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Tutorial04_-_soln - Problem Set 4 ACTSC 231 Mathematics of...

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Problem Set 4: ACTSC 231 Mathematics of Finance, Winter 2011 Q1. A deferred perpetuity-due begins payments at time n with annual payments of $1,000 per year. If the present value of this perpetuity-due is equal to $6,561 and the effective rate of interest i = 1 / 9, find n . Q1. The present value of this perpetuity-due is 1 , 000 v n ¨ a = 6 , 561; where v = 9 / 10 i.e. d = 1 / 10. We know that ¨ a = 1 /d = 10. Thus, n = ln(6 , 561 / 10 , 000) ln 0 . 9 = 4 . Q2. A loan of $50,000 is made at time 0 and is repaid by making regular annual payments of $7426.39 for 8 years in total. (a) What is the outstanding loan balance at the end of three years? (b) What is the interest payment included in the fourth regular payment? Q2. We first need to find the interest rate i on the loan, which solves 50 , 000 = 7 , 426 . 39a 8 i. Using the financial calculator, we find i = 4%. (a) We can then compute OLB 3 = 7 , 426 . 39a 5 4% = 33 , 060 . 98. (b) The interest paid in the fourth payment is i · OLB 3 = 1 , 322 . 44. Q3. Julia borrows $10,000 and repays the loan by making annual (end of the year) payments of X for 5 years. The annual effective rate on the loan is 6%. (a) Determine P and the outstanding loan balance at the end of two years. (b) Julia skips the third payment. What additional payment does she need to make at time 5 in order to repay the loan within 5 years, as originally planned? Q3. (a) The loan amount L is 10,000 and the payment P is such that L = P a 5 6% . Solving this equation, we obtain P = 2 , 373 . 96. Consequently, OLB 2 = P · a 3 6% = 6 , 345 . 63. (b) The additional payment at time 5 is the accumulated value of the missed payment at time 3, and hence is given by P (1 . 06) 2 = 2 , 667 . 39. Q4. Annual deposits are made into a fund at the beginning of each year for 10 years. The first 5 deposits are $1,000 each and deposits increase by 5% per year thereafter. If the fund earns 8% effective per year, find the accumulated value at the end of 10 years.
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