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Questions.,

1. Duration and Convexity Your research department reports continuously compounded interest rates as Maturity [Years] Interest Rate (96] 0.5 1.00 1.0 1.50 1.5 2.00 2.0 2.00 (3} Use these rates to compute the prices le0,1} and No.2) of one- and taro-year zero coupon bonds, and the price .0403] of a two-year, 3% coupon bond. Coupons are paid semi- annually, and the face value of all bonds is 100. (b) Obtain the coupon bond's duration and convexity. (cl Suppose that the monthly changes in the interest rates have a mean of zero and a standard deviation of 0.5%. Obtain the monthly 95% Value at Risk and Expected Shortfall on the coupon bond. (d) Construct a hedge portfolio of 1 coupon bond and k one-year zero coupon bonds that has zero duration. What is the value of it? What is the convexity of the hedge portfolio? is) Construct a hedge portfolio of 1 coupon bond, and k1 one-year zero coupon bonds and k2 two-year zero coupon bonds, that has zero duration and convexity. What are the values of in and k2? (f) Suppose that the yield curve shifts upward parallelly with dr = 1%. Recalculate P(0,2), le0,1) and Pz[0,2] and use this to calculate the change in the values of the hedge portfolios constructed in [cl] and {e}. Comment on the result. 85 QUANTITATIVE BUSINESS 2020-2021, MATHEMATICS WEEK 6 Exercise g): The tragedy of the commons Two farmers graze their sheep on the same patch of land. Each farmer can choose to graze 0, 1, or 2 sheep. The care for one sheep costs 20, while the care for 2 sheep costs 45 in total. The per-sheep benefits, given the total amount of sheep on the patch, are given in the table below. # sheep 2 3 benefits per sheep 100 90 60 50 ) The farmers are in fact playing a 3x 3 bimatrix game. Determine the payoff matrix of this game. ii) Show that grazing 2 sheep is a dominant strategy for each farmer. Thus, we get a unique Nash equilibrium in which both farmers graze 2 sheep, for a total of 4. iii) Show that this unique Nash equilibrium is Pareto dominated by another outcome. iv) Explain this phenomenon in your own words. So far we dealt with games where the set of strategies (or at least: the set of pure strategies) available to a player was finite. But the notion of best responses, and consequently the notion of Nash equilibrium, is applicable in a much wider context than that. We will discuss a few games in which the strategy spaces, comparable to games with mixed strategies, are continuous instead of finite. Example: Congestion and the price of anarchy Consider two routers in an electronic network that have to send data from A to B. Router #1 has to send 4 GB of data, while router #2 has to send 3 GB. e+1 As you can see, there are two routes to transport data from A to B. The first route is one on which each GB of data travels exactly 7 seconds. The second route easily gets congested. The travel time on this route when ( GB is sent over it is ( + 1 seconds for each GB. Exercise h) Both routers aim to minimize the total travel time of the packages of data under their care. Suppose that router #1 ships , GB via the lower route, and the remaining 4 -6, GB via the upper route, while router #2 ships 62 GB via the lower route, and the remaining 3-( 2 GB via the upper route. i) Explain why the total travel time for router #1, given that router #2 ships (2 GB via the lower route, is given by the formula T, = 7. ( 4 - 6 , ) + ( 6 , + 6 2 + 1 ).6 , (9) ii) Show that the best response function for router #1 is given by 6, =3-762. iii) Find the formula for the total travel time 72 of router #2, and determine its best response function. iv) Show that 6, = 62 =2 is the unique Nash equilibrium. N.B.: The structure of this congestion problem is very similar to some of the duopoly models which we will discuss in Mathematics Case W7.(a) Assume the following information represents the National Income model of an "Utopian' economy. Y= C+I+G Cza+b( Y-T) T= d + tY G =G. Where a >0;0 0; 0