Course FM / Exam 2 Introduction It wasn t very long ago that the square root key was the most advanced function of the only calculator approved by the SOA/CAS for use during an actuarial exam. Now students are faced with a variety of actuarially approved calculators. The SOA and the CAS allow the following Texas Instruments (TI) calculators: BA-35, BA II (Plus or Plus Professional Edition), TI-30 (X, Xa, XIIS solar or XIIB battery). Students may use more than one of the approved calculators during the exam, but the memory of the BA II (Plus or Plus Professional Edition) and the TI 30 (XIIS or XIIB) must be cleared before the exam. The student is responsible for the calculator to be in good working order during the exam, so it is a good idea to have a back-up calculator on hand in case the primary calculator malfunctions or its battery dies during the exam. Students often wonder which of the approved calculators is best for an actuarial exam. Let s explore the advantages and disadvantages of the approved calculators before we make our recommendation. The TI-30X Series These calculators are scientific, not financial calculators. While they are able to perform mathematical, statistical and scientific functions, they do not have time value of money functions. However, most students find that they are very powerful and easy to use calculators that can perform most every other calculation better and more quickly than the financial calculators. The TI BA-35 The BA-35 is a straightforward financial calculator that is relatively easy to master. It comes preprogrammed with financial and accounting functions, including time value of money and compound interest functions. It can perform annuity calculations, compute amortization balances, and it can convert annual percentage rates to effective interest rates. The solar version of the BA-35 has the added benefit of eliminating any concern about the batteries dying during the exam. The SOA has published a user guide to the BA-35 that should be read by students preparing for Exam FM. It can be downloaded from www.soa.org. Also, the BA-35 Quick Reference Guide and Complete Guidebook are both downloadable from http://education.ti.com/us/product/tech/ba35s/guide/ba35sguideus.html. BPP Professional Education: 2005 exams Page 1
SOA Course FM / CAS Exam 2 The BA II Plus The BA II Plus is a more sophisticated financial calculator that many students find more difficult to master. It can solve time value of money calculations involving annuities and loans, and it can generate amortization schedules. The BA II Plus can perform cash flow analysis involving up to 24 non-level, evenly spaced cash flows, and it can compute the net present value or the internal rate of return. The SOA has published a user guide to the BA II Plus that should be read by students preparing for Exam FM. It can be downloaded from www.soa.org. The BA II Plus Quick Reference Guide, which includes examples and tips, is available from TI at http://education.ti.com/us/product/tech/baii/guide/baiiguideqgus.html. A BA II Plus tutorial is also available at http://education.ti.com/us/global/freebaiiptutorials.html. Our recommendation We recommend students use a TI-30X Series calculator for the actuarial exams along with one of the financial calculators. A scientific calculator can be used even with most time value of money calculations. It forces students to understand the concepts and the formulas for Course FM, which is very helpful to pass the exam. We recommend that students prepare for the exam by working all problems by first principles with a scientific calculator, and then using a financial calculator to check your work. During the exam, a financial calculator may result in some time savings when students recognize that a particular question can be solved more easily or quickly with a financial calculator. Between the two financial calculators, we recommend the BA-35 for the actuarial exams, but the best choice depends on each student's personal preference. The BA-35 is able to help answer any question on the Course FM exam. The BA II Plus, if a student knows how to use it, is a somewhat more sophisticated calculator that might be able to answer an occasional question more quickly than the BA-35. In our opinion, this slight advantage in being able to perform complicated functions a little more quickly does not offset the many hours it will take a student to become comfortable with the BA II Plus. The SOA/CAS is interested in testing a student s knowledge of the Course FM learning objectives, not proficiency with a specific calculator. A student who has a deep understanding of the core exam topics should be able to pass the exam regardless of which calculator is used. We personally prefer the BA-35 since it is easier to learn and more straightforward to use. The BA II Plus is a great calculator, but it takes longer to learn. The BA II Plus is also less intuitive as it involves frequent use of the 2 nd function key to access its powerful operations. We advise students to focus on learning Course FM concepts and formulas rather than spending too much time learning to use a specific calculator. If a student still prefers the BA II Plus, the student should learn to use it for Course FM type questions by working the multitude of questions from our study program. It may take a little longer at first, but once you work enough problems, you will know it well enough to use during the actual exam. Page 2 BPP Professional Education: 2005 exams
Whichever calculator you choose, be very comfortable with it by the day of the exam so that no time is wasted trying to figure it out during the exam. It could be disastrous for a student to pick up a calculator for the first time a few weeks before the exam with the intention of using it during the exam. The student should learn the material on the same calculator that will be used during the exam to ensure maximum familiarity with it. Examples We have selected a few questions to illustrate how the financial calculators are used. For more instructions regarding the use of the BA-35 and BA II Plus calculators, please consult the SOA user guides at www.soa.org or the TI user guides at www.ti.com. Calculation of present values There is $100 in an account at time 10 years. The annual effective interest rate is 5%. Determine the present value at time 0. Using the BA-35, clear the calculator memory by pressing the clear memory button [CMR] ([2 nd ] followed by [CE/C]) and ensure that the calculator is in financial mode by pressing the [MODE] key until FIN appears in the display. You can also clear the memory by repeatedly pressing the [MODE] key to toggle between modes. The [N] key is the number of periods. The [%i] key is the periodic effective interest rate. The [PV] key is the present value and the [FV] key is the accumulated, or future value. In this case, enter 100 [FV], 10 [N], and 5 [%i]. Then press the compute key [CPT] followed by [PV] and the answer of 61.39 appears in the display. If the calculator does not show all the decimal places, press [2 nd ] followed by [CPT] followed by the decimal point. Using the BA II Plus, press [2 nd ] [CLR TVM] to clear the time value of money register or press [2 nd ] [CLR WORK] to reset default values. If you would like more decimal places than the default value of 2, press [2 nd ] [FORMAT] and then 5 [ENTER] to have 5 decimal places appear. Then press [2 nd ] [QUIT] to return to the calculator mode. Press 10 [N], 5 [I/Y], 100 [FV]. Press [2 nd ] [P/Y], 1 [ENTER] and then [ ] 1 [ENTER] to make sure the payments per year and the number of compounding periods per year is 1. Press [2 nd ] [QUIT] to return to calculator mode. Then press [CPT] [PV] and the answer of 61.39 appears as the present value. The BA II Plus expects negative values for cash outflows and positive values for cash inflows. Since we entered a value of 100 as a cash outflow at time 10 years, it produced a present value of 61.39 as a cash inflow at time 0. We recognize that 61.39 accumulates to 100 over 10 years at an annual effective interest rate of 5%. The compounding frequency is automatically set to match the payment frequency unless the compounding frequency is changed after the payment frequency is changed. This setting continues indefinitely until it is changed, even after the calculator has been turned off and back on again. BPP Professional Education: 2005 exams Page 3
SOA Course FM / CAS Exam 2 Calculation of accumulated values You invest $600 in an account at time 0. The annual effective interest rate is 4%. Determine the accumulated value at time 8 years. Using the BA-35, clear the calculator memory and ensure the calculator is in financial mode. Enter 600 [PV], 8 [N], 4 [%i]. Press [CPT] [FV], and the answer is 821.14. Using the BA II Plus, clear the memory and enter 600 [PV], 8 [N], 4 [I/Y]. Check P/Y and C/Y to make sure these values are 1. Then press [CPT] [FV] and the answer of 821.14 appears as the accumulated value. Present value of annuity-immediate Payments of $100 are received at times 1, 2,, 10 years. The annual effective interest rate is 4%. Find the present value of these payments at time 0. Using the BA-35, make sure the financial memories are cleared and the calculator is in financial mode. Since this is an annuity-immediate, we do not want the calculator to be in BEGIN mode. The calculator toggles between BEGIN on and off by using the [BGN] key ([2 nd ] [MODE]). The [PMT] key is the amount of each payment. Press 100 [PMT], 10 [N], 4 [%i], [CPT] [PV] and the answer is 811.09. Using the BA II Plus, clear the time value of money memory and make sure the calculator is not in BEGIN mode. To toggle between modes, press [2 nd ] [BGN] and read the display. The display reads END for end of year and BGN for beginning of year. To toggle between the two, press [2 nd ] [SET] until the desired reading. Press [2 nd ] [QUIT] to return to calculator mode. This setting continues indefinitely until it is changed, even after the calculator is turned off and back on again. Press 100 [PMT], 10 [N], 4 [I/Y], and check that C/Y and P/Y are still 1. Then press [CPT] [PV] and the display reads 811.09 as the present value. Accumulated value of annuity-immediate Payments of $100 are received at times 1, 2,, 10 years. The annual effective interest rate is 4%. Find the accumulated value of these payments at time 10 years. Using the BA-35, make sure the financial memories are cleared and the calculator is in financial mode. Since this is an annuity-immediate, we do not want the calculator to be in BEGIN mode. The BA-35 expects the payment amount to be entered as a positive number when payments are discounted backward and it expects the payment amount to be entered as a negative number when the payments are compounded forward. Press 100 [PMT], 10 [N], 4 [%i], [CPT] [FV] and the answer is 1,200.61. Page 4 BPP Professional Education: 2005 exams
Using the BA II Plus, clear the time value of money memory and make sure the calculator is in END mode for end of period and that C/Y and P/Y are set to 1. Press 100 [PMT], 10 [N], 4 [I/Y], [CPT] [FV] and the answer is 1,200.61 as the accumulated value. Present value of annuity-due Payments of $300 are received at times 0, 1, 2,, 9 years. The annual effective interest rate is 5%. Determine the present value of these payments at time 0. Using the BA-35, make sure the financial memories are cleared and the calculator is in financial mode. Since this is an annuity-due, we want the calculator to be in BEGIN mode, so press [2 nd ] [MODE] until BEGIN appears. Press 300 [PMT], 10 [N], 5 [%i], [CPT] [PV] and the answer is 2,432.35. Using the BA II Plus, clear the time value of money memory and make sure the calculator is in BGN mode for beginning of period and that C/Y and P/Y are set to 1. Press 300 [PMT], 10 [N], 5 [I/Y], [CPT] [PV] and the answer is 2,432.35 as the present value. Accumulated value of annuity-due Payments of $15 are received at times 0, 1, 2,, 6 years. The annual effective interest rate is 3%. Determine the accumulated value of these payments at time 7. Using the BA-35, make sure the financial memories are cleared and the calculator is in financial mode. Since this is an annuity-due, we want the calculator to be in BEGIN mode, so press [2 nd ] [MODE] until BEGIN appears. Press 15 [PMT], 7 [N], 3 [%i], [CPT] [FV] and the answer is 118.39. Using the BA II Plus, clear the time value of money memory and make sure the calculator is in BGN mode for beginning of period and that C/Y and P/Y are set to 1. Press 15 [PMT], 7 [N], 3 [I/Y], [CPT] [FV] and the answer is 118.39 as the accumulated value. Determining an unknown interest rate, part 1 A bank lends Deb $4,000 now. In 5 years, Deb will pay the bank a lump sum of $4,750 as repayment of the loan and interest. What is the annual effective interest rate that the bank has charged on the loan? Using the BA-35, clear the memory and press 4,750 [FV], 5 [N], 4,000 [PV], and then [CPT] [%i], and the answer is 3.497%. Using the BA II Plus, clear the time value of money memory and make sure that C/Y and P/Y are set to 1. Press 4,750 [FV], 5 [N], 4,000 [PV], [CPT] [I/Y] and the display reads 3.50, so the annual effective interest rate is 3.50%. BPP Professional Education: 2005 exams Page 5
SOA Course FM / CAS Exam 2 Determining an unknown interest rate, part 2 You invest $5,000 now in order to receive payments of $960 at times 1, 2, 3,, 7 years. Find the annual effective interest rate. Using the BA-35, clear the memory and make sure the calculator is in END mode. Press 5,000 [PV], 960 [PMT], 7 [N], [CPT] [%i] and the answer is 7.99%. Using the BA II Plus, clear the memory, make sure the calculator is in END mode, and make sure the C/Y and P/Y are set to 1. Press 5,000 [PV], 960 [PMT], 7 [N], [CPT] [I/Y] and the answer is 7.99%. Determining an unknown length of time Karen invests $47,673 now in order to receive $4,500 at the end of each year for n years, starting at the end of this year. Using an annual effective interest rate of 7%, find n. Using the BA-35, clear the memory and make sure the calculator is in END mode. Press 4,500 [PMT], 7 [%i], 47,673 [PV], [CPT] [N], and the answer is 20 years. Using the BA II Plus, clear the memory, make sure the calculator is in END mode, and make sure that C/Y and P/Y are set to 1. Press 4,500 [PMT], 7 [I/Y], 47,673 [PV], [CPT] [N], and the answer is 20 years. Deferred payments You invest $320.74 now in order to receive $40 at the start of each year for n years, with the first payment in exactly 5 years. Using an annual effective interest rate of 5.6%, find n. We must first convert the problem into one where the first payment takes place at the start or end of the first period, so we need to know the accumulated value of $320.74 when the payments start in 5 years. Using the BA-35, clear the memory and press 5.6 [%i], 320.74 [PV], 5 [N], [CPT] [FV], and the accumulated value at time 5 years is 421.185. Without clearing this value, clear the memory by pressing [CMR] ([2 nd ] [CE/C]), then press [PV] to enter 421.185 as the present value of the payments at time 5. Make sure BEGIN is displayed on the calculator, then press 40 [PMT], 5.6 [%i], [CPT] [N], and the answer is 15 years. Using the BA II Plus, clear the memory and make sure the calculator is in BGN mode. Press 5.6 [I/Y], 320.74 [PV], 5 [N], [CPT] [FV], and the accumulated value at time 5 years is 421.18. Press [PV] to enter this value as the present value at time 5. Press 40 [PMT], 5.6 [I/Y], 0 [FV], [CPT] [N], and the answer is 15 years. Page 6 BPP Professional Education: 2005 exams
Converting from effective interest rates to nominal interest rates Find the nominal interest rate convertible quarterly when the annual effective interest rate is 7%. That is, find i (4) when i = 0.07. Using the BA-35, clear the memory and enter the annual effective interest rate followed by the [APR] key ([2 nd ], 3), then the pthly conversion period to get i ( p). Press 7, [APR], 4 [=], (4) and the answer is i = 6.823%. Using the BA II Plus, go to the interest conversion worksheet by pressing [2 nd ] [ICONV] and press [2 nd ] [CLR WORK]. Press [ ] until EFF= appears. Press 7 [ENTER]. Press [ ] until C/Y= appears and press 4 [ENTER] so that C/Y= 4 conversion periods per year. Press [ ] until NOM= appears and then [CPT] to determine the nominal interest rate convertible quarterly of 6.823%. Converting from nominal interest rates to effective interest rates Find the annual effective interest rate if the nominal interest rate convertible three times a (3) year is 6%. That is, find i when i = 6%. Using the BA-35, clear the memory and enter the nominal interest rate followed by the [EFF] key ([2 nd ], 1), then the pthly conversion period to get i. Press 6, [EFF], 3 [=], and the answer is i = 6.1208%. Using the BA II Plus, go to the interest conversion worksheet by pressing [2 nd ] [ICONV] and press [2 nd ] [CLR WORK]. Press [ ] until NOM= appears. Press 6 [ENTER]. Press [ ] until C/Y= appears and press 3 [ENTER] so that C/Y= 3 conversion periods per year. Press [ ] until EFF= appears and then [CPT] to determine the annual effective interest rate of 6.1208%. Accumulated value given a nominal interest rate Find the accumulated value at time 7 years of a payment of $30 at time 0. Use a nominal interest rate of 6% a year convertible quarterly. Using the BA-35, first find the annual effective interest rate. Clear the memory and press 6, [EFF], 4 [=] to get the annual effective interest rate of 6.13636%. This should be entered as the interest rate, so press [%i]. Then press 7 [N], 30 [PV], [CPT] [FV], and the answer is 45.52. Using the BA II Plus, first determine the annual effective interest rate. Go to the interest conversion worksheet by pressing [2 nd ] [ICONV] and clear the memory. Press [ ] until NOM= appears. Press 6 [ENTER]. Press [ ] until C/Y= appears and press 4 [ENTER] so that C/Y= 4 conversion periods per year. Press [ ] until EFF= appears and then [CPT] to determine the annual effective interest rate of 6.13636%. BPP Professional Education: 2005 exams Page 7
SOA Course FM / CAS Exam 2 Press [2 nd ] [QUIT] to return to the calculator mode. Press 6.13636 [I/Y], 7 [N], 30 [PV], [CPT] [FV], and the answer is 45.517. Converting from effective interest rates to nominal discount rates Find the nominal discount rate convertible monthly if the annual effective interest rate is (12) 8%. That is, find d when i = 0.08. Using the BA-35, enter the annual effective interest rate followed by [APR], then the pthly conversion period, remembering to use a negative, to get d ( p). Clear the memory and press 8, [APR], followed by 12 [=], and the nominal discount rate convertible monthly is 7.6715%. Using the BA II Plus, it is not as straightforward to determine the nominal discount rate. It ( p) p d is better to use the relationship 1+ i = 1 to determine the nominal discount rate p compounded monthly from the annual effective interest rate. However, if an adjustment is made to one of the input items, the nominal discount rate convertible monthly can be determined using the BA II Plus. Press [2 nd ] [I CONV] then press [2 nd ] [CLR WORK]. Press the up or down arrow key until EFF= is displayed. Enter (0.08 / (1 + (0.08 / 12))) * 100 [=], and 7.94702 is displayed. Press [ENTER]. Press the up or down arrow key to make sure C/Y= 12. Press the up or down arrow key until NOM= is displayed. Press [CPT], and the answer of 7.6715 is shown. Converting from nominal discount rates to effective interest rates Find the annual effective interest rate if the nominal discount rate convertible (2) semiannually is 2%. That is, find i when d = 0.02. Using the BA-35, enter the nominal discount rate followed by [EFF], then the pthly conversion period, remembering to use a negative, to get i. Clear the memory and press 2, [EFF], followed by 2 [=], and the annual effective interest rate is 2.0304%. Using the BA II Plus, it is not as straightforward to determine the annual effective interest ( p) p d rate. It is better to use the relationship 1+ i = 1. However, if we make an p adjustment to one of the input items, we can determine the annual effective interest rate using the BA II Plus. Press [2 nd ] [I CONV] then Press [2 nd ] [CLR WORK]. Press the up or down arrow key until NOM= is displayed. Enter (0.02 / (1 (0.02 / 2))) * 100 [=], and 2.020202 is displayed. Press [ENTER]. Press the up or down arrow key to make sure C/Y= 2. Press the up or down arrow key until EFF= is displayed. Press [CPT], and the answer of 2.0304 is shown. Page 8 BPP Professional Education: 2005 exams
Accumulated value given a nominal discount rate Find the accumulated value at time 10 years of a payment of $1,000 at time 5. Use a nominal discount rate of 5% per year convertible every other month. Using the BA-35, clear the memory. We find the annual effective interest rate by pressing 5, [EFF], followed by 6 [=], and we get the annual effective interest rate of 5.149%. We need to enter this as the interest rate, so press [%i]. Then press 5 [N], 1,000 [PV], [CPT] [FV], and the answer is 1,285.37. Using the BA II Plus, we find the annual effective interest rate by pressing [2 nd ] [I CONV] then press [2 nd ] [CLR WORK]. Press the up or down arrow key until NOM= is displayed. Enter (0.05 / (1 (0.05 / 6))) * 100 [=], and 5.04202 is displayed. Press [ENTER]. Press the up or down arrow key to make sure C/Y= 6. Press the up or down arrow key until EFF= is displayed. Press [CPT], and the annual effective interest rate of 5.14914 is shown. Press [2nd] [QUIT]. Press [2nd] [P/Y] to make sure P/Y= 1 and C/Y= 1 by using the up and down arrows to navigate between the two, then [2nd] [QUIT]. Then press 5.14914 [I/Y], 1,000 [PV], 5 [N], [CPT] [FV], and the answer of 1,285.37 is shown. Selected examples from the FM text Chapter 4, Example 4.13 Find the present value of payments of $30 at the end of each quarter for 8 years. Use a nominal rate of interest of 5% a year, convertible monthly. 120 30 30 30 30 30... 30 30 payment 0 1... 8 years Working in years, the present value is 4 30a, which can be rewritten as 120 i a (4) 8 i (4) 8. Using the BA-35, we need to convert the nominal interest rate convertible monthly into an annual effective interest rate, and then convert that into a nominal interest rate convertible quarterly. Then we set the calculator to calculate the present value of the annuity. Clear the memory and make sure the calculator is in END mode. Press 5, [EFF], followed by 12 [=] to get the annual effective interest rate of 5.1162%. Press [%i], followed by [APR], (4) 4 = to get i = 5.0209%. Then press [STO]. We now have the annual effective interest rate stored in the financial memory and the nominal rate convertible quarterly stored in the memory. BPP Professional Education: 2005 exams Page 9
SOA Course FM / CAS Exam 2 Press 120 [PMT], 8 [N], [CPT] [PV] to get 771.96, which is 120a 8. The press [ ] (the multiply key) followed by [RCL] [%i], then [ ] followed by [RCL] [ MEM ] ([2 nd ], 0) then (4) [=], and we get the answer of 120a = 786.61. 8 Using the BA II Plus, first clear the financial memories by pressing [2 nd ] [CLR TVM]. Since we are working with an annuity-immediate, we want the calculator to be in END mode. To check which mode the calculator is in, press [2 nd ] [BGN]. If the calculator shows END, then press [CE/C] or [2 nd ] [QUIT]. If the calculator shows BGN, change it to END by pressing [2 nd ] [SET] and [CE/C]. For this problem, we have 4 payments of $30 per year but interest is compounded 12 times per year. Press [2 nd ] [P/Y] to see the payment frequency. If P/Y= is not set to 4, press 4 then [ENTER]. To check the compounding frequency, press the up or down arrow key until C/Y= is shown. Press 12 then [ENTER]. To leave this mode, press [2 nd ] [QUIT]. A word of caution is warranted here. If the payment frequency is changed and the mode is exited, the compounding frequency is automatically changed to match the payment frequency. If the compounding frequency is different from the payment frequency, make sure that the compounding frequency is the last setting changed before exiting this mode. If the payment frequency is changed after the compounding frequency, the compounding frequency will be automatically changed to match the payment frequency. To determine the present value, we need to calculate the number of payments first. There are 32 payments since the payments are made each quarter for 8 years. Each periodic payment is $30 and the nominal interest rate is 5% per year, convertible monthly. To determine the present value, press 32 [N], 30 [PMT], 5 [I/Y], and then [CPT] [PV]. The correct answer of 786.61 should be displayed. Since we entered the payments as a negative number, the present value appears as a positive number. Time value of money calculations require both a cash inflow and a cash outflow. Inflows are positive and outflows are negative. It doesn t matter if the payments are entered as a positive or a negative number as long as we re consistent. Chapter 5, Example 5.18 A loan of $15,000 is repaid, using the amortization method, by monthly payments at the end of each month for 8 years. The nominal rate of interest convertible quarterly is 8% a year. Find: (i) (ii) (iii) the monthly repayment the principal outstanding at the end of the 4 th year after the payment has been made the interest and principal repaid in the 49 th payment. Page 10 BPP Professional Education: 2005 exams
Using the BA-35: Part (i) Since the loan payments are in months, let s work in months. We need to determine the monthly effective interest rate. First we clear the memory and make sure the calculator is in END mode. To convert the nominal interest rate convertible quarterly to an annual effective interest rate, press 8, [EFF], 4 [=], and we get the annual effective interest rate of 8.2432%. To convert the annual effective interest rate to a nominal interest rate convertible monthly, press [APR], 12 [=], and we get the nominal interest rate convertible monthly is 7.9473%. To convert this to a monthly effective interest rate, press [ ] 12 [=], and we get the monthly effective interest rate is 0.66227%. Then press [%i], 15,000 [PV], 8 [x] 12 [=] 96 [N], and [CPT] [PMT] to get the monthly payment of 211.65. Part (ii) Press 48 [BAL] and we get 8,678.32 as the balance at time 48. Part (iii) Press 49 [I/P], and we get 57.47 as the interest portion of the 49 th payment. Using the BA II Plus: Part (i) First, clear the financial memories by pressing [2 nd ] [CLR TVM]. Since we are working with an annuity-immediate, we want the calculator to be in END mode. To check which mode the calculator is in, press [2 nd ] [BGN]. If the calculator shows END, then press [CE/C]. If the calculator shows BGN, change it to END by pressing [2 nd ] [SET] and [CE/C]. We also need to make sure that the calculator is set to monthly payments and quarterly compounding. Press [2 nd ] [P/Y] to see the payment frequency. If P/Y= is not set to 12, press 12 then [ENTER]. To check the compounding frequency, press the up or down arrow key until C/Y= is shown. Press 4 then [ENTER]. To leave this mode, press [2 nd ] [QUIT]. We have 8 years of monthly payments, so there are 96 total payments. The nominal interest rate is 8% a year, convertible quarterly. To determine the periodic payment amount, press 96 [N], 15,000 [PV], 8 [I/Y], and then [CPT] [PMT]. The correct answer of 211.65 should be displayed. Part (ii) To determine the principal at the end of year 4 after the payment has been made, we first press [2 nd ] [AMORT]. At the end of year 4, there have been 4x 12 = 48 payments. Press the down arrow key until P1= is shown. If P1 does not equal 1, press 1 [ENTER]. Press the down arrow key until P2= is shown, then press 48 [ENTER]. Press the down arrow key to see the balance displayed as BAL= 8,678.32. Of course, the loan balance is actually 8,678.32. BPP Professional Education: 2005 exams Page 11
SOA Course FM / CAS Exam 2 Part (iii) To determine the interest and principal repaid in the 49 th payment, press the up or down arrow key until P1= is shown. Then press 49 [ENTER]. Press the down arrow key until P2= is shown. Then press 49 [ENTER]. Press the down arrow key again to see that the balance after the 49 th payment is shown as BAL= 8,524.14. Press the down arrow key to see that the amount of principal in the 49 th payment is PRN= 154.17. Press the down arrow key again to see the amount of interest in the 49 th payment is INT= 57.47. Chapter 5, Example 5.1, amended (NPV and IRR) A project requires an initial investment of $50,000. The project will generate net cash flows of $15,000 at the end of the first year, $40,000 at the end of the second year, and $10,000 at the end of the third year. The project s cost of capital is 13%. Calculate the project s net present value and IRR. One of the biggest advantages of using the BA II Plus is being able to determine the IRR and the NPV quickly when the cash flows are evenly spaced. Using the BA-35, we need to use first principles to determine the NPV and the IRR. The NPV is: n net CFt 50, 000 15, 000 40, 000 10, 000 NPV = = + + + = 1, 530.71 t 0 1 2 3 t= 0 ( 1 + r) ( 1.13) ( 1.13) ( 1.13) ( 1.13) The formula for calculating IRR is: CFt NPV = = 0 t ( 1 + IRR) 15, 000 40, 000 10, 000 0 = 50,000+ + + 1 2 3 ( 1+ IRR) ( 1+ IRR) ( 1+ IRR) If we didn t have the fourth term of $10,000 at the end of the third year in the above equation, we could have solved for IRR using the quadratic equation. If this were an actual exam question, we should try plugging one of the answer choices into the formula and use trial and error to narrow down the options until the answer is found. Trial and error is a perfectly legitimate method to arrive at an answer, but it may take a little time and a few iterations before the correct answer is determined. In the above example, if we started with an IRR guess of 10%, we would get an answer of: 15, 000 40, 000 10, 000 50, 000 + + + = 4, 207.36 1 2 3 ( 1.1) ( 1.1) ( 1.1) Since we want the answer to be zero, we need a higher IRR. If our second guess for the IRR is 15%, the answer would be 135.61, which is a lot closer to zero. After a few iterations (or after using linear interpolation), we get an IRR of 14.83% that provides the right answer of zero for the NPV. Page 12 BPP Professional Education: 2005 exams
Using the BA II Plus, to go to the cash flow worksheet, press [CF]. Clear the worksheet by pressing [2 nd ] [CLR WORK]. The initial cash flow CF0= should be displayed. Press 50,000 [ENTER] to input the initial cash flow. Press the down arrow key and press 15,000 [ENTER] to input the first cash flow. Press the down arrow key and press 1 [ENTER] to indicate that this is a single cash flow. Press the down arrow key and press 40,000 [ENTER] to input the second cash flow. Press the down arrow key and press 1 [ENTER] to indicate that this is a single cash flow. Press the down arrow key and press 10,000 [ENTER] to input the third cash flow. Press the down arrow key and press 1 [ENTER] to indicate that this is a single cash flow. Press the [NPV] key. The interest rate for discounting I= should be displayed. Press 13 [ENTER] to input the interest rate. Press the down arrow key and then [CPT]. The net present value of NPV= 1,530.71 should be shown. Press the [IRR] key. Press [CPT] to determine the internal rate of return. The display should indicate IRR= 14.833. Chapter 6, Example 6.6, amended (Bond price) A 5-year $100 par value bond pays semiannual coupons at a rate of 5% per year. The annual effective yield is 7%. Determine the bond price. Using the BA-35, we need to determine the semiannual effective yield. We have: (2) 0.5 i /2 = (1.07) 1 = 3.4408% Press 10 [N], 3.4408 [%i], 2.5 [PMT], 100 [FV], [CPT] [PV], and we get a price of 92.15. Using the BA II Plus, we use the nominal interest rate convertible twice a year. This is: (2) i = 2(0.034408) = 6.8816% Press [2 nd ] [CLR TVM], 10 [N], 6.8816 [I/Y], 2.5 [PMT], and 100 [FV]. We need to make sure the calculator is set for 2 payments a year. Press [2 nd ] [P/Y], and press 2 [ENTER]. Press [2 nd ] [QUIT]. Press [CPT] [PV] to get an answer of 92.15, so the price is 92.15. Chapter 6, Example 6.7, amended (Amortization schedule) Determine the row of the amortization table for the 5 th coupon for the bond in Example 6.6. Using the BA-35, along with the information already input from Example 6.6 for this bond, enter 5 [BAL] to get a balance of 95.75. Enter 5 [I/P] to get the interest portion of the 5 th coupon of 3.27. Enter [x y] to get 0.77, so 0.77 is the amount of discount accumulated in the 5 th coupon. BPP Professional Education: 2005 exams Page 13
SOA Course FM / CAS Exam 2 Using the BA II Plus, along with the information already input from Example 6.6 for this bond, enter [2 nd ] [AMORT], 5 [ENTER] so that P1 = 5. Then press [ ] and 5 [ENTER] so that P2 = 5, then press [ ] and we see that the balance after the 5 th coupon is 95.74 ( BAL = 95.74). Press [ ] and we see that the principal portion of the 5 th coupon is 0.77 ( PRN = 0.77). Press [ ] and we see that the interest portion of the 5 th coupon is 3.27 ( INT = 3.27). Chapter 6, Example 6.12 (i) (Callable discount bond) A 10-year callable bond pays semiannual coupons at a rate of 6% per year. The bond is callable after 5 years at 102. The bond is priced to yield 7% compounded semiannually. Determine the price. With the BA-35, we use first principles and follow the logic from the text. Using the BA II Plus, the calculator can determine the price. Since g < i, this is a discount bond and the issuer will not call the bond early. Instead, the issuer will allow the bond to mature at the redemption amount on the maturity date. Press [2 nd ] [BOND] and then [2 nd ] [CLR WORK]. Since no settlement date is given, we can enter any reasonable date, so press 12.3100 [ENTER] so that the display reads SDT = 12-31-2000. Press [ ] and press 6 [ENTER] so that the annual coupon rate reads CPN= 6. Press [ ] and press 12.3110 [ENTER] so that the redemption date reads RDT = 12-31-2010. Press [ ] and then 100 [ENTER] so that the redemption value reads RV= 100. Press [ ] and [2 nd ] [SET] until ACT appears so that the day count convention is actual / actual. Press [ ] and [2 nd ] [SET] until the display reads 2/Y so that we have 2 coupons paid per year. Press [ ] and 7 [ENTER] so that that the annual yield rate reads as YLD= 6. Press [ ] and [CPT] and the price is determined as PRI= 92.89. Chapter 6, Example 6.12 (ii) (Callable premium bond) A 10-year callable bond pays semiannual coupons at a rate of 6% per year. The bond is callable after 5 years at 102. The bond is priced to yield 5% compounded semiannually. Determine the price. With the BA-35, we use first principles and follow the logic from the text. Using the BA II Plus, the calculator can determine the price. Since g > i, this is a premium bond and the issuer will call the bond early at the first call date. So the issuer will call this bond at time 5 years at the call price of 102. Page 14 BPP Professional Education: 2005 exams
Press [2 nd ] [BOND] and then [2 nd ] [CLR] {WORK]. Since no settlement date is given, we can enter any reasonable date, so press 12.3100 [ENTER] so that the display reads SDT = 12-31-2000. Press [ ] and press 6 [ENTER] so that the annual coupon rate reads CPN= 6. Press [ ] and press 12.3105 [ENTER] so that the redemption date reads RDT = 12-31-2005. Press [ ] and then 102 [ENTER] so that the redemption value reads RV= 102. Press [ ] and [2 nd ] [SET] until ACT appears so that the day count convention is actual / actual. Press [ ] and [2 nd ] [SET] until the display reads 2/Y so that we have 2 coupons paid per year. Press [ ] and 5 [ENTER] so that that the annual yield rate reads as YLD= 5. Press [ ] and [CPT] and the price is determined as PRI= 105.94. Chapter 6, Example 6.13 & 6.14 (Price between coupon dates) A corporate bond pays semiannual coupons at a coupon rate of 8%. The bond is priced to yield 6% compounded semiannually. The bond matures on 6/1/08 and is purchased with a settlement date of 10/1/03. Determine the full price of the bond as of the settlement date, long with the clean price and accrued interest. With the BA-35, we use first principles and follow the logic from the text. Using the BA II Plus, the calculator can determine the price. Press then [2 nd ] [BOND] and [2 nd ] [CLR WORK]. Press 10.0103 [ENTER] so that the display reads SDT= 10-01-2003. Press [ ] and press 8 [ENTER] so that the annual coupon rate reads CPN= 8. Press [ ] and press 6.0108 [ENTER] so that the redemption date reads RDT= 06-01-2008. Press [ ] and then 100 [ENTER] so that the redemption value reads RV= 100. Press [ ] and [2 nd ] [SET] until 360 appears so that the day count convention is 30 / 360. Press [ ] and [2 nd ] [SET] until the display reads 2/Y so that we have 2 coupons paid per year. Press [ ] and 6 [ENTER] so that that the annual yield rate reads as YLD= 6. Press [ ] and [CPT] and the clean price is determined as PRI= 108.02. Press [ ] and we see that the accrued interest is 2.67 ( AI = 2.67 ). We add the clean price and the accrued interest together and we get the full price of 108.02 + 2.67 = 110.69. BPP Professional Education: 2005 exams Page 15