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# Precautionary Savings and the Stock-Bond Covariance

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Toomas Laarits
NYU Stern
Eesti Pank, Avatud Seminar 6/2/2020.

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### Precautionary Savings and the Stock-Bond Covariance

1. 1. Precautionary Savings and the Stock-Bond Covariance Toomas Laarits NYU Stern Eesti Pank, Avatud Seminar 6/2/2020.
2. 2. Stock-Bond Covariance −8 −6 −4 −2 0 2 1990/1 2000/1 2010/1 2020/1 Cov(Tr 10y, St.) Cov(TIPS 10y, St.)=⇒ Correlations =⇒ Long History 1 / 28
3. 3. International Evidence −2 −1 0 1 2 3 1990m1 1995m1 2000m1 2005m1 2010m1 2015m1 USA Cov JPN Cov GBR Cov ESP Cov DEU Cov 2 / 28
4. 4. Stock-Bond Covariance What accounts for the high-frequency time variation in the risk exposure of a long safe bond? Inﬂation dynamics? For instance, Campbell, Sunderam and Viceira (2017). But note calculation with TIPS. Here: a shock to the price of risk aﬀects stock and bond returns in opposite directions. Therefore, times when price of risk is volatile see a more negative stock-bond covariance. 3 / 28
5. 5. Mechanism: Precautionary Savings and Risk Premium Consider asset pricing models where log SDF can be represented by mt,t+1 = −at − γtσ t+1. The log risk-free rate is rf t,t+1 = at − 1 2 γ2 t σ2 . Risk premium on a payout that depends on σd t+1 is RPt,t+1 = γtσ σd. Change in γt moves risky and safe asset prices in opposite directions. 4 / 28
6. 6. Model Intuition Two state variables: 1) aggregate dividend growth rate 2) price of risk process. Price of risk follows a CIR process, meaning its volatility is proportional to the square root of the level. The share of return volatility stemming from the two distinct sources varies over time. Both the aggregate stock market and long Treasury have constant risk loadings in the model. Yet their covariance changes with the price of risk process. 5 / 28
7. 7. Empirics Overview In the recent data, the stock-bond covariance: 1. Has a level eﬀect on the term structure of safe rates. 2. Accounts for the contemporaneous level of credit spreads. 3. Predicts with a negative sign excess stock and bond returns going forward. 4. Is low when risk-neutral probability of FTQ in Treasuries is high. 5. Predicts issuance of investment grade corporate bonds; accounts for sectoral holdings of Treasuries. 6. Co-moves with covariance calculated in other developed economies. 6 / 28
8. 8. Calibration Overview Data generated according to the model solution: 1. Matches the empirical moments of stock-bond covariance and bond beta—assuming an SDF that can match the aggregate equity risk premium. 2. Exhibits a two factor term structure with level and slope factor reﬂecting empirical counterparts. Risk compensation determines level of interest rates while dividend growth rate determines the slope of interest rates. Just like in the data, negative stock-bond covariance times see lower interest rates. 7 / 28
9. 9. Roadmap Introduction 1. Empirical Facts Rates, Spreads Return Predictability Issuance, Holdings Treasury Risk-Neutral Distribution 2. Term Structure Model Setup and Solution 3. Model Simulation Stock-Bond Covariance Term Structure of Safe Rates Literature
10. 10. Treasury Yields D.yield 1y D.yield 3y D.yield 5y D.yield 10y D.yield 30y D.Cov(Tr 10y, St.) 0.0533∗∗∗ 0.0557∗∗ 0.0543∗∗ 0.0546∗∗ 0.0793∗∗∗ (3.80) (2.85) (2.58) (2.59) (4.70) D.VIX -0.00683∗∗ -0.0128∗∗∗ -0.0117∗∗∗ -0.00729∗ -0.0114∗∗∗ (-2.86) (-3.97) (-3.38) (-2.07) (-4.03) Constant -0.0110 -0.0127 -0.0133 -0.0137 -0.0129 (-0.54) (-0.80) (-0.95) (-1.12) (-1.24) Observations 264 264 264 264 264 R2 0.069 0.087 0.070 0.044 0.137 Monthly data 1997-2018. =⇒ Forward Curve 8 / 28
11. 11. Corporate Spreads D.Aaa-Tr D.Baa-Aaa D.HY-Tr D.HY-Aaa D.GZ Spread D.Cov(Tr 10y, St.) -0.0788∗∗∗ -0.0723∗∗∗ -0.483∗∗∗ -0.404∗∗∗ -0.150∗∗∗ (-4.88) (-4.12) (-11.22) (-10.49) (-7.08) D.VIX 0.0134∗∗∗ -0.00135 0.0360∗∗∗ 0.0226∗∗∗ 0.0367∗∗∗ (5.12) (-0.45) (4.68) (3.30) (8.78) Constant 0.000857 0.00236 0.00334 0.00248 0.00596 (0.08) (0.20) (0.07) (0.07) (0.25) Observations 264 264 264 264 165 R2 0.174 0.062 0.334 0.295 0.440 Monthly data 1997-2018. =⇒ Graph 9 / 28
12. 12. Corporate Bond Returns Aaa-Tr. 12m R Baa-Tr. 12m R HY-Tr. 12m R Cov(Tr 10y, St.) -1.462∗∗∗ -1.029∗ -6.388∗∗∗ (-3.87) (-2.54) (-3.54) Constant -0.861 -0.667 -3.721 (-1.63) (-1.16) (-1.45) Observations 263 263 263 R2 0.180 0.091 0.168 Monthly data 1997-2018. =⇒ Treasury Returns =⇒ Equity Returns 10 / 28
13. 13. Issuance of Safe Corporates D.IG share D.A-rated share D.IG mat D.IG Fin/GDP D.Cov(Tr 10y, St.) -0.0286∗ -0.0348∗ -0.0272 -0.216∗ (-2.13) (-2.29) (-0.07) (-1.97) Constant 0.00106 0.00153 -0.0741 -0.00576 (0.33) (0.43) (-0.88) (-0.29) Observations 264 264 264 264 R2 0.021 0.024 0.000 0.019 Monthly data 1997-2018. 11 / 28
14. 14. Who Buys Treasuries? Share of total Treasuries held by sector. Sectors that increase holdings during negative stock-bond covariance episodes. D.Broker-Dealer D.Households D.MMF D.Cov(Tr 10y, St.) -0.369∗∗∗ -0.0988 -0.213∗∗∗ (-3.64) (-0.67) (-3.56) Constant 0.00754 -0.133 0.0177 (0.08) (-1.02) (0.33) Observations 76 76 76 R2 0.152 0.006 0.146 Quarterly data 1997-2018. =⇒ Who sells? 12 / 28
15. 15. Treasury Implied Volatility and Jump Probabilities Cov(Treasury 10y, Stocks) Implied Volatility 10y -0.169∗ -0.476∗∗∗ -0.346∗∗∗ -0.281∗∗ (-2.22) (-4.48) (-3.45) (-3.02) Jump Probability Calls (LT) -1.767∗∗ -2.091∗∗∗ -1.754∗∗∗ (-3.23) (-3.90) (-3.56) Jump Probability Puts (RT) 0.578 0.998∗ 1.118∗∗ (1.32) (2.43) (2.98) VIX -0.0479∗∗∗ (-4.09) Constant 0.984∗ 2.483∗∗∗ 1.698∗∗ 2.124∗∗∗ (2.04) (4.47) (3.21) (4.32) Observations 217 217 217 217 R2 0.446 0.402 0.471 0.543 Monthly data 1997-2014. =⇒ Graph 13 / 28
16. 16. Roadmap Introduction 1. Empirical Facts Rates, Spreads Return Predictability Issuance, Holdings Treasury Risk-Neutral Distribution 2. Term Structure Model Setup and Solution 3. Model Simulation Stock-Bond Covariance Term Structure of Safe Rates Literature
17. 17. Model Setup Setup follows Lettau and Wachter (2007): work with exogenously speciﬁed price of risk process. Log changes in the aggregate endowment process follow ∆dt+1 = g + zt + √ xtσd t+1 − 1 2 σ2 dxt. Log SDF is given by mt,t+1 = a − ρ(g + zt) − γ √ xtσc t+1. ρ is akin to 1/EIS. γ √ xt is akin to risk aversion. 14 / 28
18. 18. Price of Risk Process xt evolves according to a Cox, Ingersoll Jr. and Ross (1985) process xt+1 = (1 − φx)¯x + φxxt + √ xtσηηt+1, Dividend growth rate zt evolves according to an AR(1) process zt+1 = φzzt + σξξt+1. The ηt+1 and ξt+1 shocks are not directly priced by the SDF—only t+1 is. 15 / 28
19. 19. Solving Recursively for Risky Asset Prices In order to solve for stock prices let’s deﬁne the price dividend ratio of a maturity n dividend strip and guess the functional form Fn,t = Pn,t Dt = exp A(n) + B(n)xt + C(n)zt . Stock prices have to satisfy Fn,t = Et exp{mt,t+1} Dt+1 Dt Fn−1,t+1 . The pricing equation implies exp A(n) + B(n)xt + C(n)zt = = E exp {a − ρ(g + zt) − γ √ xtσc t+1} exp g + zt + √ xtσd t+1 − 1 2 σ2 dxt exp A(n−1) + B(n−1)xt+1 + C(n−1)zt+1 . 16 / 28
20. 20. Closed-form Expressions for Risky Asset Prices Can recursively solve the pricing equation to derive closed form expressions: =⇒ A(n) = a − ρg + g + A(n−1) + B(n−1)(1 − φx)¯x + 1 2 C2 (n−1)σ2 ξ =⇒ B(n) = B(n−1)φx − γσcσd + 1 2 B2 (n−1)σ2 η + 1 2 γ2 σ2 c =⇒ C(n) = C(n−1)φz − ρ + 1. Boundary condition implies A(0) = B(0) = C(0) = 0. The market return is calculated as Rt,t+1 = P m t+1 Dt+1 + 1 P m t Dt Dt+1 Dt . =⇒ Safe Asset Prices =⇒ Solution Graph 17 / 28
21. 21. Roadmap Introduction 1. Empirical Facts Rates, Spreads Return Predictability Issuance, Holdings Treasury Risk-Neutral Distribution 2. Term Structure Model Setup and Solution 3. Model Simulation Stock-Bond Covariance Term Structure of Safe Rates Literature
22. 22. Calibration Description Variable Value Time Discount a -0.041 Risk Aversion γ 3.000 One over IES ρ 1.200 Consumption Growth Rate g 0.020 Consumption Volatility σc 0.013 Dividend Volatility σd 0.026 Average Price of Risk Squared ¯x 50.000 Price of Risk Squared Volatility ση 2.500 Price of Risk Squared Persistence φx 0.890 Dividend Growth Rate Volatility σξ 0.002 Dividend Growth Rate Shock Persistence φz 0.800 Parameters roughly follow Lettau and Wachter (2007). Preference parameters are unitless. All other parameters reported in annualized terms. The model is simulated daily. 18 / 28
23. 23. Main Model Moments Variable Mean S.D. p10 p90 Skew Instantaneous risk premium 5.017 3.928 1.134 10.191 1.618 Annualized Sharpe ratio 0.251 0.097 0.130 0.382 0.451 Instantaneous risk-free rate 2.871 4.594 -3.193 8.669 -0.177 Precautionary savings component -3.520 2.583 -7.020 -0.842 -1.359 10-year yield 2.421 2.264 -0.608 5.110 -0.544 1,000,000 days of simulated data. 19 / 28
24. 24. Main Model Moments Variable Mean S.D. p10 p90 Skew Monthly market return, model 0.431 7.248 -8.515 9.640 0.282 Monthly market return, data 0.575 4.558 -5.660 6.110 -0.677 Monthly 10y bond return, model 0.062 3.550 -4.381 4.594 0.135 Monthly 10y bond return, data 0.138 2.696 -2.964 3.440 0.000 Stock 10y bond covariance, model -0.063 0.432 -0.584 0.363 -1.134 Stock 10y real bond covariance, data -0.332 0.589 -1.064 0.148 -2.488 Stock 10y nominal bond covariance, data -0.514 1.086 -1.833 0.482 -2.422 10y bond stock beta, model 0.026 0.194 -0.156 0.214 3.677 10y real bond stock beta, data -0.082 0.146 -0.262 0.079 0.312 10y nominal bond stock beta, data -0.088 0.263 -0.385 0.286 0.756 1,000,000 days of simulated data. 20 / 28
25. 25. Stock-Bond Covariance in the Model 0 10 20 30 40 50 Percent −.5 0 .5 1 1.5 2 10y Bond Stock Market Beta, Model 0 10 20 30 Percent −3 −2 −1 0 1 2 Stock−10y Bond Covariance, Model 0 10 20 30 Percent −.5 0 .5 1 10y Bond Stock Market Beta, Data 0 5 10 15 20 25 Percent −4 −2 0 2 4 Stock−10y Bond Covariance, Data 21 / 28
26. 26. Stock-Bond Covariance in the Model Cov(Stocks, 10y Bond) Model Price Dividend Ratio 0.0117∗∗∗ 0.0159∗∗∗ (44.20) (10.57) Price of Risk -0.0574∗∗∗ 0.00741 (-51.87) (1.39) Dividend Growth Rate 1.144 30.56∗∗∗ (1.48) (10.63) Constant -2.541∗∗∗ 0.306∗∗∗ -0.0632∗∗∗ -3.473∗∗∗ (-44.38) (51.24) (-31.91) (-9.85) Observations 47620 47620 47620 47620 R2 0.094 0.110 0.000 0.115 =⇒ Bond Beta 22 / 28
27. 27. Term Structure in the Model yield 1y yield 5y yield 10y yield 20y Level Slope Cov(Tr 10y, St.) 2.106∗∗∗ 1.710∗∗∗ 1.372∗∗∗ 0.962∗∗∗ 3.418∗∗∗ -0.282∗∗∗ (29.31) (32.90) (36.27) (39.94) (37.72) (-27.81) Constant 2.982∗∗∗ 2.775∗∗∗ 2.508∗∗∗ 2.121∗∗∗ 0.216∗∗ -0.018∗ (49.51) (63.76) (79.32) (105.45) (2.86) (-2.10) Observations 47620 47620 47620 47620 47620 47620 R2 0.046 0.057 0.068 0.081 0.073 0.042 Level and Slope refer to the ﬁrst two principal components of the simulated term structure. =⇒ Price of Risk 23 / 28
29. 29. Yield Curve Factors in the Model Level Factor Slope Factor Cov(Stocks, 10y Tr.) 3.418∗∗∗ -0.282∗∗∗ (37.72) (-27.81) Price of Risk -1.663∗∗∗ 0.143∗∗∗ (-81.26) (52.09) Dividend Growth Rate 1.350∗∗∗ 0.188∗∗∗ (54.01) (87.53) Constant 0.216∗∗ 10.70∗∗∗ 0.124 -0.0178∗ -0.923∗∗∗ 0.0172∗∗ (2.86) (75.59) (1.93) (-2.10) (-48.45) (3.13) Observations 47620 47620 47620 47620 47620 47620 R2 0.073 0.581 0.380 0.042 0.362 0.617 Level and Slope refer to the ﬁrst two principal components of the simulated term structure. 25 / 28
30. 30. Literature Bekaert and Grenadier (1999) study joint pricing of stocks and bonds in aﬃne economies. Don’t address time variation in stock-bond covariance, however. Kozak (2015) proposes an explanation using a purely real, production-based model. Two-tree model intuition. Campbell et al. (2017) attribute the stock-bond covariance dynamics to a change in the covariance between nominal rates and the real economy. Campbell et al. (2018) explore the implications of a changing covariance between inﬂation and output using a habits based model of investor preferences. Pﬂueger et al. (2018) show that the relative valuations of volatile stocks have a strong relationship with the real interest rate. Main measure—diﬀerence between average book-to-market ratios of high and low volatility stocks—is strongly correlated (.48) with the stock-bond covariance studied here. 26 / 28
31. 31. Other Measures in the Literature 1.3 1.4 1.5 1.6 1.7 BondQ −6 −4 −2 0 2 1995q1 2000q1 2005q1 2010q1 2015q1 Cov(Tr 10y, St.) PVS Bond Q Stock-Bond Covariance. PVS measure from Pﬂueger et al. (2018). Bond Market’s Q from Philippon (2009) Quarterly data 1990-2015. 27 / 28
32. 32. Precautionary Savings Inherent non-linearity in the precautionary savings term makes it cumbersome to work with. Here a particular functional form assumption allows to solve for both PS and RP in closed form. Underlines tight connection between risk dynamics and the safe term structure. Thus escapes the separation between macro quantities and risk premia studied in Tallarini (2000). Safe rates are currently low. How much on account of µc, how much on account of γt? Thank You! 28 / 28
33. 33. Citations I Bekaert, Geert and Steven R Grenadier, “Stock and bond pricing in an aﬃne economy,” 1999. Working Paper. Campbell, John Y., Adi Sunderam, and Luis M. Viceira, “Inﬂation bets or deﬂation hedges? the changing risks of nominal bonds,” Critical Finance Review, 2017, 6 (2), 263–301. , Carolin E. Pﬂueger, and Luis M. Viceira, “Monetary policy drivers of bond and equity risks,” 2018. Working Paper. Cox, John C., Jonathan E. Ingersoll Jr., and Stephen A. Ross, “An intertemporal general equilibrium model of asset prices,” Econometrica, 1985, pp. 363–384. Kozak, Serhiy, “Dynamics of Bond and Stock Returns,” 2015. Working Paper. Lettau, Martin and Jessica A. Wachter, “Why is long-horizon equity less risky? A duration-based explanation of the value premium,” The Journal of Finance, 2007, 62 (1), 55–92. 1 / 15
34. 34. Citations II Pﬂueger, Carolin, Emil Siriwardane, and Adi Sunderam, “A measure of risk appetite for the macroeconomy,” 2018. Working Paper. Philippon, Thomas, “The bond market’s q,” The Quarterly Journal of Economics, 2009, 124 (3), 1011–1056. Tallarini, Thomas D., “Risk-sensitive real business cycles,” Journal of Monetary Economics, 2000, 45 (3), 507–532. 2 / 15
35. 35. Historical Stock-Bond Covariance −1 −.5 0 .5 1 Cov(Stocks,LongTreasury) 1880 1900 1920 1940 1960 1980 2000 2020 =⇒ Back 3 / 15
36. 36. Stock-Bond Correlation −1 −.5 0 .5 1 1990/1 2000/1 2010/1 2020/1 Tr 10y Stock Correlation TIPS 10y Stock Correlation =⇒ Back 4 / 15
37. 37. Treasury Forward Curve Factors D.Level D.Slope D.Curvature D.Cov(Tr 10y, St.) 0.109∗∗∗ -0.0202 0.0200 (3.68) (-1.65) (1.87) D.VIX -0.0118∗ -0.00286 -0.00597∗∗∗ (-2.41) (-1.45) (-3.32) Constant -0.0222 0.00201 0.000477 (-1.32) (0.22) (0.09) Observations 264 264 264 R2 0.076 0.017 0.058 Monthly data 1997-2018. =⇒ Back 5 / 15
38. 38. Corporate Spreads −8 −6 −4 −2 0 2 Cov(Tr10y,St.) 0.00 5.00 10.00 15.00 20.00 HY−Tr 1995/1 2000/1 2005/1 2010/1 2015/1 2020/1 HY−Tr Cov(Tr 10y, St.) =⇒ Back 6 / 15
39. 39. Treasury Bond Returns 5y 12m ER 10y 12m ER 20y 12m ER 30y 12m ER Cov(Tr 10y, St.) -0.00203 0.000126 0.0206 0.0500∗ (-0.43) (0.01) (1.43) (2.33) Constant 0.0238∗∗∗ 0.0442∗∗∗ 0.0720∗∗∗ 0.0953∗∗ (3.53) (3.65) (3.58) (3.18) Observations 261 261 261 261 R2 0.003 0.000 0.029 0.073 Monthly data 1997-2018. =⇒ Back 7 / 15
40. 40. Equity Returns ER 12m ER 24m ER 36m ER 48m ER 60m Cov(Tr 10y, St.) -0.00353 -0.0426 -0.0751 -0.104∗ -0.143∗∗ (-0.16) (-1.14) (-1.71) (-2.27) (-3.28) Constant 0.0737∗ 0.116 0.148 0.179∗ 0.212∗ (2.28) (1.83) (1.85) (2.04) (2.30) Observations 253 241 229 217 205 R2 0.000 0.031 0.078 0.138 0.249 Monthly data 1997-2018. =⇒ Back 8 / 15
41. 41. Who Sells Treasuries? Share of total Treasuries held by sector. Sectors that decrease holdings during negative stock-bond covariance episodes. D.Depository Inst. D.Mutual Funds D.Insurance D.Cov(Tr 10y, St.) 0.0543 0.0147 0.00613 (1.94) (0.52) (0.40) Constant -0.0207 0.0376 -0.0292∗ (-0.83) (1.48) (-2.16) Observations 76 76 76 R2 0.048 0.004 0.002 Quarterly data 1997-2018. =⇒ Back 9 / 15
42. 42. Treasury Implied Vol and Jump Probabilities 0 .5 1 1.5 JumpPr.Calls(LT) −8 −6 −4 −2 0 2 Cov(Tr10y,St.) 1990/1 2000/1 2010/1 2020/1 Cov(Tr 10y, St.) Jump Pr. Calls (LT) =⇒ Back 10 / 15
43. 43. Recursive Expressions for Safe Asset Prices Can recursively solve the pricing equation to derive closed form expressions: =⇒ Af (n) = a − ρg + Af (n−1) + Bf (n−1)(1 − φx)¯x + 1 2 Cf (n−1)Cf (n−1)σ2 ξ =⇒ Bf (n) = Bf (n−1)φx + 1 2 Bf (n−1)Bf (n−1)σ2 η + 1 2 γ2 σ2 c =⇒ Cf (n) = Cf (n−1)φz − ρ Boundary condition implies A(0) = B(0) = C(0) = 0. =⇒ Back 11 / 15
44. 44. Model Solution −6.00 −4.00 −2.00 0.00 A 0 500 1000 1500 months −.002 −.0015 −.001 −.0005 0 B 0 500 1000 1500 months −10 −8 −6 −4 −2 0 C 0 500 1000 1500 months −2.00 −1.50 −1.00 −0.50 0.00 Af 0 500 1000 1500 months 0 .002 .004 .006 .008 Bf 0 500 1000 1500 months −80 −60 −40 −20 0 Cf 0 500 1000 1500 months =⇒ Back 12 / 15
45. 45. Bond Beta in the Model 10y Bond Beta in Model Price Dividend Ratio 0.00502∗∗∗ -0.0336∗∗∗ (67.01) (-89.45) Price of Risk -0.0307∗∗∗ -0.167∗∗∗ (-93.85) (-107.44) Dividend Growth Rate -0.254 -61.07∗∗∗ (-0.71) (-81.74) Constant -1.036∗∗∗ 0.223∗∗∗ 0.0258∗∗∗ 8.197∗∗∗ (-65.28) (98.96) (29.09) (91.93) Observations 47620 47620 47620 47620 R2 0.086 0.156 0.000 0.277 =⇒ Back 13 / 15
46. 46. Stock Return Predictability in the Model 1-year Returns 2-year Returns Price Dividend Ratio -0.262∗∗∗ -0.478∗∗∗ (-29.53) (-20.32) Price of Risk 0.0129∗∗∗ 0.0229∗ (3.49) (2.35) Stock-10y Bond Covariance -0.0209 -0.0387 (-1.92) (-1.55) Constant 4.632∗∗∗ -0.0627∗ 0.0186∗ 8.460∗∗∗ -0.107 0.0376 (29.62) (-2.46) (1.98) (20.39) (-1.59) (1.49) Observations 47608 47608 47608 47596 47596 47596 R2 0.165 0.003 0.000 0.151 0.002 0.000 14 / 15
47. 47. Term Structure in the Model yield 1y yield 5y yield 10y yield 20y Level Slope Price of Risk -1.014∗∗∗ -0.828∗∗∗ -0.666∗∗∗ -0.470∗∗∗ -1.663∗∗∗ 0.143∗∗∗ (-51.49) (-62.13) (-74.71) (-93.39) (-81.26) (52.09) Constant 9.372∗∗∗ 7.991∗∗∗ 6.708∗∗∗ 5.081∗∗∗ 10.701∗∗∗ -0.923∗∗∗ (68.81) (86.74) (108.72) (146.10) (75.59) (-48.45) Observations 47620 47620 47620 47620 47620 47620 R2 0.357 0.447 0.539 0.646 0.581 0.362 Level and Slope refer to the ﬁrst two principal components of the simulated term structure. =⇒ Back 15 / 15