Manual for SOA Exam FM/CAS Exam 2.


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1 Manual for SOA Exam FM/CAS Exam 2. Chapter 2. Cashflows. c 29. Miguel A. Arcones. All rights reserved. Extract from: Arcones Manual for the SOA Exam FM/CAS Exam 2, Financial Mathematics. Fall 29 Edition, available at c 29. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2. 1/1
2 Continuous payments Suppose that the payments are made very often. Then by approximation, instead of a sum, we have an integral. It is like the payments are made continuously. Let V (t) be the outstanding fund balance at time t of the cashflow. Assume that contributions are made continuously at an instantaneous rate C(t), then the equation of value is t V (t) = V ()(1 + i) t + C(s)(1 + i) t s ds. (1) 2/1 c 29. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2.
3 (1) appears as the limit of the equation of value for the cashflow: Inflow V () C(t 1 )(t 1 ) C(t 2 )(t 2 t 1 ) C(t n )(t n t n 1 ) Time t 1 t 2 t n as max 1 j n (t j t j 1 ), where = t < t 1 < t 2 < < t m = t. The equation of value at time t for this cashflow is n V (t) = V ()(1 + i) t + C(t j )(t j t j 1 )(1 + i) t t j, which tends to j=1 V (t) = V ()(1 + i) t + t C(s)(1 + i) t s ds as max 1 j n (t j t j 1 ). Recall that the Riemann integral of a function f is defined as t n f (s) ds = f (t j )(t j t j 1 ). lim max 1 j n (t j t j 1 ) j=1 c 29. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2. 3/1
4 Example 1 A continuous year annuity pays a constant rate 1 at time t where t n. Interest is compounded with an annual rate of interest of i. (i) Find the present value of the annuity at time. (ii) Find the future value of the annuity at time n. Solution: (i) The present value of this continuous annuity is PV = n (1 + i) t dt = 1 = e n ln(1+i) n e t ln(1+i) dt = 1 (1 + i) n =. e t ln(1+i) n c 29. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2. 4/1
5 Example 1 A continuous year annuity pays a constant rate 1 at time t where t n. Interest is compounded with an annual rate of interest of i. (i) Find the present value of the annuity at time. (ii) Find the future value of the annuity at time n. Solution: (i) The present value of this continuous annuity is PV = n (1 + i) t dt = 1 = e n ln(1+i) n e t ln(1+i) dt = 1 (1 + i) n =. e t ln(1+i) (ii) The future value of the continuous annuity at time n is n FV = n (1 + i) n t dt = (1 + i) n PV = (1 + i)n 1. c 29. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2. 5/1
6 If instead of compound interest, the time value of money follows the accumulation function a(t), then the future value at time t of an initial outstanding balance V () and continuous payments C(s), in the interval s t is t V (t) = V ()a(t) + C(s) a(t) a(s) ds. c 29. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2. 6/1
7 Example 2 The force of interest at time t is δ t = t3 1. Find the present value of a four year continuous annuity which has a rate of payments at time t of 5t 3. Solution: The accumulation function is ( t ) a(t) = exp δ s ds c 29. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2. 7/1
8 Example 2 The force of interest at time t is δ t = t3 1. Find the present value of a four year continuous annuity which has a rate of payments at time t of 5t 3. Solution: The accumulation function is ( t ) ( t a(t) = exp δ s ds = exp s 3 ) 1 ds c 29. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2. 8/1
9 Example 2 The force of interest at time t is δ t = t3 1. Find the present value of a four year continuous annuity which has a rate of payments at time t of 5t 3. Solution: The accumulation function is ( t ) ( t a(t) = exp δ s ds = exp s 3 ) 1 ds = e t4 4. c 29. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2. 9/1
10 Example 2 The force of interest at time t is δ t = t3 1. Find the present value of a four year continuous annuity which has a rate of payments at time t of 5t 3. Solution: The accumulation function is ( t ) ( t a(t) = exp δ s ds = exp s 3 ) 1 ds = e t4 4. The present value of the fouryear continuous annuity is 4 = C(s) a(s) ds = 4 5s 3 e s4 s 4 ds = 5e = 5 5e 6.4 c 29. Miguel A. Arcones. All rights reserved. Manual for SOA Exam FM/CAS Exam 2. 1/1
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