Expected Bond Returns

Transcription

1 Expected Bond Returns Chris Telmer May / 28

2 EH 2005 Conundrum Yield curves Yield Curve: February to April Yield Curve: February April, 1994 Nominal Yields (% CCAR) April 1994 February Maturity (Q) 2 / 28

3 Greenspan Uses the Expectations Hypothesis 1994 EH 2005 Conundrum Yield curves In early February, we thought long-term rates would move a little higher as we tightened. The sharp jump in [long] rates that occurred appeared to reflect the dramatic rise in market expectations of economic growth and associated concerns about possible inflation pressures....alan Greenspan, May 27, / 28

4 EH 2005 Conundrum Yield curves Yield Curve: May 2004 to June 2005 Yield Curve: May 2004 June Nominal Yields (% CCAR) June 2005 May Maturity (Q) 4 / 28

5 Greenspan s Conundrum 1994 EH 2005 Conundrum Yield curves The Greenspan Conundrum. Or, Alan Greenspan (finally!) Rejects the Expectations Hypothesis. Long-term interest rates have trended lower in recent months even as the Federal Reserve has raised the level of the target federal funds rate by 150 basis points. Historically, even distant forward rates have tended to rise in association with monetary policy tightening.... For the moment, the broadly unanticipated behavior of world bond markets remains a conundrum....alan Greenspan, February 16, / 28

6 Yield curves 1994 EH 2005 Conundrum Yield curves Nominal Yields Across Time and Maturity Yields (% CCAR) Nominal Yields (1970:1 to 2005:4) Maturity (Q) / 28

7 Forward rates Interpretation 7 / 28

8 Forward rates Interpretation USD-denominated price of an n-period zero-coupon bond: b n t 8 / 28

9 Forward rates Interpretation USD-denominated price of an n-period zero-coupon bond: Continuously compounded yield or spot interest rate: yt n = 1 n logbn t b n t 8 / 28

10 Forward rates Interpretation USD-denominated price of an n-period zero-coupon bond: Continuously compounded yield or spot interest rate: yt n = 1 n logbn t Forward interest rates: b n t f n t = log(b n t /b n+1 t ) 8 / 28

11 Forward rates Interpretation USD-denominated price of an n-period zero-coupon bond: Continuously compounded yield or spot interest rate: yt n = 1 n logbn t Forward interest rates: b n t f n t = log(b n t /b n+1 t ) Yields are averages of forward rates. n 1 yt n = n 1 i=0 f i t 8 / 28

12 Forward rates Interpretation USD-denominated price of an n-period zero-coupon bond: Continuously compounded yield or spot interest rate: yt n = 1 n logbn t Forward interest rates: b n t f n t = log(b n t /b n+1 t ) Yields are averages of forward rates. n 1 yt n = n 1 i=0 Short rate defined as i t. By definition, i t = y 1 t = f 0 t. f i t 8 / 28

13 Forward rates Interpretation USD-denominated price of an n-period zero-coupon bond: Continuously compounded yield or spot interest rate: yt n = 1 n logbn t Forward interest rates: b n t f n t = log(b n t /b n+1 t ) Yields are averages of forward rates. n 1 yt n = n 1 i=0 Short rate defined as i t. By definition, i t = y 1 t = f 0 t. Holding-period returns (HPRs). The continuously-compounded HPR on an n-period bond, between t and t+1 is defined as: f i t h n t+1 logb n 1 t+1 logb n t 8 / 28

14 Forward interest rates Picture Forward rates Interpretation Definition: f 1 t logb 1 t logb 2 t = i t +2yt 2 = yt 2 = 1 ( it +f 1 ) t 2 9 / 28

15 Interpretation Forward rates Interpretation Yields on zero-coupon bonds are usually called spot interest rates. They are often computed using the yields on coupon bonds ( bootstrapping ). 10 / 28

16 Interpretation Forward rates Interpretation Yields on zero-coupon bonds are usually called spot interest rates. They are often computed using the yields on coupon bonds ( bootstrapping ). What is the forward interest rate? Above, it looks like the interest rate that applies between one-year-from-now and two-years-from-now. 10 / 28

17 Interpretation Forward rates Interpretation Yields on zero-coupon bonds are usually called spot interest rates. They are often computed using the yields on coupon bonds ( bootstrapping ). What is the forward interest rate? Above, it looks like the interest rate that applies between one-year-from-now and two-years-from-now. But note that it is essentially a definition, once we know i t and y 2 t. 10 / 28

18 Interpretation Forward rates Interpretation Yields on zero-coupon bonds are usually called spot interest rates. They are often computed using the yields on coupon bonds ( bootstrapping ). What is the forward interest rate? Above, it looks like the interest rate that applies between one-year-from-now and two-years-from-now. But note that it is essentially a definition, once we know i t and y 2 t. Or is it...? On the next page we show that it is actually a market interest rate. 10 / 28

19 (continued) Forward rates Interpretation Consider a cash inflow of USD 100, to be received in one year. You know that you want to invest it for the subsequent year. You don t want to face the risk that one-year interest rates fall. 11 / 28

20 (continued) Forward rates Interpretation Consider a cash inflow of USD 100, to be received in one year. You know that you want to invest it for the subsequent year. You don t want to face the risk that one-year interest rates fall. You can arrange to lend these USD 100 at the forward interest rate! 11 / 28

21 (continued) Forward rates Interpretation Consider a cash inflow of USD 100, to be received in one year. You know that you want to invest it for the subsequent year. You don t want to face the risk that one-year interest rates fall. You can arrange to lend these USD 100 at the forward interest rate! Short sell 100 one-year bonds and invest the proceeds in two-year bonds. 11 / 28

22 (continued) Forward rates Interpretation Consider a cash inflow of USD 100, to be received in one year. You know that you want to invest it for the subsequent year. You don t want to face the risk that one-year interest rates fall. You can arrange to lend these USD 100 at the forward interest rate! Short sell 100 one-year bonds and invest the proceeds in two-year bonds. The short sale generates 100 b 1 t dollars, which will buy 100 b1 t /b2 t two-year bonds. 11 / 28

23 (continued) Forward rates Interpretation Consider a cash inflow of USD 100, to be received in one year. You know that you want to invest it for the subsequent year. You don t want to face the risk that one-year interest rates fall. You can arrange to lend these USD 100 at the forward interest rate! Short sell 100 one-year bonds and invest the proceeds in two-year bonds. The short sale generates 100 b 1 t dollars, which will buy 100 b1 t /b2 t two-year bonds. In one year we must pay the short-sale liability of USD 100. This is exactly offset by the cash inflow. 11 / 28

24 (continued) Forward rates Interpretation Consider a cash inflow of USD 100, to be received in one year. You know that you want to invest it for the subsequent year. You don t want to face the risk that one-year interest rates fall. You can arrange to lend these USD 100 at the forward interest rate! Short sell 100 one-year bonds and invest the proceeds in two-year bonds. The short sale generates 100 b 1 t dollars, which will buy 100 b1 t /b2 t two-year bonds. In one year we must pay the short-sale liability of USD 100. This is exactly offset by the cash inflow. In two years we receive USD 100 b 1 t /b2 t 11 / 28

25 (continued) Forward rates Interpretation Consider a cash inflow of USD 100, to be received in one year. You know that you want to invest it for the subsequent year. You don t want to face the risk that one-year interest rates fall. You can arrange to lend these USD 100 at the forward interest rate! Short sell 100 one-year bonds and invest the proceeds in two-year bonds. The short sale generates 100 b 1 t dollars, which will buy 100 b1 t /b2 t two-year bonds. In one year we must pay the short-sale liability of USD 100. This is exactly offset by the cash inflow. In two years we receive USD 100 b 1 t /b2 t The definition of the forward rate implies that: b 1 t b 2 t = 1/(1+i t ) 1/((1+i t )(1+f 1 t )) = (1+f1 t ) 11 / 28

26 (continued) Forward rates Interpretation Consider a cash inflow of USD 100, to be received in one year. You know that you want to invest it for the subsequent year. You don t want to face the risk that one-year interest rates fall. You can arrange to lend these USD 100 at the forward interest rate! Short sell 100 one-year bonds and invest the proceeds in two-year bonds. The short sale generates 100 b 1 t dollars, which will buy 100 b1 t /b2 t two-year bonds. In one year we must pay the short-sale liability of USD 100. This is exactly offset by the cash inflow. In two years we receive USD 100 b 1 t /b2 t The definition of the forward rate implies that: b 1 t b 2 t = 1/(1+i t ) 1/((1+i t )(1+f 1 t )) = (1+f1 t ) Therefore, our portfolio generates USD 100 (1+ft 1 ) in two years. This means that ft 1 is the implicit interest rate at which one can borrow (lend) between one year from now and two years from now. 11 / 28

27 Term Structure EH EH Exercise Testing Summary The Expectations Hypothesis for the Term Structure of Interest Rates 12 / 28

28 The Expectations Hypothesis The expected return on any bond strategy is the same* EH Exercise Testing Summary 13 / 28

29 The Expectations Hypothesis EH Exercise Testing Summary The expected return on any bond strategy is the same* Example: HPR on 2-year zero minus HPR on one-year zero: h 2 t+1 h 1 t+1 = logb 1 t+1 logb 2 t ( 0 logb 1 ) t 13 / 28

30 The Expectations Hypothesis EH Exercise Testing Summary The expected return on any bond strategy is the same* Example: HPR on 2-year zero minus HPR on one-year zero: h 2 t+1 h 1 t+1 = logb 1 t+1 logb 2 t ( 0 logb 1 ) t = logb 1 t+1 + ( logb 1 t logb 2 ) t 13 / 28

31 The Expectations Hypothesis EH Exercise Testing Summary The expected return on any bond strategy is the same* Example: HPR on 2-year zero minus HPR on one-year zero: h 2 t+1 h 1 t+1 = logb 1 t+1 logb 2 t ( 0 logb 1 ) t = logb 1 t+1 + ( logb 1 t logb 2 ) t = logb 1 t+1 +ft 1 13 / 28

32 The Expectations Hypothesis EH Exercise Testing Summary The expected return on any bond strategy is the same* Example: HPR on 2-year zero minus HPR on one-year zero: h 2 t+1 h 1 t+1 = logb 1 t+1 logb 2 t ( 0 logb 1 ) t = logb 1 t+1 + ( logb 1 t logb 2 ) t = logb 1 t+1 +f 1 t EH = E t (h 2 t+1 h1 t+1 ) = 0 so that 13 / 28

33 The Expectations Hypothesis EH Exercise Testing Summary The expected return on any bond strategy is the same* Example: HPR on 2-year zero minus HPR on one-year zero: h 2 t+1 h 1 t+1 = logb 1 t+1 logb 2 t ( 0 logb 1 ) t = logb 1 t+1 + ( logb 1 t logb 2 ) t = logb 1 t+1 +f 1 t EH = E t (h 2 t+1 h1 t+1 ) = 0 so that f 1 t = E t logb 1 t+1 13 / 28

34 The Expectations Hypothesis EH Exercise Testing Summary The expected return on any bond strategy is the same* Example: HPR on 2-year zero minus HPR on one-year zero: h 2 t+1 h 1 t+1 = logb 1 t+1 logb 2 t ( 0 logb 1 ) t = logb 1 t+1 + ( logb 1 t logb 2 ) t = logb 1 t+1 +f 1 t EH = E t (h 2 t+1 h1 t+1 ) = 0 so that f 1 t = E t logb 1 t+1 = E t y 1 t+1 = E t i t+1 13 / 28

35 The Expectations Hypothesis EH Exercise Testing Summary The expected return on any bond strategy is the same* Example: HPR on 2-year zero minus HPR on one-year zero: h 2 t+1 h 1 t+1 = logb 1 t+1 logb 2 t ( 0 logb 1 ) t = logb 1 t+1 + ( logb 1 t logb 2 ) t = logb 1 t+1 +f 1 t EH = E t (h 2 t+1 h1 t+1 ) = 0 so that f 1 t = E t logb 1 t+1 = E t y 1 t+1 = E t i t+1 *By return we will mean continuously-compounded return. Note that, if we also were to consider discretely-compounded returns, the various ways of writing the EH would be inconsistent with each other, and inconsistent with what s above. The reason is basically Jensen s inequality. In FOREX this gets referred to as Siegel s paradox: if the EH holds in USD units, it can t in GBP units. Sticking with continuous compounding avoids much of this. Quantitatively none of this amounts to much. There s an old paper by Cox-Ingersoll-Ross that spells-out all of this. 13 / 28

36 Exercise EH Exercise Testing Summary Consider the two-period HPR on three-period bond: h 3 t,t+2 ( logb 1 t+2 logb 3 t) /2 14 / 28

37 Exercise EH Exercise Testing Summary Consider the two-period HPR on three-period bond: h 3 t,t+2 ( logb 1 t+2 logb 3 t) /2 Subtract it from the two-period HPR on two-period bond: h 2 t,t+2 ( 0 logb 2 t) /2 14 / 28

38 Exercise EH Exercise Testing Summary Consider the two-period HPR on three-period bond: h 3 t,t+2 ( logb 1 t+2 logb 3 t) /2 Subtract it from the two-period HPR on two-period bond: h 2 t,t+2 ( 0 logb 2 t) /2 Use the same arithmetic as above to demonstrate that ft 2 = E t yt / 28

39 Exercise EH Exercise Testing Summary Consider the two-period HPR on three-period bond: h 3 t,t+2 ( logb 1 t+2 logb 3 t) /2 Subtract it from the two-period HPR on two-period bond: h 2 t,t+2 ( 0 logb 2 t) /2 Use the same arithmetic as above to demonstrate that ft 2 = E t yt+2 1 Therefore, in general, f n t = E t y 1 t+n 14 / 28

40 Exercise EH Exercise Testing Summary Consider the two-period HPR on three-period bond: h 3 t,t+2 ( logb 1 t+2 logb 3 t) /2 Subtract it from the two-period HPR on two-period bond: h 2 t,t+2 ( 0 logb 2 t) /2 Use the same arithmetic as above to demonstrate that ft 2 = E t yt+2 1 Therefore, in general, f n t = E t y 1 t+n The EH says that forward interest rates are expected future short rates. 14 / 28

41 the EH EH Exercise Testing Summary Forward rates = E(future spots) f n t = E t y 1 t+n 15 / 28

42 the EH EH Exercise Testing Summary Forward rates = E(future spots) ft n = E t yt+n 1 Expected HPRs equal for all n: E t h n t+1 = E t h m t+1 = yt 1 15 / 28

43 the EH Forward rates = E(future spots) EH Exercise Testing Summary f n t = E t y 1 t+n Expected HPRs equal for all n: E t h n t+1 = E t h m t+1 = y 1 t Long yields are expected averages of future short rates: yt n ( = E t y 1 t+1 +yt yt+n 1 1 ) 15 / 28

44 Testing EH Exercise Testing Summary Unconditional test: average HPRs equal by maturity. E t h n t+1 = E t h m t+1 = y 1 t = Eh n t+1 = Eh m t+1 = y 1 t 16 / 28

45 Testing EH Exercise Testing Summary Unconditional test: average HPRs equal by maturity. E t h n t+1 = E t h m t+1 = y 1 t = Eh n t+1 = Eh m t+1 = y 1 t Source: Cochrane, John, New facts in finance, Economic Perspectives, Chicago Fed, April / 28

46 (continued) Conditional tests: EH Exercise Testing Summary f n t = E t y 1 t+n 17 / 28

47 (continued) EH Exercise Testing Summary Conditional tests: ft n = E t yt+n 1 = ft n yt 1 ( = E t y 1 t+n yt 1 ) 17 / 28

48 (continued) EH Exercise Testing Summary Conditional tests: ft n = E t yt+n 1 = ft n yt 1 ( = E t y 1 t+n yt 1 ) = ft n yt 1 = ( yt+n 1 yt) 1 εt+n 17 / 28

49 (continued) EH Exercise Testing Summary Conditional tests: ft n = E t yt+n 1 = ft n yt 1 ( = E t y 1 t+n yt 1 ) = ft n yt 1 = ( yt+n 1 yt) 1 εt+n where ε t+n is defined as the forecast error, s.t., yt+n 1 yt 1 ( E t y 1 t+n yt) 1 +εt+n 17 / 28

50 (continued) EH Exercise Testing Summary Conditional tests: ft n = E t yt+n 1 = ft n yt 1 ( = E t y 1 t+n yt 1 ) = ft n yt 1 = ( yt+n 1 yt) 1 εt+n where ε t+n is defined as the forecast error, s.t., yt+n 1 yt 1 ( E t y 1 t+n yt) 1 +εt+n This suggests the regression: y 1 t+n y 1 t = a+b ( f n t y 1 t) +residuals 17 / 28

51 (continued) EH Exercise Testing Summary Conditional tests: ft n = E t yt+n 1 = ft n yt 1 ( = E t y 1 t+n yt 1 ) = ft n yt 1 = ( yt+n 1 yt) 1 εt+n where ε t+n is defined as the forecast error, s.t., yt+n 1 yt 1 ( E t y 1 t+n yt) 1 +εt+n This suggests the regression: y 1 t+n y 1 t = a+b ( f n t y 1 t) +residuals The EH null is a = 0, b = 1 17 / 28

52 Results: Panel A EH Exercise Testing Summary Source: Cochrane, John, New facts in finance, Economic Perspectives, Chicago Fed, April / 28

53 Regression EH Exercise Testing Summary If the EH is violated then, necessarily, there are excess expected returns available. The only issue is how much risk? 19 / 28

54 Regression EH Exercise Testing Summary If the EH is violated then, necessarily, there are excess expected returns available. The only issue is how much risk? A different (complementary) regression is useful. Recall, the EH says that, for all n, So, how about, for n > 1, h n t+1 y 1 t E t h n t+1 = y 1 t = a+b ( f n 1 t y 1 t) +residuals 19 / 28

55 Regression EH Exercise Testing Summary If the EH is violated then, necessarily, there are excess expected returns available. The only issue is how much risk? A different (complementary) regression is useful. Recall, the EH says that, for all n, So, how about, for n > 1, h n t+1 y 1 t E t h n t+1 = y 1 t = a+b ( f n 1 t y 1 t) +residuals The EH null is a = 0, b = / 28

56 Regression EH Exercise Testing Summary If the EH is violated then, necessarily, there are excess expected returns available. The only issue is how much risk? A different (complementary) regression is useful. Recall, the EH says that, for all n, So, how about, for n > 1, h n t+1 y 1 t E t h n t+1 = y 1 t = a+b ( f n 1 t y 1 t) +residuals The EH null is a = 0, b = 0. If b 0 then the n-period forward premium predicts the excess HPR on the n-period bond. 19 / 28

57 Results: Panel B EH Exercise Testing Summary Source: Cochrane, John, New facts in finance, Economic Perspectives, Chicago Fed, April / 28

58 Fixed Income EH Exercise Testing Summary For n = 2,3,..., run the regression, get estimates of a and b, â and ˆb h n t+1 y 1 t = a+b ( f n 1 t y 1 t) +residuals Carry trade: at each date (month), t: If â+ˆb ( ft n 1 If â+ˆb ( ft n 1 y 1 t y 1 t ) ) > 0 borrow short, invest long < 0 borrow long, invest short Out-of-sample backtesting, etc., important 21 / 28

59 Summary of the Evidence EH Exercise Testing Summary Unconditional: EH holds (pretty much) It doesn t matter if you buy-and-hold n year bonds or m years bonds. 22 / 28

60 Summary of the Evidence EH Exercise Testing Summary Unconditional: EH holds (pretty much) It doesn t matter if you buy-and-hold n year bonds or m years bonds. Conditional: EH violated There are times when the yield curve is steep enough to justify borrowing short and investing long There are other times when the reverse is true If you average across these times, you get the unconditional evidence An analogous result is true for currencies 22 / 28

61 Summary of the Evidence EH Exercise Testing Summary Unconditional: EH holds (pretty much) It doesn t matter if you buy-and-hold n year bonds or m years bonds. Conditional: EH violated There are times when the yield curve is steep enough to justify borrowing short and investing long There are other times when the reverse is true If you average across these times, you get the unconditional evidence An analogous result is true for currencies Bottom line: there are risk premiums inherent in the behavior of the term structure 22 / 28

62 (continued) EH Exercise Testing Summary Here s another clever way to think about the conditional evidence: Suppose that the yield curve is strongly upward-sloping The idea of the EH is that if you invest long, future short rates will increase. This will depress the HPRs on your long bonds and make the (implicit alternative) portfolio of rolled-over shorts look good. Your (apparent) excess yield gets gobbled up. Does this happen? Yes and no / 28

63 (continued) How different are realized short rates from predicted short rates EH Exercise Testing Summary Source: Cochrane, John, New facts in finance, Economic Perspectives, Chicago Fed, April / 28

64 EH Exercise Testing Summary Evidence in favor of excess expected returns on certain bond portfolios is evidence against risk neutrality One uses an arbitrage-free (aka risk neutral) term structure model for asset allocation and/or risk management at one s peril. The evidence indicates that m t is stochastic and that we must model it to do a serious job of fixed-income asset allocation and risk management Approaches: Affine term structure models Factor models 25 / 28

65 Discussion of Fama-Bliss EH Exercise Testing Summary Discussion of Fama-Bliss assignment 26 / 28

66 What is the EH? 27 / 28

67 What is the EH? People sometimes think that the EH is specific to bonds and currencies. This isn t quite right. What is the EH? 28 / 28

68 What is the EH? What is the EH? People sometimes think that the EH is specific to bonds and currencies. This isn t quite right. The EH is just the statement that expected returns are equal across different assets. 28 / 28

69 What is the EH? What is the EH? People sometimes think that the EH is specific to bonds and currencies. This isn t quite right. The EH is just the statement that expected returns are equal across different assets. Bonds and currencies just happen to be fixed-income, so things are nice and clean. 28 / 28

70 What is the EH? What is the EH? People sometimes think that the EH is specific to bonds and currencies. This isn t quite right. The EH is just the statement that expected returns are equal across different assets. Bonds and currencies just happen to be fixed-income, so things are nice and clean. But we could just as easily write: E t q j t+1 +δj t+1 q j t = E t q k t+1 +δk t+1 q k t for two stocks, j and k. This would be the EH for equities. If there are forwards on stocks the same sort of forwards equal expected future spots would pop out. 28 / 28