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Chicago Booth BUSF 35150 - Interest rates II. Factor Models notes

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25 Interest rates II. Factor Models notes• The “Investments notes” contain a matrix review. If you forgot how to multiply matrices youshould go read that now.25.0.4 Motivation and idea1960 1970 1980 1990 2000 201051015Yields of 1−5 year zeros and fed funds1960 1970 1980 1990 2000 2010−2−1012Yield spreads y(n)−y(1)• Look at the graph. Look at the movie.24Most movements in 1-5 year bonds are a) “levelshifts” (top graph) b) c h anges in slope of the term structure (bottom graph). How can wecapture this behavior?• How about()= + level+ slope+(otherstuff).If⎡⎢⎢⎢⎢⎢⎢⎢⎣(1)(2)(3)(4)(5)⎤⎥⎥⎥⎥⎥⎥⎥⎦=⎡⎢⎢⎢⎢⎢⎣11111⎤⎥⎥⎥⎥⎥⎦level+⎡⎢⎢⎢⎢⎢⎣−2−1012⎤⎥⎥⎥⎥⎥⎦slope+(otherstuff)Then a 1% change in “level” moves all yields up 1%. A ch ange in “slope” moves the shortrates down and the long rates up; it changes the slope of the yield curve. Two “factors”describe the movements of five yields.24lecture9.m426• Analogy: This is a lot like the FF 3F model.+1= + +1+ +1+ +1+ +1We express each yield (return) as a sum of coefficients (betas) specific to that security times“factors” (level, slope; rmrf, hml, smb) common to all securities. In turn the factors arespecific combinations (portfolios) of the same underlying securities. rmrf is the sum of allreturns, hml is long value stoc ks short growth stocks etc.; level is an ave ra ge of all yields,slope is long high maturity yields and short low maturity yields etc.• We’re describing va riance, the movement of ex-post yields, not yet expected returns,etc. Nowwe do care about time series regressions, R2significance of coefficien ts, etc. We’ll tie togethermeans and variances in a bit.25.0.5 A simple approach to a factor model• Why don’t we try exactly the FF approach? Let’s define a level factor as the average of allyields (like rmrf) and the slope factor as a high-low portfolio just lik e hmllevel≡15h(1)+ (2)+ (3)+ (4)+ (5)islope=12h(4)+ (5)i−12h(1)+ (2)iClearly, when all yields go up, level will rise. When the yield curve slopes up slope will be abig number, and when it slopes down, it will be a negative n umber.• Now, we can just run regressions to find the “betas.”()= + × level+ × slope+ ()I did it,2(1)0.09 0.9998 -0.77 0.97(2)-0.08 1.0003 -0.28 0.97(3)-0.04 0.9999 0.10 0.97(4)-0.02 1.0008 0.38 0.98(5)0.04 0.9993 0.56 0.98This ought to look pretty familiar and just about what you expected. (The units are /%, sothe intercepts are truly tiny, i.e. 9 bp for (1) These are yields not returns, so the interceptsare not “alphas.” )• The 2are very high here. The factor model is almost a perfect fit. There are really onlytwo portfolios that change, and then everything else is just a linear combination of these.Here is a plot of actual yields andfit ()= × level+ × slopeThe lines are indistinguishable, meaning that the tw o-factor model is an excellent fit.4271955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010051015Actual Yields1955 1960 1965 1970 1975 1980 1985 1990 1995 2000 2005 2010051015Fitted Yields −− Two factor model• The good fit means that we are very close to an “arbitrage pricing theory.” Once where youknow where level and slope are (or any t wo yields) on any date, leveland slope,youknowexactly where all the other yields must be! We will end up with “no-arbitrage term structuremodels” in which there are exactly n factors, and the rest are priced by arbitrage.• Notice the two steps in constructing a factor model. Any factor model.1. How do y ou form the factors (level, slope) from yields or other observable variables?level= ,slope= 2. What are the loadings or betas; if the factor mo ves, how much does each bond move?()= level+ slope25.0.6 The eigenvalue decomposition / principal components analysis• That’s nice, but ad-hoc. Can you do better — get a closer fit—withadifferent constructionof level and slope? What’s the best way to form another factor (curvature) to get a betterfit?• Objective:()= + level+ slope+ curve+(otherstuff, as small as possible)How do you construct factors that are uncorrelatedwitheachother,soastomaximize2/minimize the variance of the error.428• The answ er is: the eigenvalue decomposition/principal components. (OLS regression picksthe    to minimize the variance of the error given the factors. The question here is, how doyou pick the factors so that after you have run the regressions you get the smallest possibleerror.)• Here’s the procedure. First a very short version for people comfortable with matrices1. Form the covariance matrix of yields. Using the notation=⎡⎢⎢⎢⎢⎢⎣(1)(2)(3)...⎤⎥⎥⎥⎥⎥⎦Σ = (0)This is, for the Fama Bliss data, a 5 x 5 matrix.2. Take its eigenvalue decomposition. (The matlab function eig does this, and you don’thave to know anything about how it w orks. If yo u’re curious see the appendix to thenotes.) This decomposition forms three matrices from Σ,Σ = Λ0in which Λ is diagonal, and250 = 0= .3. That’s it. In matlab, there are two steps. With yields =a ×  matrix as usual,form Sigma = cov(100*yields) andthenform[Q,L] = eig(Sigma).That’s all you have to do. What does this mean?4. The columns of  represent the weights by which you form factors from yields,= 05. The columns of  also represent the loadings or betas, which answer “how much does mo ve when a factor moves,”= Proof: is defined as = 0,so=(0)= .6. The factors are uncorrelated with each other, and the diagonals of Λ are their vari-ances,(0)=ΛProof: (0)=(00)=0(0) = 0Σ0 = Λ7. We form approximate (but often a very good approximation) factor models by leavingout factors with small


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