. Answer the following attachments with adequate explanations.

Assume that a five-year, government-issued coupon bond at par value NOK 100,000 and 10% coupon rate was issued today. The bond pays one annual coupon at the end of each year, while the par value is redeemed exactly five years from today including the last coupon payment. In the market, the bond is considered default-free. Government-issued, zero-coupon bonds (Bo,r) with maturities ranging from exactly one to five years from today (T = 1, 2, ...,5) are currently trading at the NOK-prices listed below. Assume that the bonds are default-free. Maturity Market price 1 year 93,023.16 2 year 84,167.99 3 year 75,131.42 4 year 67,073.49 5 year 60,017.97 To the following questions, please provide numerical answers based on the unbiased expectations theory of the term structure of interest rates: (a) (7 points) Calculate the current, risk-free spot-rates for one, two, three, four, and five years from today. (b) (7 points) Calculate the one-year forward rates for year two, year three, year four and year five. (c) (8 points) Calculate the equilibrium market price of the five-year, government-issued coupon bond issued today. (d) (8 points) What must be the equilibrium rate of interest on a one-year investment (or loan) in year five?This result follows from causality because 17-2 involves {wy-2. no-s. . . . ), which are all uncorrelated with w, and way,. The correlation between , and x, 2 is not zero, as it would be for an MA( 1), because x, is dependent on x,-2 through , . Suppose we break this chain of dependence by removing (or partial out) the effect t,. That is, we consider the correlation between x, - dry- and ,-2 - der,-1, because it is the correlation between x, and x,-2 with the linear dependence of each on x, | removed. In this way. we have broken the dependence chain between x, and x,_2. In fact. cover, x,-1. X,-2 -$5,-1) =cov(Wrex, 2 - 6x,_1) =0 Hence, the tool we need is partial autocorrelation, which is the correlation between t, and x, with the linear effect of everything "in the middle" removed. To formally define the PACF for mean-zero stationary time series, let f,+, for 4 2 2. denote the regression" of x,4/ on (X4-4-1. 2,th-2. . ... ,+1 ). which we write as Rith = Bixith-1 + 82xith-2 + . . .+ Bh-1.1/+1. (3.53) No intercept term is needed in (3.53) because the mean of x, is zero (otherwise. replace x, by x, - #, in this discussion). In addition, let f, denote the regression of ty on ( . /+1, -x,+2.. . ..X,th-1). then $ = Bixler + Bakitz + + Bh-1X1th-1- (3.54) Because of stationarity, the coefficients. 81. .... Pa- are the same in (3.53) and (3.54): we will explain this result in the next section, but it will be evident from the examples. Definition 3.9 The partial autocorrelation function (PACF) of a stationary process, My, denoted ohn. for h = 1, 2,. .., is 1 = Corr( X,+1. 3,) = p(1) (3.55) and Ohh = COIT(X,th - (yah. X - 8,). h 2 2. (3.56) The reason for using a double subscript will become evident in the next section. The PACE. dan. is the correlation between x,+4 and x, with the linear dependence of (trel..... 1ren-1) on each, removed. If the process x, is Gaussian, then dha = com(,th. 4, Artie.. ..k/4 1); that is. sun is the correlation coefficient between .t,, and ., in the bivariate distribution of (x, 4. .x, ) conditional on (X/+1. ...,+h-1)Question 2 Consider a finite society ] = (1,..., I), where the preferences of the individuals are repre- sented by the utility functions (u' : R4 -+ R)iej. Assume that all of these functions are continuous, strictly quasiconcave and strictly monotone. Unlike in class, we are going to treat individual endowments as variables. For individual i, her endowment is denoted by WERK Define the functions x' : R4 x R4 - R4 by x' (p, w') = argmaxxent {u'(x) |p . x

RL by Z (p, w) = [lx'(p, w') - w'] (2) and the set M = {(p, w) ER! x RY | Z(p, w) =0}. (3) Here, x' is individual i's demand function and Z is the aggregate excess demand function. Ob- viously, p is a vector of competitive equilibrium prices of exchange economy (3, (ut, wthe} if, and only if, (p, w) E M. Set M is hence called the equilibrium set, or equilibrium manifold, of preference profile (u' )ies. In this exercise you are going to prove that an observer loses no information when working with the equilibrium set, in comparison to the profile of individual demands. In technical language, you will argue that the equilibrium manifold identifies both the excess demand function and all the individual demand functions: given set M, there is one and only one function Z that generates M by Eq. (3); and there is one and only one profile (x' heg that generates that Z by Eq. (2).' 1. Suppose that p, w and ware such that: (a) (p, w) and (p, w) are both in M; and (b) for all i, p . wi = p . wi. Argue that ), wi = _, wi. 2. Fix p and w, and suppose that w is such that: (a) (p, w) is in M; and (b) for all i, p . W = p . w. Argue that Z(p, w) = Z(wi - wi ). 1 One can go further and show that, in fact, the set identifies all the individual preferences, in the sense that there is one and only one profile (u' lies that generates (x' )ies by Eq. (1). Asking this further step in a prelim exam would probably qualify as a human rights violation, though. Page 2 of 7 Micro Prelim June 25, 2018 3. Argue that for any p and any w, there exists we Ry such that (a) (p, w) is in M; and (b) for all i, p. wi = p. w. (Hint: Think of a profile of endowments such that at prices p each individual i demands x' (p, w' ) and which guarantees that markets clear.) 4. Use the previous steps to explain how an analyst who only observes M can construct function Z in a unique manner. 5. Argue that there exists a sub-profile of individual endowments for all agents other than i = 1, say (we,..., w ), such that for all p and all wi, x' (p, w' ) = Z(p, w', w/,..., w') + w. (Hint: Think of a way of kicking all agents but i = 1 out of the market.) 6. Explain how, once the analyst of part 4 has constructed function Z, she can construct function x' in a unique manner. 7. Conclude