2 Theory of Interest

1. (A nice inheritance) Use the "72 rule". Years = 1994-1776 = 218 years. (a) i = 3.3%. Years required for inheritance to double = Zf = 8 :'=! 21.8. Times doubled= Hi = 10 times. $1 invested in 1776 is worth 210 :'=! $1,000 today. (b) i = 6.6%. Years required to double = ~ :'=! 10.9. Times doubled = ~ times. $1 invested in 1776 is worth 220 :'=! 000, 000 today. $1, 2. (The 72 rule) Using (1 + r)n = 2 gives nIn (1 +r) In2 = 0.69. We have nr :'=! 0.69 and thus n :'=! ~ = 20

= In2. Using In (1 + r) :'=! and r :'=! PI.

Using instead In(1 + r) :'=! r- !r2 = r(1 -!r) we have nIn(1 + r) = In2 or equivalently nr :'=! ~. For r :'=! 0.08, we have (1 -r /2)-1 :'=! 1.042. Therefore, n:'=! !(0.069)(1.042) r 3. (Effective rates) (a) 3.04% (b) 19.56% (c) 19.25%. 4. (Newton's method) We have I(") i "k 0 1 1 2/3 2 13/21 3 0.618033 4 0.618034

= ~

r

= ~ t

I("k) = -1 + " + " 2 , I , (,,) = 1 + 2" , "k+1 = "k -f' I' ("k) 1 3 1/9 7/3 0.00227 2.23810 -2.2 x 10-6 2.23607 0 2.23608

I("k)

"k+1 2/3 13/21 0.618033 0.618034

0.618034

5. (A prize) PV = $4, 682, 460.

1

2

CHAPTER mE BASIC 2. mEORY OFINTEREST 6. (Sunk cost) The payment stream for apartment A is 1,000, 1,000, 1,000, 1,000 1,000, 1,000 while for B it is 1,900, 900, 900, 900, 900, 900. At any interest rate PVA l1(x) = = = = = = ~ --IT x 1 )'2X)'-1 )'2 ()' -1) X)'-2 1 -)'

(b) U(x) = )'X)'-l

Relative risk aversion coefficients, 11,are constant for both utility fW1ctions. 5. (Equivalency) If results are consistent, we have that V(x) = aU(x) + b, and since V(A') = A' and V(B') = B' we must have A' B' = = aU(A') + b aU(B') + b

So solving both equations simultaneously we find parameters a and b: a = A' -U(B') U(A') -B' B'U(A') -A'U(B') U(A') -U(B')

b

=

6. (HARA) The hyperbolic absolute risk aversion function is given by: U(x) = y 1-)'

(~+b ax

)' , b>O.

(a) Linear: We can write the HARA as: U(x) =

l=l. 1 y ax(1 -)') )' + b(l-

(

)')y

1

)'

Choosing)' = 1 and a = 1 and using L'Hopital's rule we can write: U(x) = = IJ!!- ° such that A Ty = 0. Note that this is the same as: DIIYl D21Yl DNIYl + + + D12Y2 D22Y2 + + D13Y3 D23Y3 + + ...+ ...+ ...+ D1SYS -'llYS+l D2SYS -'l2YS+l DNSYs -'lNYS+l = = = 0 0 0

DN2Y2 +

DN3Y3 +

Note that by dividing each element of y by YS+l we get positive state prices tfJi = Yi/Ys+l such that: D 11 tfJl D21tfJl DNltfJl + + + D12tfJ2 + D22tfJ2 + DN2tfJ2 + D13tfJ3 + D23tfJ3 + DN3tfJ3 + ...+ ...+ ...+ DlstfJS -'11 D2stfJS -'12 DNstfJS -'IN = = = 0 0 0

Therefore, if there is no arbitrage, there are positive state prices. 13. (Quadratic pricing) From the earlier exercise we have E[Ut(x*)(ri U(x) = x -C/2X2 then, Ut (x) = 1- cx, so in this case we have E[(lor equivalently E[(lWritten out we have Hi -R = = cW[E(RMRi) -HMR] cW[COV(RM,Ri) + HM(Ri -R)]. CWRM)(Ri -R)] = 0. cWRM)(ri -rf)] = 0 -rf)] = 0. If

This implies, Hi -R = )1COV(RM, for Ri) )1 = 1 -cwHM. AppIing this to RM yields HM -R = )1var(RM) which shows that RM-R )1 = var(RM )

and hence finally Ri -R = fJi(RM -R).

46

CHAPTER GENERAL 9, THEORY

14. (At the track) Gavin Jones will choose the fraction (Xof his money m to bet on the horse so as to maximize his expected utility: max E[U] = ~~ (a) The first order necessary condition is: 1 m ---=0 2 .Jm + 4(Xm which yields: 3 m 4 ..;(1- (X)m + ~~,

7 (X= 52 = .1346

Gavin's maximizing choice is to bet 13.46% of his money and keep the rest in his pocket. (b) We can summarize Gavin's world by the following three alternatives:

1 C(KI). Buy option 1 and short option 2. Use option 1 to cover the obligations of option 2, since max[O, 5-KI] ;:: max[O, 5-K2] for all5. Keep profit of C(K2) -C(KI). (b) Assume K2 > KI and suppose to the contrary that K2 -KI < C(KI) -C(K2). Buy option 2 and short option 1 to obtain K2 -KI + E profit (where E > 0). Use option 2 and profits K2 -KI to cover option 1 since max[0,5 -K2] + (K2 -KI) = max[K2 -KI,5...