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ACTSC 231 tutorial

ACTSC 231 tutorial - Problem Set 6 ACTSC 231 Mathematics of...

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Problem Set 6: ACTSC 231 Mathematics of Finance, Winter 2011 Q1. Payments are made into an account continuously at a rate of 8 Y + tY per year, for 0 t 10. At time T = 10, the account is worth $20, 000. Find Y if the account earns interest according to a force of interest δ t = 1 / (8 + t ) at time t , for 0 t 10. Q1. The accumulated value at T = 10 of the annuity is 10 0 (8 Y + tY ) m ( t ) dt, where m ( t ) is the accumulated factor over period [ t, 10] so that m ( t ) = exp 10 t δ r dr = exp 10 t ( r + 8) - 1 dr = exp ln( r + 8) 10 r = t = 18 t + 8 . Thus, its accumulated value is equal to 10 0 (8 Y + tY ) 18 t + 8 dt = Y 10 0 18 dt = 180 Y. Equating 180 Y to 20,000, we obtain Y = 20 , 000 180 = 111 . 11 . Note: Students may first develop an expression for the present value and then compute the accumulated value by the formula AV = PV · a (10). Q2. A 2-year deferred annuity paying continuously at a rate of 300 + 200 t per year, for 2 t 10. Find the present value of this annuity, if the force of interest function δ t = 5 / (3 + 2 t ) for 0 t 10.
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