Fixed-Income Securities, Interest Rates and Term Structure of Interest Rates

Transcription

1 I Fixed-Income ecurities, Interest ates and Term tructure of Interest ates Dipl. Kaufm. alentin Popov tatistics and Econometrics Department University of aarland ofia, pril 2012 I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 1

2 I These lecture notes are based on: the lecture notes on Portfolio election by Prof. Friedmann, University of aarland The Econometrics of Financial Markets, Campbell, Lo and MacKinlay, 1997, Princeton University Press (Chapter 10) I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 2

3 I Outline Introduction and otation Basic concepts Zero-coupon bonds Coupon-bonds Estimating the Zero-Coupon Term structure The Expectation Hypothesis and Interest ate Forecasts I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 3

4 I Outline Introduction and otation Basic concepts Zero-coupon bonds Coupon-bonds Estimating the Zero-Coupon Term structure The Expectation Hypothesis and Interest ate Forecasts I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 4

5 I Definition Fixed-income securities are default-free bonds with payments that are fully specified in advance. fixed-income markets equity markets Examples for fixed-income securities include Treasury securities - bills, notes, bonds (U), taatsanleihen - Bundesobligationen/-anleihen (Germany), etc. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 5

6 I otation Price of a security at time t: P t, P t > 0 Future payment at time t + i: C t+i, i I I, C t+i 0 Payments take place at discrete times t I 0 Length of a time interval from t to t + n (n I): n (basis) periods I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 6

7 I Discrete vs. continuous returns: Discrete returns If not explicitly specified, returns are based on a single holding period. The discrete return of a bond with price P t > 0 at time t and future payments C t+i, i I I, C t+i 0, i I C t+i > 0, is defined as the interest rate, for which the price P t equals the present value of the future payments: P t = i I C t+i (1 + ) i, > 1, Interest payments take place at discrete points of time. pecial case: Discrete return for a single payment at time t + n 1 + = ( Ct+n P t ) 1 n and C t+n = P t (1 + ) n. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 7

8 I Discrete vs. continuous returns: Continuous returns How much is the interest rate x k, which equals the single-period return 1 +, when the interest payments take place pro rata within the period after 1 k, 2 k,..., k 1 k, 1 sub-periods? e.g. = 0.21 : (1 + x k k )k = 1 + x k = k((1 + ) 1 k 1), x 2 = 0.2, x 4 = , x 12 = , x 365 = I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 8

9 I Discrete vs. continuous returns: Continuous returns Continuous return: r lim (1 + r k k )k = 1 + e r = 1 + r = ln(1 + ), e.g. for = 0.21 : r = ln(1.21) = It holds r = ln(1 + ), Equality is reached when = 0; when 0 the difference is minor. pecial case: Continuous return for a single payment at time t + n r = ln(1 + ) = 1 n (ln C t+n ln P t ) and C t+n = P t e nr. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 9

10 I Discrete and continuous returns eturns r I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 10

11 I Discrete vs. continuous returns: one period returns pecial case: Discrete and continuous one-period return at time t + 1 (no additional payment - buy at P t, sell at P t+1 = C t+1 ): t+1 = P t+1 1 = P t+1 P t. P t P ( t ) Pt+1 r t+1 = ln(1 + t+1 ) = ln P t = ln P t+1 ln P t. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 11

12 I Multiple-period returns over k periods Discrete return t (k) and continuous return r t (k): with 1 + t (k) = P t P t k r t (k) = ln(1 + t (k)) = ln 1 + t (k) = r t (k) = k 1 (1 + t i ) i=0 k 1 r t i. i=0 ( Pt P t k ) = ln P t ln P t k I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 12

13 I Comparison of returns for different period lengths nnualisation: (Example: daily return t resp. r t, 5 trading days per week, 250 trading days p.a.) ssumption: Daily resp. weekly returns are constant over the period (a week resp. an year) Discrete returns Daily return t annualized with: (1 + t ) Weekly return t (5) annualized with: (1 + t (5)) 52 1 Continuous returns Daily return r t annualized with: 250 r t Weekly return r t (5) annualized with: 52 r t (5) I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 13

14 I Outline Introduction and otation Basic concepts Zero-coupon bonds Coupon-bonds Estimating the Zero-Coupon Term structure The Expectation Hypothesis and Interest ate Forecasts I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 14

15 I Outline Introduction and otation Basic concepts Zero-coupon bonds Coupon-bonds Estimating the Zero-Coupon Term structure The Expectation Hypothesis and Interest ate Forecasts I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 15

16 I Definition Zero-coupon bonds, also called discount bonds, make a single payment at a date in the future known as maturity date. The size of the payment is the face value of the bond. For convenience in the following we assume that the face value is always 1 monetary unit (say EUO). The length of time until maturity is the maturity of the bond. n example for a zero-coupon bond are U Treasury bills. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 16

17 I Yield to maturity Length of time from today (t) until maturity (t + n): maturity n Payment at t (Price): Payment at t + n (face value): P nt P 0,t+n = 1 EUO Yield to maturity: Y nt = ( 1 P nt ) 1 n 1 with P nt = 1 (1+Y nt) n log yield: y nt = ln(1 + Y nt ) with ln P nt = ny nt I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 17

18 I Properties Elasticity of a variable B with respect to a variable is defined to be the derivative of B with repect to, times /B: db d B Equivalently, it is the derivative of lnb with respect to ln. Elasticity of the price with respect to the yield (maturity n) n = d ln P nt d ln(1 + Y nt ), i.e. dp nt P nt = n dy nt 1 + Y nt n dy nt I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 18

19 I Yield spread and term structure Yield spread: nt = Y nt Y 1t resp. s nt = y nt y 1t. Term structure of interest rates at t: Y nt (or y nt ) as a function of n Yield curve (plot of the term structure): normal yield curve : upward sloping, inverse yield curve : downward sloping. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 19

20 I Discount bonds: holding-period returns The holding-period return on a bond is the return over some holding period less than the bond s maturity. For convenience, the holding-period is set to a single period. In that way, the discrete return n,t+1 is given through the price change of an n period bond purchased at time t at price P nt and sold at time t + 1 at price P n 1,t+1 : 1 + n,t+1 = P n 1,t+1 P nt = (1 + Y nt ) n (1 + Y n 1,t+1 ) n 1 resp. for the continuous holding-period return: r n,t+1 = ln P n 1,t+1 ln P nt = y nt (n 1)(y n 1,t+1 y nt ). I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 20

21 I Properties The excess return r n,t+1 y 1t depends on the spread s nt and the change in the yield over the holding period: It holds: r n,t+1 y 1t = s nt (n 1)(y n 1,t+1 y nt ) y nt = 1 n 1 r n i,t+i+1 (average holding-period return), n i=0 due to the compensation of purchasing and selling activities, n 1 r n i,t+i+1 = ln P 0,t+n ln P nt = ln 1 + ny nt = ny nt. i=0 I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 21

22 I Forward rate from t + n until t + n + 1 with discount bonds The forward rate is an interest rate on a fixed-income investment to be made in the future. In particular, the forward rate F nt stands for the interest rate negotiated at time t for an investment in a 1-period discount bond, which (the investment) takes place after n periods, i.e. from t + n until t + n + 1. Consider the following strategy: 1) Buy a discount bond (maturity n + 1) at price P n+1,t 2) Finance the purchase by going short on x discount bonds (maturity n) at price P n,t : xp n,t = P n+1,t x = P n+1,t P n,t 3) Implied forward rate F nt with (1 + F nt )x = 1. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 22

23 I Forward rate rbitrage is, loosely speaking, the possibility of a risk-free profit at zero cost. On an arbitrage-free market, it hods for the forward rate F nt : 1 + F nt = 1 = (1 + Y n+1,t) n+1 P n+1,t /P nt (1 + Y nt ) n, resp. for the continuous forward rate f nt : f nt = ln P nt ln P n+1,t = y nt + (n + 1)(y n+1,t y nt ) as well as y nt = 1 n = y n+1,t + n(y n+1,t y nt ) n 1 i=0 f it (average forward rate). I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 23

24 I emarks 1) f nt > 0 (and also F nt > 0), if the discount price P nt falls with increasing maturity n. 2) f nt > y nt and f nt > y n+1,t (resp. F nt > Y nt and F nt > Y n+1,t ), when the term structure is normal. 3) F nt (known and fixed at t) is different from Y 1,t+n (unknown at t). I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 24

25 I Outline Introduction and otation Basic concepts Zero-coupon bonds Coupon-bonds Estimating the Zero-Coupon Term structure The Expectation Hypothesis and Interest ate Forecasts I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 25

26 I Definition Coupon bonds make (fixed) coupon payments of a given fraction of face value at equally spaced dates up to and including the maturity date, when the face value is also paid. Coupon bonds can be though of as packages of discount bonds. Examples include U Treasury notes and bonds. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 26

27 I Formula Given face value 1 and a (constant) coupon C, the payments at t + i, i = 1,..., n are: K i = C, i = 1,..., n 1, and K n = 1 + C. The price at time t is denoted by P Cnt. P Cnt = n K i P it = i=1 n i=1 K i (1 + Y it ) i I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 27

28 I Per-period yield to maturity The per-period yield to maturity (or internal rate of return) on a coupon bond, Y Cnt is defined as: P Cnt = n i=1 K i (1 + Y Cnt ) i = C n i=1 1 (1 + Y Cnt ) i + 1 (1 + Y Cnt ) n In general the equation above cannot be inverted to get an analytical solution for Y Cnt. Exceptions in two special cases (in each case with solution Y Cnt = C/P Cnt ): 1) P Cnt = 1, the bond is sold at face value (at par) Y Cnt = C. 2) n = Y C t = C/P C t. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 28

29 I Zero-coupon and coupon term structure For coupon bonds payments take place not only at maturity but also at shorter periods. s a result Y Cnt = C (at par) differs generally from Y nt. Generally it holds: min Y it Y Cnt max Y it. i=1,...,n i=1,...,n In particular, for P Cnt > 1 and large C, Y Cnt diverges from Y nt in direction short-term interest rates, whereas for P Cnt < 1 and small C the per-period yield lies nearer to Y nt. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 29

30 I Zero-coupon and coupon term structure The definition of the coupon term structure requires P Cnt = 1. The coupon term structure assigns (as a function of the maturity) every n the coupon C of the the at par traded bond: C(n, t) := Y Cnt P=1 = C with n 1 1 = C(n, t) (1 + Y it ) i + 1 (1 + Y nt ) n, resp. i=1 C(n, t) = 1 1 (1+Y nt) n n, n = 1, 2,... 1 i=1 (1+Y it ) i I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 30

31 I Duration (effective maturity) of a coupon-bond Example: Consider two coupon-bonds with: identical maturity n = 5, identical return per period yield to maturity 10% and coupons C1 = 0.10 and C2 = The respective prices are then P 0.10,5,t = 1, P 0.01,5,t = For which of the two bonds is the percentage price change due to a change in per period yield larger? I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 31

32 Definition Basic idea: a measure for the length of time that a holder of a coupon bond has invested his money. Macaulay s Duration D Cnt of a coupon bond : D Cnt = In particular: n w i i n, mit w i = i=1 P Cnt = 1 D Cnt = n ( i= C(n, t) K i (1+Y Cnt ) i P Cnt, n w i = 1. i=1 ) i C(n, t). D Cnt resp. D Cnt = D Cnt /(1 + Y Cnt ) (modified duration) serves the function of a measure of the negative of the elasticity of a coupon bonds s price with respect to its gross yield: dp Cnt P Cnt = D Cnt d(1 + Y Cnt ) (1 + Y Cnt ) = D Cnt dy Cnt. I E I T I Dipl. Kaufm. alentin Popov Fixed-Income ecurities 32

33 I Extensions Modified duration measures the proportional sensitivity of a bond s price to a small absolute change in its yield. Example: Let D Cnt = 10. Then the increase in the yield of 1 basis point causes a 10 basis points drop in the bond price. The concept of (modified) duration implies a linear relationship between the change in yield and the change in price. However, the relationship between the log price and the yield is convex improvement through quadratic approximation. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 33

34 I Quadratic approximation econd-order Taylor series approximation: P(Y + dy) P(Y) + P (Y) dy P (Y)(dY) 2, i.e. dp Cnt dp Cnt 1 dy Cnt + 1 d 2 P Cnt 1 P Cnt dy Cnt P }{{ Cnt 2 dy } Cnt 2 P Cnt }{{} D Cnt K Cnt with the so-called convexity K Cnt of the coupon bond: (dy Cnt ) 2, K Cnt := d 2 P Cnt dy 2 Cnt 1 P Cnt = n i=1 i(i+1) (1+Y Cnt ) i+2 K i P Cnt > 0. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 34

35 I Example P 0.04,5,t = 1; Price change following an increase of the per period yield to maturity from 4 % to 5 %, i.e. dy Cnt = (1) With Y Cnt = 0.05 and C = 0.04, the exact price of the coupon bond is with dp Cnt /P Cnt = (2) Lin. pprox.: D Cnt = , D Cnt = and dp Cnt /P Cnt D Cnt dy Cnt = (3) Quadr. pprox.: K Cnt = , so that dp Cnt /P Cnt D Cnt dy Cnt K Cnt (dy Cnt ) 2 = I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 35

36 I Outline Introduction and otation Basic concepts Zero-coupon bonds Coupon-bonds Estimating the Zero-Coupon Term structure The Expectation Hypothesis and Interest ate Forecasts I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 36

37 I Problem statement Basic problem: extract an implied zero-coupon term structure from the coupon term structure. Justification: From empirical point of view, there is more data on coupon bonds. In the following the time subscripts t are omitted to economize on notation. The theoretical relationship between coupon bonds with prices P Cj,j, coupons C j and maturity j = 1, 2,..., n and the discount bond prices P 1,..., P n resp. the zero-coupon term structure is given as follows: P Cj,j = C j P 1 + C j P (1 + C j )P j, j = 1, 2,..., n. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 37

38 I Linear regression approach Problem: coupon term structure may be more-than-complete. Idea: consider a stochastic, bond-specific error term u j in the linear relationship above. In particular, let J be the number of all the bonds outstanding at a particular date with prices P Cj,n j, coupons C j and maturities n j, j = 1, 2,..., J. Then the relationship between the coupon bond and zerocoupon bond prices can be modeled by the following cross-sectional linear regression: P Cj n j = C j P 1 + C j P (1 + C j )P nj + u j, mit j = 1,..., J. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 38

39 I Details Classical assumption: u j iid with E (u j ) = 0 and ar (u j ) = σ 2 j σ 2. Modification: Heteroskedasticity with σ j = σs j, where s j can be the bid-ask spread or duration. Let = max j n j, then β = (P 1, P 2,..., P ) is the vector with the coefficients. The rang condition for the (J )-regression matrix X implies J. Problem : Too many, unrestricted parameters P 1,..., P. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 39

40 I pproach 1 Discount bond prices lie on a polynomial, e.g. P n = P(n) = 1 + θ 1 n + θ 2 n 2 + θ 3 n 3, from which follows P C j n j = X nj 1θ 1 + X nj 2θ 2 + X nj 3θ 3 + u j, j = 1,..., J, with P C j n j = P Cj n j 1 n j C j n j X nj k = n k j + C j i k, k = 1, 2, 3 With ˆθ 1, ˆθ 2, ˆθ 3 one can calculate ˆP n and the estimated zero-coupon bond term structure. i=1 I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 40

41 I pproach 2 Discount bond prices lie on a spline function. n rth-order spline is a piecewise rth-order polynomial with r 1 continuous derivatives; its rth derivative is a step function. The points where the rth derivative changes discontinuously are known as knot points. For K 1 subintervals there are K knot points: K 2 junctions and 2 endpoints. The spline has K 2 + r free parameters, r for the first subinterval and 1 (that determines the unrestricted rth derivative) for each of the following K 2 subintervals. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 41

42 I pproach 3 Functions with a better fit for larger n. elson/iegel-approach for a continuous yield y(n), where P(n) = e ny(n) : y ( 1 e α 1 ) ( n 1 e α 1 ) n (n) = β 0 + β 1 + β 2 e α 1n. α 1 n α 1 n Parameter interpretation: β 0 = lim n β 0 + β 1 = lim y(n) n 0 (long-term yield), (short-term yield). I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 42

43 I vensson s extention: y(n) = y (n) + β 3 ( 1 e α 2 n α 2 n ) e α 2n. The new term converges for n and for n 0 to zero. Estimation of ˆβ 0, ˆβ 1, ˆβ 2, ˆβ 3, ˆα 1, ˆα 2 : J (P Cj n j ˆP Cj n j ) 2 min j=1 s 2 j with ˆP Cj n j = C jˆp(1) + C jˆp(2) + + (1 + C j )ˆP(n j ), ˆP(n) = e n ŷ(n). pplication: Deutsche Bundesbank I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 43

44 I Term structure according to vensson s approach: y(n) y(n) n β 0 = 0.04, β 1 = 0.01, β 2 = 0.1, β 3 = 0.1, α 1 = 0.5, α 2 = 0.2 I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 44

45 I Outline Introduction and otation Basic concepts Zero-coupon bonds Coupon-bonds Estimating the Zero-Coupon Term structure The Expectation Hypothesis and Interest ate Forecasts I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 45

46 I The Expectation Hypothesis The term structure depends on the expectations of the investors about the future yields in such a way that (Hypothesis I): ny nt! = E t (y 1,t + y 1,t y 1,t+n 1 ). E t ( ) denotes the mean conditional on the information I t available at time t. (ssumption: the market expectations coincide with E t ( ) (rational expectations).) I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 46

47 I Interpretation and problems Interpretation: The fixed yield on a long-term bond equals the expected yield of consecutive short-term bonds, i.e. the expected change in the short-term interest rates determines the term structure. ote that the hypothesis is defined for continuous returns. It holds: (1 + Y nt ) n = e nynt = e Et( n 1 i=0 y 1,t+i) ( ( < E t e ) n 1 n 1 i=0 y 1,t+i = E t (1 + Y 1,t+i ) This result is due to Jensen s Inequality, according to which: E(g(X)) g(e(x)), i=0 ). where X is a random variable and g is a convex function. I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 47

48 I n alternative specification In terms of continuous returns, the expectation hypothesis above is equivalent to the following one: The fixed yield on a short-term bond equals the expected one-period holding-period return on a long-term bond, i.e. the expected change in the long-term bonds determines the term structure. Formally (Hypothesis II): y 1t = E t (r n,t+1 ) = E t (ln P n 1,t+1 ln P n,t ) = E t (ny nt (n 1)y n 1,t+1 ). Turning to discrete returns and applying Jensen s Inequality, one gets: 1 + Y 1t = e y 1t = e Et(r n,t+1) < Et (e r n,t+1 ) = E t (1 + n,t+1 ). I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 48

49 I Equivalence of both hypotheses in the continuous case For the proof of the equivalence of both hypotheses one uses the rule E t (E t+1 ( )) = E t ( ), which is a special case of the law of total expectation E(Y) = E X (E(Y X)). From Hypothesis I to Hypothesis II: From ny nt = E t (y 1,t + y 1,t y 1,t+n 1 ) and (n 1)y n 1,t+1 = E t+1 (y 1,t y 1,t+n 1 ) resp. E t ((n 1)y n 1,t+1 ) = E t (y 1,t y 1,t+n 1 ), ubtracting the third row from the first one yields: y 1t = E t (ny nt (n 1)y n 1,t+1 ) = E t (r n,t+1 ). I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 49

50 I n alternative proof: From Hypothesis II to Hypothesis I: n iterative application of y 1t = E t (ny nt (n 1)y n 1,t+1 ) resp. ny nt = y 1t + E t ((n 1)y n 1,t+1 ) : delivers the following result: ny nt = y 1t + E t ((n 1)y n 1,t+1 ) = y 1t + E t (y 1,t+1 + E t+1 ((n 2)y n 2,t+2 )) =... = y 1t + E t (y 1,t y 1,t+n 1 ). I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 50

51 I The yield spread s nt as a forecasting tool The relevance of the yield spread for the forecast of the (short-term) change in the long-term interest rates (large n) follows from Hypothesis II. In particular: or y 1t y nt = (n 1)E t (y nt y n 1,t+1 ), resp. E t (y n 1,t+1 y nt ) = y nt y 1t n 1 = s nt n 1, with E t (ε nt ) = 0. y n 1,t+1 y nt = s nt n 1 + ε nt, I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 51

52 I elationship with the forward rate From the original expression ny nt = E t (y 1,t + y 1,t y 1,t+n 1 ), it follows (with the acceptance of this expectation hypothesis) that f nt = (n + 1)y n+1,t ny nt = E t (y 1,t+n ). In that way the forward rate f nt is a (longterm) forecast for the shortterm interest rate y 1,t+n (and also a forecast for f n 1,t+1, since f nt = E t (E t+1 (y 1,t+n )) = E t (f n 1,t+1 )). I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 52

53 Models and tests with time constant term premia Forecast for the future long rates: E t (y n 1,t+1 y nt ) = α n + s nt resp. n 1 s nt y n 1,t+1 y nt = α n + β n n 1 + ε nt, t = 1,..., T. Testing the hypothesis β n = 1; empirical evidence for β n < 1, quite often β n < 0, see Campell, Lo, MacKinley (1997), The Econometrics of Financial Markets, Princeton Unversity Press, ) Forecast for the future short rates: E t (y 1,t+n y 1t ) = δ n + (f nt y 1t ) resp. y 1,t+n y 1t = δ n + β n (f nt y 1t ) + ε nt, t = 1,..., T. Testing the hypothesis β n = 1. (ee Eugene F. Fama (1976): Forward rates as predictors of future spot rates, Journal of Financial Economics, ol. 3, ) Dipl. Kaufm. alentin Popov Fixed-Income ecurities 53 I E I T I

54 I ote When the expectation hypothesis are defined for the discrete instead of the continuous returns, Hypothesis I and Hypothesis II are contradictory. On the one side: (1 + Y nt ) n! = E t ((1 + Y 1t )(1 + Y 1,t+1 ) (1 + Y 1,t+n 1 )) = (1 + Y 1t ) E t (E t+1 ((1 + Y 1,t+1 ) (1 + Y 1,t+n 1 ))) = (1 + Y 1t ) E t ( (1 + Yn 1,t+1 ) n 1), I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 54

55 I ote On the other side: (1 + Y 1t ) ( )! Pn 1,t+1 = E t (1 + n,t+1 ) = E t P nt ( ) = (1 + Y nt ) n 1 E t (1 + Y n 1,t+1 ) n 1, However : ( ) 1 E t (1 + Y n 1,t+1 ) n 1 > (Jensen s Inequality) 1 E t ((1 + Y n 1,t+1 ) n 1 ). I E I T Dipl. Kaufm. alentin Popov Fixed-Income ecurities 55