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1、 1Financial Mathematics Second Edition A Practical Guide for Actuaries and other Business Professionals By Chris Ruckman, FSA & Joe Francis, FSA, CFA Published by BPP Professional Education Solution 9.1 The table below describes the probability distribution of the accumulated amount at the end of 10
2、 years: 1,00010AV Probability =91,000 1.05 1.051,628.8946 0.25 =91,000 1.05 1.071,930.3822 0.40 =91,000 1.05 1.102,475.8451 0.35 The expected accumulated value is: +=1,628.8946 0.251,930.38220.402,475.8451 0.35$2,045.92 Solution 9.2 The table below shows the probability for each possible value of 21
3、0(1,000)AV. 1,00010AV 2(1,000)10AV Probability 1,628.8946 21,628.89462,653,297.71= 0.25 1,930.3822 21,930.38223,726,375.33= 0.40 2,475.8451 22,475.84516,129,808.84= 0.35 The expected value of the square of the accumulated value after 10 years is: Solutions to practice questions Chapter 9 Financial M
4、athematics Solutions to practice questions Chapter 9 BPP Professional Education 2210(1,000) 0.25 2,653,297.710.403,726,375.330.35 6,129,808.844,299,307.65EAV=+= The standard deviation of the accumulated value is: 0.50.52221010Standard deviation(1,000) ( 1,000)4,299,307.652,045.92$336.91EAVEAV=Soluti
5、on 9.3 The mean and the variance of the annual yield are: 22222 (0.2)(0.10)(0.5)(0.12)(0.3)(0.15)0.125(0.2)(0.10 )(0.5)(0.12 )(0.3)(0.15 )0.1250.000325tiE is=+=+=The mean of the accumulated amount at the end of 6 years is: 6665,0005,0005,000(1.125)$10,136.43EAVE AV= The variance is: 266222 62 622612
6、(5,000)5,000()5,000(1)(1)5,000(1.125)0.000325(1.125)158,408.56VarAVVar AVisi=+=+ =The standard deviation is therefore: Standard deviation158,408.56$398.01= Solution 9.4 The mean of the accumulated value at the end of three years is: 333100,000100,000 100,000(1.14)$148,154.40EAVE AV= Solution 9.5 Sta
7、tements III and IV are the only correct statements. I. 1nnE AVEi=+II. (1)nnE AVEi=+III. ()1n nE AVEi=+ IV. ()=+1 n nE AVE i Statement I is clearly not correct since its value goes to one as n increases (assuming that the possible values of i are positive values less than 1). Statement II is the form
8、ula for the fixed interest rate model. Statement III simplifies to the correct formula for independent and identically distributed interest rates: ()()()11 1nnn nE AVEiE ii=+=+=+ Solutions to practice questions Chapter 9 Financial Mathematics BPP Professional Education 3Statement IV also simplifies
9、to the correct formula for independent and identically distributed interest rates: ()()=+=+1 1nn nE AVE ii Solution 9.6 The expected value of the accumulated value is: 123233123233Expected value10,000(1)(1)(1)8,000(1)(1)32,000(1)10,000 (1) (1) (1)8,000 (1) (1)32,000 (1)10,000(1.04)(1.10)(1.07)8,000(
10、1.10)(1.07)32,000(1.07)$55,896.80EiiiiiiEiEiEiEiEiEi=+=+=+=Solution 9.7 Since the interest rates are independent and identically distributed, we have: ?2334233412(1)(1)(1)(1)(1)(1)(1)1(1) (1)(1)(1) (1)(1)(1)1(1)(1)(1)1nnnnnnnnnn iE sEiiiEiiiEiEEiEiEiEiEiEiEiEiiis=+=+=+=If the interest rates are inde
11、pendent and identically distributed, then the expected accumulated value of an annuity immediate is equal to an annuity immediate calculated at the mean interest rate. Solution 9.8 Part (i) 222(1)(1.05)$1.1025E AVi=+= Part (ii) 22 22 22 224()(1)(1)(1.05)0.0002(1.05)0.000441Var AVisi=+=+=Financial Ma
12、thematics Solutions to practice questions Chapter 9 BPP Professional Education 4Solution 9.9 We can use the formula for the expected value of the 1-year accumulation factor to find the value of 2/2+: 22/2/2211.05/2ln(1.05)tEiee +=+=This can then be put into the formula for the variance to solve for
13、2: 222222222(/2)2ln(1.05)2221(1)0.0002(1)0.0002(1)0.00021.05 (1)0.0002/1.0510.0001814tVarieeeeeeee +=+=Now we can find : 2/2ln(1.05)ln(1.05)0.0001814/20.0486995+=Solution 9.10 Part (i) The general formula is: 2/2nnnE AVe+= For 2n =, we have: 222 2(0.0486995) (0.0001814)1.1025E AVee +=Part (ii) The g
14、eneral formula is: 222(1)nnnnVar AVee+= For 2n =, we have: 224222 4(0.0486995) 2(0.0001814)2(0.0001814)(1)(1)0.000441Var AVeeee+=Solutions to practice questions Chapter 9 Financial Mathematics BPP Professional Education 5As expected, the expected accumulated value and the variance are the same as th
15、e expected accumulated value and variance found in Solution 9.8, since both stochastic models are independent and identically distributed. Even if their underlying distributions are different from one another, the fact that the interest rates have the same mean and variance means that the mean and variance of the accumulated values are also the same. Solution 9.11 Only statements I and III are correct. We are given that: (1)1.10tEi+= and 21 0.025ttVariVar i+= Further
