Learning Module 5: Economics and Investment Markets

Learning Module 5 Economics and Investment Markets EXAMPLE 1 The Inter-Temporal Rate of Substitution (1 of 3) 1. Suppose the investor's willingness to trade present for future consumption can be represented as Here, is a random shock to the economy that affects the cash flows of the marginal investor, 2 is what makes a random variable. The exponential form is consistent with assumptions about investor risk aversion and consumption growth often made in finance; a and b are typically negative given those same assumptions. Parameters of the distribution of z consistent with observed market data can also be established. Suppose that 2, assumed to have a mean of zero, takes on one of two values a negative value indicating a bad state or a positive value indicating a good state. The probabilities of bad and good states are 0.4 and 0.6, respectively. Using market-consistent values in the exponent in the expression for we can calculate the price of a bond promising $1 for sure in one year as the expected value of the investor's willingness to trade present for future consumption: E(1)=0.4ea+b2 for a bad state) +0.6eb (2 for a good state) -0.4×0.954676 +0.6×0.954379 0.954498 In the calculation, 0.954676 and 0.954379 are the asset's prices in the bad and good states, respectively. Note that the values for the random shock are consistent with the level of the yield curve in the United States from January 1999 to January 2014. The derivation of these numbers is beyond the scope of this reading. Also note the higher value of $1 received in the bad state. Following Equation 3, the investor is willing to buy the risk-free bond today for $0.954498 in exchange for $1 in one year. Also notice that the willingness to invest is smaller for the positive shock (z for a good state), because an investor is willing to pay less for the bond in the case of a good state. Thus, the positive shock is associated with a higher level of consumption today by the investor.3 EXAMPLE 2 The Inter-Temporal Rate of Substitution (2 of 3) In Example 1, suppose the current market price of the real default-free bond is $9,540 per $10,000, but the investor's inter-temporal rate of substitution is $0.954498 per $1 promised. The investor would then value the guarantee of $10,000 in one period more than the market, so she would purchase it. As she buys more of the bond, her future income will be higher and its marginal utility lower, leading to a fall in her marginal willingness to invest in the risk-free asset. Only if there are many investors with the willingness to trade at $9,544,98 would the market price increase until all investors have the same marginal willingness to invest. EXAMPLE 3 The Inter-Temporal Rate of Substitution (3 of 3) Following the circumstances in Example 1, the one-period real risk-free interest rate is 1 = 1-0.954498 0.954498 = 0.047671, or 4.7671%. EXAMPLE 4 The Case of Increasing Wealth 1. This idea can be illustrated by raising the economic shock by a fixed amount regardless of whether the economy is good or bad, which has the effect of increasing the individual's resources and making her wealthier. For example, suppose we add 0.1 to 2 and thus to the resources of the investor relative to the shock in Example 1. The expected inter-temporal rate of substitution for the investor is now lower for this safe asset (the default-free bond). The expected value of the investor's willingness to trade present for future consumption would then be E()=0.4×0.954528+0.6×0.954231 = 0.954350. Compare this result with Example 1. The inter-temporal rate of substitution is lower under the good and bad shock to the economy. As a result, the expected inter-temporal rate of substitution E, (1) is lower for the wealthier investor by 0.000148 (=0.9544980.954350). Thus, the wealthier investor will buy the safe bond only at a lower price, and if this lower price is not the equilibrium price, the investor will substitute away from riskless assets to risky assets. Because of decreasing absolute risk aversion with wealth and because their fundamental consumption needs are met, wealthy investors will demand a lower premium than poorer investors for holding risky assets, all else being equal. Pricing a Two-Period Default-Free Bond 1. In this example, we illustrate how the pricing formula in Equation 5 leads to a risk premium on a two-period default-free bond that is not present in the one-period default-free bond. In these calculations, we use five or six digits to the right of the decimal point because the risk premium is small for a two-period bond relative to a one-period default-free bond. Suppose the price at Time 1 of the two-year default-free bond is given by P12-1 In this case, the future price can be shown to be P12-1-0.839181 for $1 at Time 2 with probability p=0.4 and P12-1-0.954840 for $1 at Time 2 with probability p=0.6 The expected price at time t+ 1 of a $1 bond maturing at time t+ 2 is 0.4 x 0.839181 0.60.954840 $0.908576. Without considering the investor's willingness to invest, the current value of the two-period bond is the simple present value using the one-period real risk-free interest rate of 4.7671% (from Example 3) as the discount rate. Thus, under the assumption stated, the bond would be worth (1) 0908576$0.867234. But the actual 1047671 price in the financial markets based on Equation 5 is PE(P) 0.4x0.839181 x 0.9546760.6 x 0.954840 x 0.954379 = 0.867226, where 0.95-4676 and 0.954379 are the asset's prices in the bad and good states, as determined in Example 1. The price based on Equation 5 is smaller than the present discounted value at the risk-free rate, the difference is 0.000008 per 1 principal value (ie., 0.8672340.8672260.000008). Thus, the holder of a two-year bond earns a risk premium. The reason for this result can be seen by calculating E(P1)E()=0.908576(0.4×0.954676 +0.6 x 0.954379) 0.867234. where 0.908576 is the Time 1 price of the bond as determined earlier. consequently, we see that E(PHS)()() To summarize, the price uncertainty of the two-period bond at t 1 gives rise to a risk premium, although the bond is default-risk-free. EXAMPLE 6 An Alternative Method to Evaluate the Price Discount for Risk 1. The covariance between the investor's willingness to invest and the price of the two-year bond next period can also be computed as follows (recall the standard formula for covariance is cov(x, y) = p[x-E(x)][y,-E(y)]: cov (m+12-1) = 0.4(0.954676-0.954498) × (0.839181 -0.908576) + 0.6(0.954379 0.954498) × (0.954840-0.908576) =-0.000008 In the bad state of the economy, the willingness to invest (0.954676) is above its average (0.954498), yet the bond price (0.839181) is below its average (0.908576). The reverse is true in the good state. Thus, the covariance between the inter-temporal rate of substitution and the price of the asset is negative. This result means the investor finds this investment inferior to one with a payoff that is independent of her willingness to invest. In particular, we have P E(PL) 1+41 = -0.000008. With a lower price, the return on the two-year bond is higher. In this example, the price discount is not too large because the risk between a one- and two-year US government bond is not that crucial. However, riskier assets, such as equity, will have a higher discount. In addition, the higher risk premium on equity still follows from the covariance between the cash flow and the investor's willingness to invest over the time horizon of the investment. Thus, a higher risk premium for stocks arises from a larger value for this covariance. EXAMPLE 7 The Effect of Volatility on Prices 1. One can see the effect of volatility by doubling the standard deviation of the random variable from what was assumed in Example 1. In this case, the price of the one-period bond in Example 1 would be E, (m) 0.4×0.954855 +0.6×0.954260 = 0.954498. Notice that the expected value is the same as in Example 1 but that the prices in each state are more dispersed, reflecting the doubling of the standard deviation. For the two-period default-free bond, continuing with the parameter values (a' and b') from Example 5, we would compute E(P12-1)=0.4×0.776625 +0.6 × 1.005451 = 0.913921. Notice that doubling the volatility leads to a somewhat unrealistic price greater than 1 (implying a negative yield) in the good state, even though the expected price is less than 1. Then, PE(P12-11) 0.40.776625×0.954855+0.6×1.005451×0.954260 0.872303. The risk neutral price is E(P12-1) 1+141 0.913921 0.872336. So, from Equa- 1.047671 tion 6, cov, (P12-1)=0.8723030.872336-0.000033. As a result, the holding period return on a two-period bond for one year is higher [i.e., (0.913921 -0.872303)/0.8723034.771%, compared with 4.768% in Example 6], and because of the higher volatility, investors require a higher premium. EXAMPLE 11 The Risk Premium for Inflation Uncertainty 1. Suppose that an analyst estimates that the real risk-free rate is 1.25% and that average inflation over the next year will be 2.5%. If the analyst observes the price of a default-free bond with a face value of £100 and one full year to maturity as being equal to £95.92, what would be the implied premium embedded in the bond's price for inflation uncertainty? Solution The (approximate) implied premium can be calculated as follows: 0.504% 100 (1+0.0125+0.025). EXAMPLE 12 Level, Slope, and Curvature of the Yield Curve 1. The yield curves shown in Exhibit 8 all have three distinct characteristics. These characteristics are referred to as level, which indicates whether rates are high or low, on average: slope, which is an indicator of the steepness of the curve, or how quickly or slowly rates change with maturity, and curva ture, an indicator of how much the curve is different from a straight line. These characteristics were first noted in rates by Steeley in 1990 and Litter- man and Scheinkman in 1991. All three components can change over time. We can calculate how much of the change in any yield curve is attributable to each of these factors over time. We have performed this calculation for the UK and US yield curves using data with maturities ranging from 3 months to 10 years, spanning the period from January 1999 to January 2014. The principal components analysis technique was used to perform the analysis. These results are shown in Exhibit 9. The majority of movement of the US (92.7%) and UK (95.2%) yield curves is in the level-that is, shifts up and down in the yield curve over time. The slope component accounts for a much smaller proportion of the change over time-6.9% for the US curve and 4.5% for the UK curve, Curvature changes account for only 0.3% in each case (note that we have rounded these figures to the nearest decimal point for convenience). Taken together then, changes in these three factors explain the vast majority of changes to these two yield curves over time. These re- sults are typical for government bond yield curves in developed economies Exhibit 9: Percentage of Yield Curve Movement Explained by Three Components United States Level Slope Curvature United Kingdom 92.7% 95.2% 6.9% 45% 0.3% 0.3% Sources: Based on data from Blisomberg and the authors calculations. Exhibit 10 presents another way of viewing the dynamics of the US yield curve for 15 years starting in 1999. The chart shows the way in which the level, slope, and curvature of the US Treasury curve has changed over time. EXAMPLE 13 Interest Rates, the Yield Curve, and the Business Cycle 1. What financial instrument is best suited to the study of the relationship of real interest rates with the business cycle? A. Default-free nominal bonds B. Investment-grade corporate bonds C Default-free inflation-indexed bonds Solution: C is correct. These bonds' prices are sensitive to changes in real interest rates because the payments are adjusted for changes in the price of goods 2. Suppose investors forecast an unanticipated increase in real GDP growth and the volatility of GDP growth for a particular country. The effect of such a forecast would be for the coupon payments of an inflation-indexed bond issued by the government of the country: A. to rise. B. to fall. C to be indeterminate. Solution: A is correct. The coupon payments would be expected to increase, reflecting an increase in the real interest rate. 3. The yield spread between non-inflation-adjusted and inflation-indexed bonds of the same maturity is affected by: A. a risk premium for future inflation uncertainty only. B. investors inflation expectations over the remaining maturity of the bonds. C both a risk premium for future inflation uncertainty and investors" inflation expectations over the remaining maturity of the bonds. Solution: C is correct. The difference between the yield on a zero-coupon default-free nominal bond (such as a US government STRIP) and on a zero-coupon default-free real bond of the same maturity (such as a US government TIPS) is called the break-even inflation rate. The break-even inflation rate should incorporate investors' inflation expectations over the remaining maturity plus a risk premium for uncertainty about future inflation, as in Equation 10. 4. State an economic reason why inverted yield may predict a recession. Solution: The late stages of business cycles are often characterized by relatively high inflation and high short-term interest rates. To the extent that longer-term yields reflect expectations of declining inflation and a slackening in demand for credit, the yield curve would be expected to flatten or invert. EXAMPLE 14 The Japanese Yield Curve and Business Cycle, 1990-2018 1. During the 1980s, real Japanese GDP growth averaged more than 4.60% per year. During the 1990s and 2000s, it averaged around 1.5%. The catalyst for the decline in growth was the collapse of Japan's property bubble and the stock crash in the early 1990s. Exhibit 14 shows Japan's real annual GDP growth over the post-bubble period, which incorporates the global financial crisis of 2008-2009. The exhibit also shows how real short-term policy rates fell steadily from a level of 4%-5% in 1990, finally becoming negative by 1997. Since that time, real rates have generally been negative as Japan's cen- tral bank, the Bank of Japan, has tried to stimulate its economy by cutting its nominal policy rate to (near) zero. Exhibit 14 also shows the term spread over this period-that is, the slope of the yield curve (created by subtracting the policy rate from 10-year Japanese government bonds). At the start of the period, as the Bank of Japan cut its policy rate from just over 8.0% at the end of 1990 to 0.45% by the end of 1995, the curve steepened from 1.63% at the end of 1990 to 2.59% by the end of 1995. The sharp steepening of the curve in response to the cuts in the policy rate arguably indicated the market's view that the policy stimulus would work, that is, the yields on longer-dated Japanese government bonds embedded some expectation of positive growth and inflation in the future. However, since the mid-1990s, despite cutting its policy rate to zero and enacting programs of quantitative easing, the term spread has fallen steadily. By the end of 2018, it was virtually 0.00%. The weak economic growth and inflation environment in Japan since the collapse of the property bubble is reflected in the fall and flattening of the Japanese yield curve. EXAMPLE 15 The Credit Risk Premium 1. Suppose that an analyst estimates that the real risk-free rate is 1.25%, average inflation over the next year will be 2.5%, and the premium required by investors for inflation uncertainty is 0.50%. If the analyst observes the price of a corporate bond with a face value of £100, with one full year to maturity, as being equal to £94.21, what would be the implied credit premium embedded in the bond's price for inflation uncertainty? Solution The (approximate) implied premium can be calculated as follows: 1.90% 100 94.21 (1+1.25%+2.50% +0.50%). EXAMPLE 16 The Credit Premium for the Royal Bank of Scotland The global financial crisis had an impact on the prices of all financial assets. Exhibits 15 and 17 show the impact of the crisis on credit spreads derived from indexes, whereas Exhibit 19 demonstrates how the same crisis caused the credit spreads on some sovereign issuers to rise dramatically too. In Exhibit 20, we focus on the impact of the crisis on the CDS premium (a close proxy for the credit spread) on five-year Royal Bank of Scotland senior unsecured debt. The exhibit shows the same increase in the perception of credit risk. The eventual decline in the premium was a consequence of the UK government's nationalization of this systemically important global bank. EXAMPLE 18 Valuation and the Business Cycle 1. Describe how real estate valuation is distinguished from valuation of public equities. Solution Real estate does not trade in public markets (the exception being REITs). Compared with the valuation of public equities, the valuation of real estate should reflect a discount for relative lack of liquidity.