Final_Package for Actsc

Final_Package for Actsc - SOS ActSci 231 Final Review...

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891.67+ 300 SOS ActSci 231 Final Review Package The growth of money Accumulation function The function a(t) is the accumulation at time t of $1 invested at time 0 (how much $1 at t = 0 is worth at time t) a(0) = 1 and a(t) ≥ a(0) for all t ≥ 0 An investment of K at time 0 accumulates to K*a(t) at time t A k (t) is the accumulation at time t of K investment at time 0 (how much $k at t = 0 is worth at time t) A 1 (t) = a(t) for all t ≥ 0 Interest is added continuously at an annual rate of 3% on total accumulated value A(t) = 1.03t, t ≥ 0 Simple interest a(t) = 1+it, t ≥ 0 Pays interest only on original capital Example : At a particular rate of simple interest, $1,200 invested at time 0 will accumulate to $1,320 in T years. Find the accumulated value of $100 invested at t = 0 at the same rate of simple interest, but for years. Solution : Let i be the simple interest rate, then we have 1200(1 + it ) = 1320, so it = 0 : 1 Thus, A 100( ) = 100(1 + i* ) = 100(1.05) = 105 Example : A invests $100 for 3 years, at 8% simple interest per year. B also invests $100 for 1 year at 8% simple interest. B then withdraws the proceeds and re-invests the whole proceeds for 2 more years, again at 8% simple interest. Who will have more money at the end of 3 years? Solution:
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SOS ActSci 231 Final Review Package A: 100(1+0.08*3) = 124 B: 100(1+0.08*1) = 108 and then 108(1+0.08*2) = 125.28 Hence, B will have more money at the end of 3 years. Simple interest is inconsistent if we allow withdrawal and reinvestment Compound interest a(t) = (1+t) t t = 0,1,2,3,… Pays interest on balance earned so far Consistent with withdrawals and reinvestments a(t)*a(s) = a(t+s) Example : A invests $100 for 3 years, at 8% compound interest per year. B also invests $100 for 1 year at 8% compound interest. B then withdraws the proceeds and re-invests the whole proceeds for 2 more years, again at 8% compound interest. Who will have more money at the end of 3 years? Solution: A: 100(1.08) 3 = 125.9712 B: 100(1.08) = 108 and then 108(1.08) 2 = 125.9712 Effective rate of interest The effective rate of interest for a given period is the amount of interest payable over the period as a proportion of the balance at the beginning of the period i[n,n-1] = For one period n-1 to n: Simple interest: -> decreasing function of n Compound interest: i -> constant with n Example : Given that A k (t) = t 3 + 2t 2 + 4t + 16, find and i 3 .
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SOS ActSci 231 Final Review Package Solution : i3 = Ak3 - ( ) Ak 2 A ( ) k 2 = 73 - 4040 = 0.825 Example: Robin deposits $4,300 into an account on March 1, 1998. The bank guarantees that the annual effective rate for the balance under $5,000 is 3.5% and for the balance over $5,000 is 5%. Suppose that there is no other deposits or withdrawals except for one withdrawal of $1,000 on March 1, 2003 and a deposit of $500 on March 1, 2004. Then find his account balance on March 1, 2006. Solution:
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Final_Package for Actsc - SOS ActSci 231 Final Review...

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