Asset Pricing Prof. Lutz Hendricks Econ520 April 29, 2021 1 / 49 Objectives In this section you will learn: 1. how asset prices are determined as present discounted values of “dividends.” 2. how asset prices are affected by policy. 3. how the yield curve can be used to predict future interest rates. 2 / 49 Bond Yields The Yield Curve The Yield Curve 7 November 2000 6 5 June 2001 Yield (percent) Yield 4 3 3 months 6 months 1 year 2 years 3 years 5 years 10 years 30 years Maturity The yield curve plots interest rates (yields) against bond maturities. Here: treasury bonds 4 / 49 Bond Prices A discount bond pays $100 in T years. The price of a bond is the discounted expected present value of its payments I complication: risk 1 year discount bond: P1t = $100=(1 + i1t) I this actually defines i1t, the 1 year yield from t to t + 1 $100 1 + i1t = (1) p1t 5 / 49 2 year bond 2 year discount bond: I price P2t I payoff tomorrow: P1;t+1 (it’s now a 1 year bond) I without risk: it’s 1 year return must be i1t P1;t+1 $100 P2t = = (2) 1 + i1;t (1 + i1t)(1 + i1;t+1) Of course, i1;t+1 is not known at t – we must use its expected value 6 / 49 Bond Prices Yield on 2 year bond: the constant interest rate that produces the bond price as present value: $100 P2t = 2 (3) (1 + i2t) Therefore: 2 (1 + i2t) = (1 + i1;t)(1 + i1;t+1) (4) Or approximately 1 i = (i + i ) (5) 2;t 2 1;t 1;t+1 7 / 49 The Yield Curve The yield curve plots yields against maturities If the yield curve is upward sloping: i1;t+1 > i1;t I agents expect rising short term interest rates We can interpret the yield on a T year bond as the average short term interest rate agents expect over the next T years 8 / 49 The Yield Curve Long bond yields reveal investor expectations about future interest rates. Qualifications: I risk I liquidity 9 / 49 Example: The 2001 Recession 7 November 2000 6 5 June 2001 Yield (percent) Yield 4 3 3 months 6 months 1 year 2 years 3 years 5 years 10 years 30 years Maturity Nov-2000: the tail end of the tech bubble expansion 10 / 49 2001 Recession Nov-2000 I the Fed tries to slow growth with unemployment below the natural rate I high short term interest rates I expected slowdown of activity ! lower long term interest rates Jun-2001 I the tech bubble burst ! recession I the Fed lowers short term interest rates I investors expect a recovery, but not soon I the yield curve slopes upwards 11 / 49 The 2001 Recession in the Model LM i i A Nov-2000: Y > Yn iЈ IS expected to shift gradually left Nominal interest rate, Yield curve is downward sloping IS ISЈ (forecast) Yn Y Output, Y 12 / 49 The 2001 Recession Adverse shift LM in spending i LMЈ i A Jun-2001: B Monetary expansionA larger than expected AЈ reduction in demand. iЈ The Fed responds by lowering i Nominal interest rate, IS ISЉ (realized) YЈ Y Output, Y 13 / 49 The 2001 Recession LMЈ (forecast) i LM iЈ AЈ Jun-2001: Expections of gradual demand A recovery i plus Fed tightening Nominal interest rate, ISЈ (forecast) IS Y Yn Output, Y 14 / 49 The Current Yield Curve Source: treasury.gov 15 / 49 The Current Yield Curve Key features: 1. Liquidity trap short rates are essentially 0 2. Rates are below 2% up to maturities of 10 years investors expect low interest rates for a long time 3. Long yields are quite low (2.7% nominal) investors expect low inflation (or low real returns) 16 / 49 The Stock Market The S&P 500 Oct 2005: ^GSPC 1248.29 1.60K 1.40K 1.20K 1.00K 0.80K 0.60K 0.40K 0.20K © 2013 Yahoo! Inc. 1970 1975 1980 1985 1990 1995 2000 2005 2010 1D 5D 1M 3M 6M YTD 1Y 2Y 5Y Max FROM: Jan 2 1970 TO: Jul 1 2013 +1,748.67% Source: yahoo.com 1950 1960 1970 1980 1990 2000 2010 18 / 49 A Simple Model of Stock Prices I Abstract from risk premia for now and assume I the return on stocks = the return on bonds I Return on bonds: i1;t I Return on stocks: I t: invest Qt I t + 1: earn dividend Dt+1 and sell stock for Qt+1 I rate of return: (Dt+1 + Qt+1)=Qt 19 / 49 A Simple Model of Stock Prices Equal rates of return Dt+1 + Qt+1 1 + i1;t = (6) Qt Solve Dt+1 + Qt+1 Qt = (7) 1 + i1;t Now apply the same equation for Qt+1 Dt+1 Dt+2 + Qt+2 Qt = + (8) 1 + i1;t (1 + i1;t)(1 + i1;t+1) Now apply the same for Qt+2, then for Qt+3, etc... 20 / 49 A Simple Model of Stock Prices We end up with Dt+1 Dt+2 Dt+3 Dt+n Qt+n Qt = + + + ::: + + (9) Zt;1 Zt;2 Zt;3 Zt;n Zt;n where Zt;n = (1 + i1;t)(1 + i1;t+1):::(1 + i1;t+n) (10) discounts payoffs in t + n to t. How does this change with inflation? 21 / 49 A Simple Model of Stock Prices Result Stock prices equal the present discounted value of future dividends. This is called the fundamental value. Implications: 1. High interest rates (now or in the future) depress stock prices. 2. Low dividends (now or in the future) depress stock prices. 3. Stock returns are generally unpredictable 3.1 really: the difference between stock and bond returns is unpredictable 3.2 this is called the equity premium 22 / 49 Caveats 1. Stocks are risky, so they generally earn higher returns than bonds 1.1 We should discount by r1;t plus a risk premium. 2. Stock prices often deviate from fundamental values for reasons that are not well understood 2.1 The deviations are called bubbles 23 / 49 Shocking the Stock Market LM i LMЈ Monetary expansion: i A 1. Low interest rates AЈ 2. High future output and dividends Nominal interest rate, Both raise stock prices IS Y Output, Y 24 / 49 Rise in Consumption Steep i i LM LM AЉ AЈ Flat Two offsetting effects AЈ A LM (a) A (b)1. higher Y 2. higher i ISЈ Change in stock prices is ISЈ ambiguous Nominal interest rate, IS Nominal interest rate, IS YA YA Output, Y Output, Y The Fed LMЉ tightens 25 / 49 i AЉ LM LMЈ A AЈ The Fed (c) accommodates ISЈ Nominal interest rate, IS YA Output, Y Steep i i LM LM AЉ AЈ Flat AЈ A LM (a) A (b) ISЈ ISЈ Nominal interest rate, Nominal interest rate, IS Rise in ConsumptionIS YA YA Output, Y Output, Y The Fed LMЉ tightens i AЉ LM Possible Fed reactions: 1. Accomodate LMЈ A AЈ The Fed 2. Stabilize (c) accommodates One implication: No stable ISЈ relationship between output Nominal interest rate, IS and stock prices YA Output, Y 26 / 49 Stocks Are Too Volatile How volatile should stocks be? A rough approximation: I go to a period t (before today) I collect data on future interest rates and dividends I compute the present value of dividends I compare with actual stock prices 27 / 49 Stocks Are Too Volatile Comparing Actual Real Stock Price with Three Alternative PDVs of Future Real Dividends 10000 PDV Dividends Using Actual Real Stock Price (P) Actual Future Interest Rates 1000 PDV of Dividends (P* A=0) Price 100 Consumption-Discounted Dividends, (P*, A=4) 10 1860 1880 1900 1920 1940 1960 1980 2000 2020 Year Source: Robert Shiller (http://www.econ.yale.edu/~shiller/data.htm) 28 / 49 Stocks Are Too Volatile Notes on the Shiller graph: 1. Actual real price is the raw stock price, deflated with a price index 2. P*, A=0: present value of dividends, assuming a constant interest rate 3. P*, actual future interest rates: present value of dividends, discounted by realized future long-term interest rates 4. P*, A=4: uses the Consumption Asset Pricing Model with risk aversion of 4 Each line is scaled such that 1. the terminal value equals that last observed stock price 2. the geometric mean of the growth rates is the same Pt+1+Dt+1 The values are actually computed backwards as Pt = . 1+rt+1 29 / 49 Why Might Stocks Be so Volatile? 1. Tail risk: stock prices include the risk of rare (bad) events e.g. major depressions; financial crises 2. Long-term risk: the long-run growth rate of dividends (or productivity) could fluctuate 3. Bubbles: fluctuations are random; not related to fundamentals 30 / 49 Asset Pricing With Risk What Is Risk? Possible answers (with counter-examples): 32 / 49 A Simple Model of Risk A household lives for 2 periods. In period 1, he receives income y and eats c1. In period 2, there are 2 possible states: 1. good state: income yH with probability pH 2. bad state: income yL with probability pL Prefences: u(c1) + pLu(cL) + pHu(cH) (11) | {z } expected utility in pd. 2 33 / 49 A Simple Model of Risk There are 2 assets: 1. asset L pays 1 unit of consumption in state L 2. asset H pays 1 unit of consumption in state H Budget constraints: c1 = y − pLsL − pHsH (12) cL = yL + sL (13) cH = yH + sH (14) 34 / 49 Household Problem maxu(y − pLsL − pHsH) + pju(yj + sj) (15) s ;s ∑ L H j=L;H | {z } expected utility in pd 2 First-order conditions: 0 0 u (c1)pj = pju (yj + sj); j = L;H (16) This solves for the household’s willingness to pay for the assets: 0 u (cj) pj = pj 0 ; j = L;H (17) u (c1) 35 / 49 Asset Prices 0 pL pL u (cL) = 0 > 1 (18) pH pH u (cH) Simplify by assuming that pj = 1=2 Result An asset is valuable, if it pays in a state with low consumption (high marginal utility).