versus cash payments in the future? • Future payments are uncertain? • When should we undertake a new project – now, later or never? • How do we manage resources over time? – When do we end a profitable project? – How do we use up a non-renewable resource? Intertemporal Prices • Interest rate r, Today is t = 0: $1 invested today becomes $(1 + r) at t = 1 $1 invested today becomes $(1 + r)2 at t = 2, etc. • Today’s price of ($1 at t = 1) is 1/(1+r) (I.e. $ 1/(1+r) invested today becomes $1 at t = 1) • Today’s price of ($1 at t = 2) is 1/(1+r)2 , etc. Present Value • Present Value of a stream of cash flows is the value in today’s prices PV = C0 + C1/(1+r) + C2/(1+r)2 + ….+ CT/(1+r)T where Ct is the (positive or negative) cash flow at time t • PV Criterion: Invest in projects with PV > 0 • r is “discount rate,” PV often computed for many values Example Consider two projects, A and B Present Value t=0 1 2 3 r = 1% 10% Project A -200 50 50 120 15 -23 Project B 100 50 50 -220 -15 21 -30 45 Difference B - A (Timing of payments matters, with discount rate very important) Choice under Uncertainty • Another aspect of future cash flows is uncertainty. This is modeled via random variables with a distribution. • How do you react to uncertainty? – Cover yourself; avoid big losses at all costs – Make decisions using average (mean) values, ignoring the randomness. – Take big risks, relishing in the thrill of the unknown (“the wonder of it all”) Risk Aversion • Suppose you are offered a job with a financial firm, and there are two alternative compensation packages. A. $ 100,000 Salary $ 100,000 Bonus You expect to receive the bonus with probability .5. B. • $ W Salary only, where W > 100,000. What is the smallest value of W that would cause you to take B over A? Risk Aversion (continued) • If your answer is W = 150,000 = E(package A), You are risk neutral W < 150,000, You are risk averse. W > 150,000, You are risk loving` • Risk Premium: what you would pay to avoid facing the risk, e.g. W = 130,000 gives risk premium of 20,000 = E(package A) - W. Production Technology Choice • Choice of a technology commits a firm to a production process – Risks arise from uncertainty in input prices – Risks arise from uncertainty in quantity or output prices • Consider choosing a ‘high FC + low MC’ technology over a ‘low FC + high MC’ technology – This is a bet on high quantity or high output prices, enough to cover the high FC. – If substantial chance of low quantity or low prices, low FC choice is safer. Example: Production Technology • A risk neutral firm must choose between two available technologies – Technology 1: FC = 400 and MC = 9 – Technology 2: FC = 4,000 and MC = 4 (low FC + high MC) (high FC + low MC) • Technology installed at cost FC today (year 0) and production occurs in year 1, with r = .1. • In year 1, quantity is either 200 with probability p and 1000 with probability 1 – p – We consider p = .1, .2 and .5 • Price P = 12 Example: Production Technology (1) • We must compute PV for each technology in each possible situation. • For instance, with Technology 1 – Q = 200: Variable profits: (P – MC)*200 = (12-9)*200 = 600 • Present value with r = .1 - 400 + 600 / (1+ .1) = 145 – Q = 1000: Variable profits:(P – MC)*1000 = (12-9)*1000 = 3000 • Present value with r = .1 - 400 + 3000 / (1+ .1) = 2,327 • Technology 1: Expected Present Value at p = .2 and r = .1 EPV = .2 * 145 + .8 * 2,327 = 1,891 Example: Production Technology (2) • Expected Present Values Probability p of Low Quantity 0.1 0.2 0.5 Discount Rate r 10% Tech 1 Tech 2 2109 2691 * 1891 2109 * 1236 * 364 25% Tech 1 Tech 2 1808 1888 * 1616 * 1396 1040 * -160 * denotes preferred choice Irreversibility and Option Value • Many investment decisions are irreversible – Once committed, costs are (at least partially) sunk • With uncertainty, there is a value to waiting – There is an ‘option’ value to flexibility – postponing decision while uncertainty resolves. • Consider pricing with – Season tickets – Rent-to-buy arrangements Example: Irreversibility • Two possible technologies, B and V; it is uncertain which one will become the standard • If you develop the right technology, then profits are 100. If not, your profits are 40 (since you have to license from someone else). • Your market research suggests that there is a 80% probability that V will be the standard. • How much do you want to pay to keep the B option alive until uncertainty resolves? Example: Irreversibility (2) • If you research only one technology, then you should research V, and your expected profits are: π = 0.8 * 100 + 0.2 * 40 = 88 • If you research both, then expected profits are: π = 0.8 * 100 + 0.2 * 100 = 100 • Value of keeping both options open is 12. This is what you are willing to pay. Issues for Discussion 1. (“When to cut down the tree?” problem.) Suppose I have a process that is increasing in value, when do I halt it? 2. (“When to sell the oil?” problem.) Suppose we have a non-renewable resource, how do we best use it up? When to cut down a tree? • We assume that process initially increases rapidly in value and then slows down. • Essential Logic: At any moment, you can halt the process and invest the value at interest r. Optimal to keep the process going when it’s value is growing at a rate greater than r, and halt it when the growth rate drops below r. When to sell the oil? • Consider two time periods: t = 0 and t = 1, fixed amount of oil to sell, price taker. • Essential Logic: Sell when you get the highest profit for each unit. If you sell in both periods, marginal unit must have same profit PV, namely P0 – MC0 = (P1 – MC1)/(1+r) • Note, if MC flat or rising, P must rise. • P – MC increases at rate of interest. Take Away Points • Money today and money tomorrow are different things. Present value is the correct way to combine such cash flows. • People tend to be risk-averse. This is an important consideration for e.g. incentives. • Flexibility has value (option value) which can be priced.