Questions and Answers of Introduction To Actuarial And Financial

Determine the duration of the investment proceeds in Example 4.15 at general δ. Interpret your expression.Example 4.15An investment project is expected to return $3m in 2 years’ time and $4m in 4
Calculate the duration for each of the following investments at δ = 5% per annum. Interpret your answers.a. Purchase at t = 0 resulting in receipts of $30 at t = 1, 2, 3, and 4.b. Purchase at t = 0
State the rule xn for n ∈ N+ for the following sequences. Use this rule to extend the sequences by a further three terms. 9' a. {1,1,16) b. (3, 6, 11, 18,...} c. {-2, 4,-8, 16, ...} d. {2,7, 8,
You are given thatDetermine values for the following summations. 100 Σ(0.11)2 = = i=1 6767 2 and 10 Σ(-0.2) = i=1 1,627,604 9,765,625
A fixed-interest investment has price £145 and duration 4.3 years at i = 5.64% per annum. Calculate an approximation to price in the following situations.a. The required return, expressed as
Calculate the value of the finite series formed from the following sequences. a. All terms of [{10, 8, 6, 4,...,0} b. The first five terms of {1,2,3,4, ...}
Express the following series in terms of sigma notation. a. 2¹ +22 +2³+...+ 2100 3+5+7+9+11+... b. c+六++‥+立 13
The following rules give the nth term in an infinite sequence, where n ∈ N+. Determine which associated infinite series are convergent. a. X = n +1 b. xn = 2n+1 c. xn d. xn = (-1)"0.99n-2 2n +0.2"
Determine the range of x over which the Taylor expansion of ln(1 + x) about x = 1.5 will converge. Note that this result was stated but not proved in Example 5.19.Example 5.19Develop a third-order
Determine the following limit algebraically.Note that this limit was considered using a tabular approach in Question 3.4.Question 3.4Confirm the following limits using a tabular approach. lim 8-0 cos
Determine the Maclaurin expansions of the following functions and determine the range of x for which each expansion is convergent. a. f(x) = 2x-³ b. g(x) = sin x X
Use Properties i-iv to simplify the following expressions as far as possible. + 1) + Σ 1 26 – 1) j=1 a. ΣΩz(2i 400 410 b. Στov + Σο(4i – 1) i=10 50 c. 245 2m + Σ1 41 d. Σ5 14-10 + Σ (3)
The notation is used to denote the present value of an n-year level unit annuity paid in advance (i.e., at the start of each year). Derive an expression for in terms of some fixed compounding
Classify the following as either arithmetic or geometric progressions (or some combination of both). State the starting value and common difference/factor in each case. a. {10, 14, 18, 22, ...,
Use an algebraic method to evaluate the following summations. a. Σ1 (50 + 4(n – 1)) b. 4+3+... - 15 200 C. Σ32 10n d. Σ?516(2 + n) n=16
Determine an expression for the following series for general k ∈ {2, 3, 4, . . .}.Evaluate he Σ1n(4") n=1
Use an algebraic method to evaluate the following summations. a. Σ–, 10 × 2"-1 b. 1+10+100++100,000 500 C. Σ (3)" d. Σ35–1 0.4"+1 25. =-10
Evaluate the following summations. a. Σ1 (2n + 2") 57 b. 10 (3+0.1n + (0.9) −2)
Determine whether each of the following are convergent or divergent. Calculate the converged value of the sum where appropriate. a. 1+3 +5+... b. 5+5/3+5/3² +... c. Σ1 0.01 × 2 d. Σ{(-3) + 26 –
Use Wolfram Alpha to perform the following indefinite integrals. a. fx²e³ dx b. fy² cos(2y) dy

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