# Is the amount that matures in three years ie is least

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Chapter 15 / Exercise 35
College Physics
Serway/Vuille
Expert Verified
is the amount that matures in three years (i.e., is least liquid)We define liquidity ratios as the ratio of these amounts to the total wealth:L1(t) = (B1(t) +B2(t-1) +B3(t-2))/w(t),L2(t) = (B2(t) +B3(t-1))/w(t),L3(t) =B3(t)/w(t).For the two strategies above, do the liquidity ratios converge ast→ ∞? If so, to whatvalues?Note:as above, simple numerical simulation alone isnotacceptable.d) Suppose you could change theinitialinvestment allocation for the 35-35-30 strategy,i.e., choose some other nonnegative values forB1(0),B2(0), andB3(0)that satisfyB1(0) +B2(0) +B3(0) = 1.What allocation would you pick, and how would it bebetter than the(0.35,0.35,0.30)initial allocation? (For example, would the asymptoticgrowth rate be larger?) How much better is your choice of initial investment allocations?Hint for part d:thinkvery carefullyabout this one.Hint for whole problem:watchout for nondiagonalizable, or nearly nondiagonalizable, matrices.Don’t just blindlytype in matlab commands; check to make sure you’re computing what you think you’recomputing.9
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Chapter 15 / Exercise 35
College Physics
Serway/Vuille
Expert Verified
Solution.a) We take as state vectorx(t) =B1(t)B2(t)B2(t-1)B3(t)B3(t-1)B3(t-2)..The components consist of the six critical quantities:the amount of one-, two-, andthree-year CDs held of each possible maturity date (i.e., one, two, and three years). Wecan express the system asx(t+ 1) =Fx(t) +Gu(t)whereu(t)R3gives the amountof one, two, and three year CDs purchased att+ 1, andF=000000000000010000000000000100000010,G=100010000001000000.The payout is given byp(t) =hx(t), whereh= [1.05 0.06 1.06 0.07 0.07 1.07].The allocation of the payout is given byu(t) = [0.35 0.35 0.30]Tp(t)for the 35-35-30strategy and byu(t) = [0.60 0.20 0.20]p(t)for the 60-20-20 strategy. Finally, putting itall together, we end up withx(t+ 1) =Ax(t)whereA35=F+G[0.35 0.35 0.30]Thfor the 35-35-30 case, and similarly for the 60-20-20 case. To be fully explicit, we haveA35=0.36750.02100.37100.02450.02450.37450.36750.02100.37100.02450.02450.374501.000000000.31500.01800.31800.02100.02100.32100001.00000000001.00000,A60=0.63000.03600.63600.04200.04200.64200.21000.01200.21200.01400.01400.214001.000000000.21000.01200.21200.01400.01400.21400001.00000000001.00000.The following matlab code calculates the matrixAfor each allocation strategy.AA = [0 0 0 0 0 0 ;10
0 0 0 0 0 0 ;0 1 0 0 0 0 ;0 0 0 0 0 0 ;0 0 0 1 0 0 ;0 0 0 0 1 0 ];BB = [1 0 0;0 1 0;0 0 0;0 0 1;0 0 0;0 0 0];PP = [1.05 0.06 1.06 0.07 0.07 1.07]; % PP*x gives payoutalloc35 = [.35 .35 .3]’;% for 35-35-30 allocationA35 = AA+BB*alloc35*PP;alloc60 = [0.60 0.20 0.20]’;% for 60-20-20 allocationA60 = AA+BB*alloc60*PP;WW=ones(1,6); %WW*x gives total wealthThis yields:A35 =0.36750.02100.37100.02450.02450.37450.36750.02100.37100.02450.02450.374501.000000000.31500.01800.31800.02100.02100.32100001.00000000001.00000A60 =0.63000.03600.63600.04200.04200.64200.21000.01200.21200.01400.01400.214001.000000000.21000.01200.21200.01400.01400.21400001.00000000001.00000There are several other correct answers.For example, some people decided to use astate vector that included the past three samples of each of theBi’s,i.e., a state ofdimension9.Provided no errors were made, this works fine.At the other end, somepeople found a state description that has a state dimension of4.b)Asymptotic wealth growth rate.This is going to depend on the eigenvalues ofA, so firstwe check the eigenvalues, which turn out to be1.0627,-0.3266±0.4421i,0,0,0forA35, and1.0598,-0.2019±0.4015i,0,0,011
forA60