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Arbitrage and Financial Decision Making Corporate Finance Dennis Ng 1 Outline I. Introduction II. Valuing Costs and Benefits III. Interest Rates and the Time Value of Money IV. Present Value and the NPV Decision Rule V. Arbitrage and the Law of One Price VI. No-Arbitrage and Security Prices VII. The Price of Risk 2 I. Introduction To begin understanding how a company should finance and how its money should be allocated to investment projects, we need to understand the basics of valuation Valuation is simply putting a price on something. We usually use the Dollar, Euro, Pound, etc. to price things, but prices can be expressed in many ways. To make comparisons, we need to get all prices in the same units ... this is really what valuation is all about. 3 II. Valuing Costs and Benefits Identify Costs and Benefits May need help from other areas in identifying the relevant costs and benefits Marketing Economics Organizational Behavior Strategy Operations 4 Using Market Prices to Determine Cash Values Suppose a jewellery manufacturer has the opportunity to trade 600 ounces of silver and receive 10 ounces of gold today. To compare the costs and benefits, we first need to convert them to a common unit. 5 Using Market Prices to Determine Cash Values (cont'd) Suppose gold can be bought and sold for a current market price of $1230 per ounce. Then the 10 ounces of gold we receive has a cash value of: (10 ounces of gold) ($1230/ounce) = $12,300 today 6 Using Market Prices to Determine Cash Values (cont'd) Similarly, if the current market price for silver is $18 per ounce, then the 600 ounces of silver we give up has a cash value of: (600 ounces of silver) ($18/ounce) = $10,800 today 7 Using Market Prices to Determine Cash Values (cont'd) Therefore, the jeweller's opportunity has a benefit of $12,300 today and a cost of $10,800 today. In this case, the net value of the project today is: $12,300 - $10,800 = $1500 8 III. Interest Rates and the Time Value of Money Time Value of Money Consider an investment opportunity with the following certain cash flows. Cost: $10,000 today Benefit: $10,500 in one year The difference in value between money today and money in the future is due to the time value of money. 9 The Interest Rate: An Exchange Rate Across Time The rate at which we can exchange money today for money in the future is determined by the current interest rate. Suppose the current annual interest rate is 7%. By investing or borrowing at this rate, we can exchange $1.07 in one year for each $1 today. Risk-Free Interest Rate (Discount Rate), rf: The interest rate at which money can be borrowed or lent without risk. Interest Rate Factor = 1 + rf Discount Factor = 1 / (1 + rf) 10 IV. Present Value and the NPV Decision Rule The net present value (NPV) of a project or investment is the difference between the present value of its benefits and the present value of its costs. Net Present Value 11 The NPV Decision Rule Accepting or Rejecting a Project Accept those projects with positive NPV because accepting them is equivalent to receiving their NPV in cash today. Reject those projects with negative NPV because accepting them is equivalent to reducing the wealth of investors. Equivalent to burning up the cash today. 12 The NPV Decision Rule (cont'd) When making an investment decision, take the feasible alternative with the highest NPV. Note: one feasible alternative is usually to \"do nothing\". This has an NPV of 0. Thus the best of the feasible alternatives should have an NPV > 0 unless the \"do nothing\" alternative is not possible. 13 Calculate the NPV What was the NPV of our previous investment example? Consider a second project that costs $50,000 to invest in now and it gives a benefit of $60,000 in one year (Assume no risk and a 5% interest rate). 14 NPV and Individual Preferences Although the second project has the highest NPV, what if we do not want to spend the $50,000 for the cash outlay now? Would the first project be a better choice? Would \"do nothing\" be a better choice? Should this affect our choice of projects? NO! As long as we are able to borrow and lend at the risk-free interest rate, the second project is superior whatever our preferences regarding the timing of the cash flows. What are our net cash flows if we borrow $50,000 to invest in the project? 15 NPV and Individual Preferences (cont'd) Regardless of our preferences for cash today versus cash in the future, we should always maximize NPV first. We can then borrow or lend to shift cash flows through time and find our most preferred pattern of cash flows. This is the \"separation\" of individual preferences from the investment decision. Financial markets allow for this separation and for us to consume and invest differently than what we could do if we just spent our income as it was earned. 16 V. Arbitrage and the Law of One Price Another concept, in addition to time value, is important for understanding the relative values of assets. The concept is \"arbitrage\". Arbitrage The practice of buying and selling equivalent goods in different markets to take advantage of a price difference. An arbitrage opportunity occurs when it is possible to make a profit without taking any risk or making any investment. 17 Arbitrage and the Law of One Price (cont'd) Law of One Price If equivalent investment opportunities trade simultaneously in different competitive markets, then they must trade for the same price in both markets. Normal Market A competitive market in which there are no arbitrage opportunities. 18 VI. No-Arbitrage and Security Prices Valuing a Security Assume a security promises a risk-free payment of $1000 in one year. If the risk-free interest rate is 7%, what can we conclude about the price of this bond in a normal market? 19 No-Arbitrage and Security Prices (cont'd) What if the price of the bond is lower? Assume the price is $900. 20 No-Arbitrage and Security Prices (cont'd) What if the price of the bond is higher? Assume the price is $990. 21 Determining the No-Arbitrage Price Unless the price of the security equals the present value of the security's cash flows, an arbitrage opportunity will appear. No Arbitrage Price of a Security: Consider a security that will pay the owner $100 today and $200 in one year with certainty. If the current price of the security is $270 what arbitrage opportunity is available? 22 Determining the Interest Rate From Bond Prices If we know the price of a risk-free bond, we can use to determine what the risk-free interest rate must be if there are no arbitrage opportunities. 23 Determining the Interest Rate From Bond Prices (cont'd) Suppose a risk-free bond that pays $1000 in one year is currently trading with a competitive market price of $929.80 today. The bond's price must equal the present value of the $1000 cash flow it will pay. 24 The NPV of Trading Securities In a normal market, the NPV of buying or selling a security is zero. 25 The NPV of Trading Securities (cont'd) Separation Principle We can evaluate the NPV of an investment decision separately from the decision the firm makes regarding how to finance the investment or any other security transactions the firm is considering. Always holds for a normal market as the financing side has a zero NPV and thus does not add to or detract from the NPV of the investment. 26 Valuing a Portfolio The Law of One Price also has implications for packages of securities. Consider two securities, A and B. Suppose a third security, C, has the same cash flows as A and B combined. In this case, security C is equivalent to a portfolio, or combination, of the securities A and B. This is called Value Additivity 27 Example Problem Moon Holdings is a publicly traded company with only three assets: It owns 50% of Due Beverage Co., 70% of Mountain Industries, and 100% of the Oxford Bears, a football team. The total market value of Moon Holdings is $200 million, the total market value of Due Beverage Co. is $75 million and the total market value of Mountain Industries is $100 million. What is the market value of the Oxford Bears? 28 VII. The Price of Risk Risky Versus Risk-free Cash Flows Assume there is an equal probability of either a weak economy or strong economy. 29 The Price of Risk (cont'd) What is the \"expected cash flow\" from the risk-free bond? What is the \"expected cash flow\" from the market index? Although both investments have the same \"expected\" cash flows, the market index has a lower value since it has a greater amount of risk. What is the risk-free interest rate? (1+rf) = 1100/1057.69 , rf= 4% For the index, what rate would equate the PV of the expected cash flow to today's market price? 30 Risk Aversion and the Risk Premium Risk Aversion Investors prefer to have a safe income rather than a risky one of the same \"expected\" amount. Risk Premium The additional return that investors expect to earn to compensate them for a security's risk. When a cash flow is risky, to compute its present value we must discount the cash flow we expect on average at a rate that equals the risk-free interest rate plus an appropriate risk premium. What was the example's market risk premium? 31 The No-Arbitrage Price of a Risky Security Return to 43 If we combine security A with a risk-free bond that pays $800 in one year, the cash flows of the portfolio in one year are identical to the cash flows of the market index. By the Law of One Price, the total market value of the bond and security A must equal $1000, the value of the market index. 32 The No-Arbitrage Price of a Risky Security (cont'd) Given a risk-free interest rate of 4%, the market price of the bond is: ($800 in one year) / (1.04 $ in one year/$ today) = $769.23 today Therefore, the initial market price of security A is $1000 - $769.23 = $230.77. What is the price of security B that pays $600 in a weak economy and nothing in a strong economy? 33 Risk Premiums Depend on Risk If an investment has much more variable returns and adds to the risk of investors, it must pay investors a higher risk premium. On the other hand, if an investment insures investors (lowers their risk), then it should have a negative risk premium. 34 Risk Is Relative to the Overall Market The risk of a security must be evaluated in relation to the fluctuations of other investments in the economy. A security's risk premium will be higher the more its returns tend to vary with the overall economy and the market index. If the security's returns vary in the opposite direction of the market index, it offers insurance and will have a negative risk premium. Calculate the expected return and risk premium for each security in our example. 35 Risk, Return, and Market Prices When cash flows are risky, we can use the Law of One Price to compute present values by constructing a portfolio that produces cash flows with identical risk. 36 Converting Between Dollars Today and Dollars in One Year with Risk Computing prices in this way is equivalent to converting between cash flows today and the expected cash flows received in the future using a discount rate rs that includes a risk premium appropriate for the investment's risk: 37 Example Recalculate the price of the risk-free bond, the market index, security A, and security B using what you know about their expected cash flow and the appropriate risk premium for each. If you had a potential project that would cost $16,000 now and give back either $16,000 or $28,000 in one year depending on whether the market was weak or strong, then should it be accepted? 38 BUS/ECON-2819-002 - Corporate Finance I Assignment 1 Due: Wednesday, Sep. 30Th, In class Please show your work for all questions. Total /20 1) Go to the Toronto Stock Exchange website at tmx.com. Look up the stock market quote for New Flyer Industries (go to the top right corner, scroll over \"Get Quote\" and enter \"NFI\". Click \"Go\".) Scroll down and click on the Financials tab. This will bring up the most recent balance sheet. You can also access the cash flow and income statements by clicking the appropriate links. Using the information from the 2014 financial statements, as well as the information under the \"Quote\" tab (market cap, shares outstanding), calculate the following ratios (show your work or you will lose marks. Each question is worth 1 mark): a) Market-to-book ratio b) Current ratio c) Gross Margin d) Operating Margin e) Net Profit Margin f) ROA g) ROE h) P/E ratio 2) A bond promises a risk-free payment of $1000 in one year. The risk-free rate of interest is 2.19%. a) What is the price of the bond? (1 mark) b) If the price of the bond is actually $950, what is the arbitrage strategy? Illustrate all cash flows. (3 marks) c) If the price of the bond is actually $1000, what is the arbitrage strategy? Illustrate all cash flows. (3 marks) 3) Assume that there are two possible future states of the economy: weak and strong. Two securities, A and B, are available for trading. Prices (at t=0) and future payoffs (at t=1) in both states are given in the following table: Payoffs ($)at t=1 Price ($) at t=0 weak state Strong state A 50.48 0 200 B 70.00 100 0 Assume that both states are equally likely. Answer the following questions. a. There is another security, call it C, whose payoff at t=1 is equal to $200 in the weak state and $400 in the strong state. Find the no-arbitrage price (at t=0) of security C. (1 mark) b. What is the risk free rate of return in this economy? (2 marks) c. Find expected rate of return and expected risk premium for security C from part a. (2 marks) 1