Week 4 Lecture - Interest Rates and Risk Aversion (PDF)

Week 4 Interest rates and risk aversion BANK2011 Banking and the Financial System The University of Sydney Page 1  Learning Objectives 1. Apply present value to a stream of payments using bond valuation. 2. Interpret risk as a measure of uncertainty about pay...

Week 4 Interest rates and risk aversion BANK2011 Banking and the Financial System The University of Sydney Page 1  Learning Objectives 1. Apply present value to a stream of payments using bond valuation. 2. Interpret risk as a measure of uncertainty about payoffs. 3. Explain how to quantify risk. 4. Define risk aversion and explain the role risk premium plays in the risk- return trade-off. 5. Demonstrate how to reduce risk through hedging and diversification. The University of Sydney Page 2  Bond Basics – A bond is a promise to make a series of payments on specific future dates. – Bonds create obligations, and are therefore thought of as legal contracts that: – Require the borrower to make payments to the lender, and – Specify what happens if the borrower fails to do so. The University of Sydney Page 3  Bond Basics – The most common type of bond is a coupon bond. – Issuer is required to make annual payments, called coupon payments. – The annual interest the borrower pays (ic), is the coupon rate. – The date on which the payments stop and the loan is repaid (n), is the maturity date or term to maturity. – The final payment is the principal, face value, or par value of the bond. The University of Sydney Page 4  Coupon Bond Called a coupon bond as buyer would receive a certificate with a number of dated coupons attached. Coupons The University of Sydney Page 5  Valuing the Principal – Assume a bond has a principal payment of $100 and its maturity date is n years in the future. – The present value of the bond principal is: F $100 PBP = = (1+ i)n (1+ i)n – The higher the n, the lower the value of the payment. The University of Sydney Page 6  Valuing the Coupon Payments – These resemble loan payments. – The longer the payments go, the higher their total value. – The higher the interest rate, the lower the present value. – The present value expression gives us a general formula for the string of yearly coupon payments made over n years. C C C C PCP = + + +...... + (1+ i)1 (1+ i)2 (1+ i)3 (1+ i)n The University of Sydney Page 7  Valuing the Coupon Payments Plus Principal – We can just combine the previous two equations to get:  C C C C  F PCB = PCP + PBP =  + + +...... + +  (1+ i) (1+ i) (1+ i) 1 2 3 (1+ i)n  (1+ i)n – The value of the coupon bond, PCB, rises when – The yearly coupon payments, C, rise and – The interest rate, i, falls. The University of Sydney Page 8  Bond Pricing – The relationship between bond prices and interest rates is very important. – Bonds promise fixed payments on future dates, so the higher the interest rate, the lower their present value. – The value of a bond varies inversely with the interest rate used to calculate the present value of the promised payment. The University of Sydney Page 9  Defining Risk – According to the dictionary, risk is “the possibility of loss or injury.” – For outcomes of financial and economic decisions, we need a different definition. Risk is a measure of uncertainty about the future payoff to an investment, assessed over some time horizon and relative to a benchmark. The University of Sydney Page 10  Defining Risk 1. Risk is a measure that can be quantified. – The riskier the investment, the less desirable and the lower the price. 2. Risk arises from uncertainty about the future. – We do not know which of many possible outcomes will follow in the future. 3. Risk has to do with the future payoff of an investment. – We must imagine all the possible payoffs and the likelihood of each. 4. Definition of risk refers to an investment or group of investments. – Investment described very broadly. 5. Risk must be assessed over some time horizon. – In general, risk over shorter periods is lower. 6. Risk must be measured relative to some benchmark - not in isolation. – A good benchmark is the performance of a group of experienced investment advisors or money managers. The University of Sydney Page 11  Possibilities, Probabilities, and Expected Value – Probability theory states that considering uncertainty requires: – Listing all the possible outcomes. – Figuring out the chance of each one occurring. – Probability is a measure of the likelihood that an event will occur. – It is always between zero and one. – Can also be stated as frequencies. The University of Sydney Page 12  Possibilities, Probabilities, and Expected Value – We can construct a table of all outcomes and probabilities for an event, like tossing a fair coin. The University of Sydney Page 13  Possibilities, Probabilities, and Expected Value – If constructed correctly, the values in the probability column will sum to one. – Assume instead we have an investment that can rise or fall in value. – $1,000 stock which can rise to $1,400 or fall to $700. – The amount you could get back is the investment’s payoff. – We can construct a similar table and determine the investment’s expected value - the average or most likely outcome. The University of Sydney Page 14  Possibilities, Probabilities, and Expected Value – Expected value is the mean - the sum of their probabilities multiplied by their payoffs. The University of Sydney Page 15  Possibilities, Probabilities, and Expected Value What if $1,000 Investment could 1. Rise in value to $2,000, with probability of 0.1 2. Rise in value to $1,400, with probability of 0.4 3. Fall in value to $700, with probability of 0.4 4. Fall in value to $100, with probability of 0.1 The University of Sydney Page 16  Possibilities, Probabilities, and Expected Value Expected Value = 0.1 × ($100) + 0.4 × ($700) + 0.4× ($1,400) +0.1× ($2,000) = $1,050 The University of Sydney Page 17  Possibilities, Probabilities, and Expected Value – Using percentages allows comparison of returns regardless of the size of initial investment. – The expected return in both cases is $50 on a $1,000 investment, or 5 percent. – Are the two investments the same? – No - the second investment has a wider range of payoffs. – Variability equals risk. The University of Sydney Page 18  Measures of Risk – It seems intuitive that the wider the range of outcomes, the greater the risk. – A risk-free asset is an investment whose future value is known with certainty and whose return is the risk-free rate of return. – The payoff you receive is guaranteed and cannot vary. – Measuring the spread allows us to measure the risk. The University of Sydney Page 19  Variance and Standard Deviation The variance is the average of the squared deviations of the possible outcomes from their expected value, weighted by their probabilities. 1. Compute expected value. 2. Subtract expected value from each of the possible payoffs and square the result. 3. Multiply each result by the probability. 4. Add up the results. The University of Sydney Page 20  Variance and Standard Deviation 1. Compute the expected value: ($1400 × ½) + ($700 × ½) = $1,050. 2. Subtract this from each of the possible payoffs and square the results: $1,400 – $1,050 = ($350)2 = 122,500(dollars)2 and $700 – $1,050 = (–$350)2 =122,500(dollars)2 3. Multiply each result by its probability and add up the results: ½ [122,500(dollars)2] + ½ [122,500(dollars)2] =122,500(dollars)2 4. The standard deviation is the square root of the variance: = Variance = 122, 500 dollars 2 = $350 The University of Sydney Page 21  Variance and Standard Deviation – Standard deviation is the (positive) square root of the variance standard deviation = 𝑉𝑎𝑟𝑖𝑎𝑛𝑐𝑒 – The standard deviation is more useful because it deals in normal units, not squared units (like dollars-squared). – We can convert the standard deviation into a percentage of the initial investment. – We can compare other investments to this one. – Given a choice between two investments with equal expected payoffs, most will choose the one with the lower standard deviation. – The greater the standard deviation, the higher the risk. The University of Sydney Page 22  Variance and Standard Deviation – We can see Case 2 is more spread out - higher standard deviation - therefore it carries more risk. The University of Sydney Page 23  Value at Risk – Sometimes we are less concerned with spread than with the worst possible outcome – Example: We don’t want a bank to fail – Value at Risk (VaR): The worst possible loss over a specific horizon at a given probability. – For example, we can use this to assess whether a fixed or variable-rate mortgage is better. The University of Sydney Page 24  Value at Risk – For a mortgage, the worst-case scenario means you cannot afford your mortgage and will lose you home. – Expected value and standard deviation do not really tell you the risk you face, in this case. – VaR answers the question: how much will I lose if the worst possible scenario occurs? – Sometimes this is the most important question. The University of Sydney Page 25  The Impact of Leverage on Risk – Leverage is the practice of borrowing to finance part of an investment. – Although leverage does increase the expected return, it increases the standard deviation. – Leverage magnifies the effect of price changes. – If you borrow to purchase an asset, you increase both the expected return and the standard deviation by a leverage ratio of: Leverage Ratio = Cost of Investment/ Owner’s contribution to the purchase The University of Sydney Page 26  The Impact of Leverage on Risk Source: Cecchetti and Schoenholtz (2021: 111) – Leverage compounds the worst possible outcome. The University of Sydney Page 27  Systemic Risk – Systemic risks are threats to the system as a whole, not to a specific household, firm or market. – Common exposure to a risk can threaten many intermediaries at the same time. – A financial system may contain critical parts without which it cannot function. – Obstacles to the flow of liquidity pose a catastrophic threat to the financial system. The University of Sydney Page 28  Risk Aversion, the Risk Premium, and the Risk-Return Trade-off – Most people do not like risk and will pay to avoid it because most of us are risk averse. – Insurance is a good example of this. – A risk averse investor will always prefer an investment with a certain return to one with the same expected return but any amount of uncertainty. – Therefore, the riskier an investment, the higher the risk premium. – The compensation investors require to hold the risky asset. The University of Sydney Page 29  Figure 5.2: The Tradeoff between Risk and Expected Return The University of Sydney Page 30  Reducing Risk through Diversification – Some people take on so much risk that a single big loss can wipe them out. – Traders call this “blowing up.” – Risk can be reduced through diversification, the principle of holding more than one risk at a time. – This reduces the idiosyncratic risk an investor bears. – One can hedge risks or spread them among many investments. The University of Sydney Page 31  Hedging Risk – Hedging is the strategy of reducing idiosyncratic risk by making two investments with opposing risks. – When one does poorly, the other does well. – Let’s compare three strategies for investing $100: – Invest $100 in 3M. – Invest $100 in Texaco. – Invest half in each company. The University of Sydney Page 32  Hedging Risk – Investing $50 in each stock to ensure your payoff. – Hedging has eliminated your risk entirely. The University of Sydney Page 33  Spreading Risk – You can’t always hedge as investments don’t always move in a predictable fashion. – The alternative is to spread risk around. – Find investments whose payoffs are unrelated. – We need to look at the possibilities, probabilities and associated payoffs of different investments. The University of Sydney Page 34  Spreading Risk – Let’s again compare three strategies for investing $100: – Invest $100 in 3M. – Invest $100 in Microsoft. – Invest half in each company. The University of Sydney Page 35  Figure 5.4: Spreading Risk Payoffs from Two Investment Strategies – We can see the distribution of outcomes from the possible investment strategies. – This figure clearly shows spreading risk lowers the spread of outcomes and lowers the risk. The University of Sydney Page 36  Spreading Risk – The more independent sources of risk you hold in your portfolio, the lower your overall risk. – As we add more and more independent sources of risk, the standard deviation becomes negligible. – Diversification through the spreading of risk is the basis for the insurance business. The University of Sydney Page 37  Homework problems – CS chapter 4 – Problems 3, 12-14 – CS chapter 5 – Problems 4, 5, 7, 8, 14, 19 The University of Sydney Page 38