Module 2 - with solutions(2).doc

University of ConnecticutMath 3615: Financial Mathematics Problems1. For each of the following calculations, the annual effective interest rate is 5%.(a)What is the present value of an annuity-immediate consisting of a payment of 100 at the end of each year for 40 years?(b)What is the present value of an annuity-immediate consisting of a payment of 25 at the end of each quarter for 40 years?(c)What is the present value of an annuity-immediate consisting of a payment of 400 at the end of each fourth year for 40 years?(d)What is the present value of an increasing annuity-immediate consisting of a payment of 100 at the end of the first year, increasing by 6 each year for 40 years?(e)What is the present value of a geometric annuity-immediate consisting of a payment of 100 at the end of the first year, increasing by 6% each year for 40 years?2. (a) 100 is deposited at the beginning of each year for 30 years, and earns a positive level compound rate of interest.The balance at the end of 30 years is 13 times the balance at the end of 15 years.Find the effective annual interest rate.Find the balance at the end of 15 years.(b) 100 is deposited at the beginning of each year for 30 years, and earns a positive level compound rate of interest.The balance at the end of 30 years is 13 times the balance at the end of 10 years.Find the effective annual interest rate.Find the balance at the end of 10 years.3.A 20-year annuity-immediate has a first payment of 500 (payable at t=1).Each subsequent payment for the next 10 years increases by 100, until a payment of 1500 is made at t=11.Annual payments continue at 1500 until the final payment at t=20.At an 8% effective annual interest rate, what is the present value of this annuity?What is the value at time t=20?4.You purchase a 1,000 par value 10-year bond for 1,000.The bond pays semi-annual coupons and has an 8% (annual) coupon yield.You invest the coupon payments in an account that pays 6% annual interest, convertible quarterly.What is the total value available to you at the end of 10 years?What overall annual effective interest rate have you earned during the 10 years on your 1,000 investment?5. An arithmetically increasing perpetuity-immediate makes annual payments of 1, 2, 3, etc.A geometric perpetuity-immediate makes annual payments of 1, (1.05), (1.05)2, etc.If these two perpetuities have the same present value at an annual effective interest rate of i, what is the value of i?6. Suppose that you deposit 100 into an account at the end of each 6month period.If the account earns interest at a rate of 6% per annum, convertible quarterly, after how many years will your account balance reach 2,180.30?If the account earns interest at a rate of 6% per annum, convertible quarterly, after how many years (to 2 decimal places) will your account balance reach 2,200.00?7. A, B, and C are to share the value of a perpetuity. Each of the three is to receive payments for a period of time such that each receives one-third of the total value of the perpetuity.Under the agreement, A will receive the payments from the perpetuity for the first myears. Then B will receive the payments for the next n years. Finally, C (and Cs heirs) will receive all payments after m+n years.If m and n have been properly determined using a level interest rate in all years, and n is equal to 12 years, what is the value of m?8. Extra Practice problem taken from my first test, Michelle is a new actuarial student at Storrs Life Insurance Company. Because she understands the importance of saving and the value of compound interest, she starts to save for her retirement immediately after starting her new job. Michelle makes annual deposits into her new retirement savings account at the beginning of each year for 30 years. She makes no other deposits into nor withdrawals out of this account. Michelles first 10 deposits are for $5000 each. After that, Michelle increases her deposits by $500 each year for 5 years, until they reach $7500. Then Michelle increases her next deposit by 5%, to $7875, and she continues to increase each remaining deposit by 5% per year. If her retirement savings account earns interest at a 7% annual effective rate throughout the period, how much money will be in Michelles retirement account at the end of 30 years?Answers1. (a) Calculator problem: answer = 1715.9140 N I/Y 5 100 PMT 0 FV CPT PV (b) 1747.75 Convert the annual effective rate to a quarterly rate.Calculator: 160 N; 1.227223 I/Y; 25 PMT; 0 FV CPT PV(c) 1592.44Calculate the 4 year interest rate: Calculator: 10 N; 21.5506 I/Y; 400 PMT; 0 FV CPT PV (d) 3093.18 Solve this using the formula: P=100; Q=6; n=40; i=.05(e) 4610.42From first principles, the PV is:Solve using this formula: = with 2. (a) 18.01723% $7,205.27 (b) 11.612% 1922.31Detailed solution:(a) 100 x = 1300 x 100 x = 1300 x Let X=Divide by 100 and use quadratic formula to solve equations of the form: X=12 (or 1 but X=1 gives nonsensical result)i=18.01723% and 100 = $7205.27This approach doesnt work for part (b)!For part (b) 100 x = 1300 x 100 x ( = 1300 x Divide by And let X=100(1+X+) =1300100(1+X+) =1300Divide by 100 and Solve using quadratic formulaX=3 i=11.612% 100 = 1922.313. PV = 10,614.83 FV = 49,475.27To do this problem, split the annuity into two segments.The first segment is an arithmetically increasing annuity immediate that pays 500 at time t=1; 600 at t=2; up to 1500 at time t=11Solve this using the formula: P=500; Q=100; n=11; i=8.0%PV=3569.48 + 3026.57 = 6596.05And this is really the PV at time t=0, which we need.The second segment is a level pay annuity paying 1500 at time 12, 13, . . .20.This is just 1500 x This value is 9,370.33But this value is at time t=11 (since we did an annuity immediate and first payment is at time t=12; i.e. the end of the period)So, the PV = 9370.33 x =4018.78So, the PV of this annuity = 6596.05 + 4018.78=10,614.83Of course, the FV= 4. 2077.28 7.584% as follows:You collect 40 at the end of each 6 month period which you invest at 6% annual interest convertible quarterly.So, your accumulated value at the end of 10 years from the coupon payments is40 with n = 20 and an interest rate of: (you need the semiannual interest rate)Calculator: 20 N; 3.0225 I/Y; 40 PMT; 0 PV CPT FVGives you 1077.28In addition you get the bonds maturity value of 1000 at the end of 10 years. So, you have a total of $2077.28 after 10 years.You invested 1000 and ended up with 2077.28 after 10 years.What rate did you earn?5. 5.263% determined as follows:PV of first annuity is with P=Q=1=Present Value of a geometrically increasing perpetuity due is But this is a perpetuity immediate so PV= where PV = So, 6. (a) 8.5 years (b) 8.651 years6(a) is just a calculator problem:First we have to determine the correct interest rate:We need the interest rate for the six month payment periods.Here are the key strokes on your calculator:Key strokes to determine the interest rate and properly put it into your calculator:2nd CLR TVM1.015 2 = (answer will be 1.030225) 1 = (answer .030225) x 100 (answer is 3.0225)I/Y (that is after you get 3.0225 press the I/Y key- you have now entered the proper interest rate)100 PMTFV -2180.30 (Note for annuities PMT has different sign than PV and FV)0 PVCPT N (answer 16.999986) 17 half years, or 8.5 years.As long as the answer comes out to an integer (with rounding) you are fine. Problem (b) shows you have to careful if the answer for n is not an integer!You should do part (b) the same way we did part (a) above, except FV=-2200The answer you will get is N=17.12So it appears that the answer is 17 half years (periods) plus .12 half years (periods) which would be 8.5 years plus .06 years =8.56 years.But this is incorrect. Lets see why:After 8.5 years we have 2180.30 as computed in part (a). Payments are made at the end of each period, so no more money will be deposited in the account until time 9.So, from time 8.5 years to time 8.56 years our account only grows with interest at a rate of 3.0225% for .12 periods. (This is 0.12 half years-using the semiannual rate)So, at that time our account s only worth: So, when we get a non integer for n in problems like this with discrete payments, we have to calculate the balance to the whole number only (not rounded-truncate to 17 payments)In this case after 17 payments we have 2180.30How long before that grows to 2200?Correct answer is it takes 17 half years plus .302 half years which is8.5 years + .151 years=8.651 years!7. 7 yearsAt time 0 As share has PV = = So, Bs share has PV= x So so i=.0594631Back to A: 8. 681,780.2