Part 1 – Simple Interest



Mathematics of FinanceCourse BookletCopyright held by Evan van Dyk on a CC by 4.0 licence. Last updated August 2020Contents TOC \o "1-3" \h \z \u Part 1 – Simple Interest PAGEREF _Toc50383872 \h 4Introduction PAGEREF _Toc50383873 \h 4I = PRT and variations PAGEREF _Toc50383874 \h 5Introducing Maturity Value Formula PAGEREF _Toc50383875 \h 8Introducing Principal Value PAGEREF _Toc50383876 \h 9Equal Payments Formula PAGEREF _Toc50383877 \h 11Part 2 – Simple Interest Applications PAGEREF _Toc50383878 \h 13Introduction PAGEREF _Toc50383879 \h 13Promissory Notes PAGEREF _Toc50383880 \h 14Promissory Notes – Finding the Purchase Price of a non-interest bearing note PAGEREF _Toc50383881 \h 15Promissory Notes – Finding the purchase price of an interest-bearing note. PAGEREF _Toc50383882 \h 16Treasury Bills PAGEREF _Toc50383883 \h 17Part 3 – Compound Interest PAGEREF _Toc50383884 \h 18Introduction PAGEREF _Toc50383885 \h 18Introducing the Future Value formula. PAGEREF _Toc50383886 \h 20Introducing Present Value PAGEREF _Toc50383887 \h 22Equal payments and compound interest PAGEREF _Toc50383888 \h 26Part 4 – Further Topics in Compound Interest PAGEREF _Toc50383889 \h 28Introduction PAGEREF _Toc50383890 \h 28Compound interest formula and n PAGEREF _Toc50383891 \h 29Finding i in compound interest PAGEREF _Toc50383892 \h 33Part 5 – Ordinary Annuities PAGEREF _Toc50383893 \h 35Introduction PAGEREF _Toc50383894 \h 35Ordinary Annuities – Future Value PAGEREF _Toc50383895 \h 39Ordinary Annuities – Present Value PAGEREF _Toc50383896 \h 41General Annuities – Future Value PAGEREF _Toc50383897 \h 43General Annuities – Present Value PAGEREF _Toc50383898 \h 45Part 6 – Ordinary Annuities: Payment Amount and Number of Payments PAGEREF _Toc50383899 \h 48Introduction PAGEREF _Toc50383900 \h 48Calculating payment (ordinary annuity - future value known) PAGEREF _Toc50383901 \h 50Calculating payment (ordinary annuity – present value known) PAGEREF _Toc50383902 \h 51Calculating N (ordinary annuities) PAGEREF _Toc50383903 \h 52Calculating payment (general annuities – Future Value known) PAGEREF _Toc50383904 \h 54Calculating payment (general annuities – Present Value known) PAGEREF _Toc50383905 \h 55Calculating n (general annuities) PAGEREF _Toc50383906 \h 56Part 7 – Annuities Due PAGEREF _Toc50383907 \h 59Introduction PAGEREF _Toc50383908 \h 59Ordinary Annuities Due PAGEREF _Toc50383909 \h 61General Annuities Due PAGEREF _Toc50383910 \h 63Part 8 – Annuities: Special Situations (Deferred Annuities, Perpetuities, and Constant-Growth) PAGEREF _Toc50383911 \h 65Introduction PAGEREF _Toc50383912 \h 65Deferred Annuities PAGEREF _Toc50383913 \h 67Ordinary Perpetuities PAGEREF _Toc50383914 \h 69Perpetuities Due PAGEREF _Toc50383915 \h 70General Perpetuities PAGEREF _Toc50383916 \h 71Constant-Growth Annuities PAGEREF _Toc50383917 \h 72Part 9 – Loan Amortization / Loan Repayment Schedules PAGEREF _Toc50383918 \h 74Amortization Schedule PAGEREF _Toc50383919 \h 75Part 10 - Bonds PAGEREF _Toc50383920 \h 77Introduction PAGEREF _Toc50383921 \h 77Bond Pricing - Market Rate Equals Coupon Rate PAGEREF _Toc50383922 \h 79Bond Pricing on Interest Payment Date PAGEREF _Toc50383923 \h 80Bond Pricing between Interest Payment Date PAGEREF _Toc50383924 \h 81Part 11 – Net Present Value PAGEREF _Toc50383925 \h 83Introduction PAGEREF _Toc50383926 \h 83Calculating Net Present Value PAGEREF _Toc50383927 \h 84Midterm 1 Review Questions PAGEREF _Toc50383928 \h 89Midterm 1 Review Answers PAGEREF _Toc50383929 \h 96Midterm 2 Review Questions PAGEREF _Toc50383930 \h 97Midterm 2 Review Answers PAGEREF _Toc50383931 \h 107Final Exam Review Questions PAGEREF _Toc50383932 \h 108Final Exam Review Answers PAGEREF _Toc50383933 \h 120Formula Sheet PAGEREF _Toc50383934 \h 121Part 1 – Simple InterestIntroductionIn this first part, you will learn about interest for periods of less than a year. It is called simple interest because it is an easy method of calculating the amount of interest that accrued on a loan. However, it is only appropriate for short periods of time.New Formulas:Calculating interest: I = PrtCalculating maturity value (simple interest):S = P(1 + rt)Calculating principal (simple interest):P=S(1+rt)Equal payments formula (simple interest):P=S(1+rt1) + S(1+rt2) … S(1+rtn)Each of the variables have the following meaning:I = Amount of interest payableP = Principle (or “Present Value”)R = Rate of interest (expressed as a decimal)T = Time expressed in years (ex: 3 months is expressed as 3/12, 45 days as 45/365)S = Maturity value (or “Future Value”)Note: You can rearrange any of the formulas to solve for the individual variable.I = PRT and variationsCalculate the amount of interest for $7,000 at 4.0% p.a. for 9 months.Calculate the amount of interest for $1,500 at 3% p.a. for 180 days.What principal will earn interest of $50.00 at 9.00% in 10 months?Find the annual rate of interest required for $700 to earn $33.00 in 6 months.Determine the number of months required for a deposit of $1,500 to earn $20.00 interest at 12%Introducing Maturity Value FormulaFormula: S = P(1 + rt).Note: This is a re-writing of the formula, which is S = P + PRTCameron’s Burgers put $22,000 into a short-term investment that matures in 300 days. An investment of this length earns 2.0% p.a. How much will the investment be worth at maturity? (what is the future value?)Introducing Principal ValueFormula: P=S1+rtCalculate the value of an investment 6 months before the maturity date that earns interest at 5% p.a. and has a maturity value of $bining everything into one questionYou are owed payments of $500 due 3 months ago, $1,000 due today, and $1,000 due in 6 months. You have been approached to accept a single payment 2 months from now with interest allowed at 8% p.a. How much will the single payment be?Equal Payments FormulaIntroducing equal payments and the formula P=S(1+rt1) + S(1+rt2) … S(1+rtn)Note: You’ll be solving for S.Jack Robinson owes $2,000 today to a friend. He doesn’t have the money currently, so he asks his friend if he can make two equal payments in 3 and 8 months with interest at 5%. Find the amount of the equal payments.TipsWhen first learning the formulas, you may find it helpful to write down what each variable means on the corner of the page.Always make a timeline. Getting in the habit of making a timeline now will make future content easier.Part 2 – Simple Interest ApplicationsIntroductionPromissory notes are written notes to pay money to another. An example of one can be found below:Source: Mathematics of Finance, with Canadian Applications, 11th Edition, S.A. Hummelbrunner and K.S. Coombs, Prentice Hall, Toronto, Canada. (2018)In this case, the “Face Value” of the note is the principal. It is the amount literally written on the face of the note.Treasury bills, or “T-Bills,” are promissory notes issued by the government of Canada via the Bank of Canada. They do not carry an interest rate. Rather, they have their maturity value stated on the front and bidders purchase the bills at auction.Promissory NotesFind the maturity value of a $2,000, six-month (183 days) promissory note with interest at 2.5%Promissory Notes – Finding the Purchase Price of a non-interest bearing noteTo find the purchase price of a non-interest bearing note, you need to find the present value on the remaining life of the note (180 days – 70 days). The interest rate of the notes in this question is 0%.A 180-day promissory note for $5,000 is discounted at 8%. Calculate the value of the note 70 days after it has been issued.Promissory Notes – Finding the purchase price of an interest-bearing note.A 95-day note for $10,000 at 5% interest has been issued. Find the value of the note 20 days after it has been issued if the rate money is worth is 8%Treasury BillsTreasury Bills do not bear interest. Rather, they are bought and sold at auction. Therefore, the “Face Value” of the T-Bill (the amount written on the face of the treasury bill itself) is the maturity value, or “future value.” Investors bid on the note at a discounted price, and the “yield” is the effective interest rate. An investment dealer bought a 182-day Canadian T-Bill with a face value of $100,000 to yield an annual return of 2.65%. Find the purchase price at the date of issue.Part 3 – Compound InterestIntroductionCompound interest is when you earn interest on interest. The following chart illustrates this comparison ($10,000 loan at 10%. On the right, interest is compounded annually).The interest is “compounded” at certain points in time. In the above example, it is compounded once a year, or annually. It can be compounded at any different interval, including semi-annually, quarterly, monthly, daily, hourly, etc. This affects our calculations. A chart to help you understand this follows:Compounding FrequencyLength of Compounding PeriodNumber of Compounding Periods per Year (m)Annual12 months (1 year)1Semi-annual6 months2Quarterly3 months4Monthly12 months12Daily1 day365Charts source: Mathematics of Finance, with Canadian Applications, 11th Edition, S.A. Hummelbrunner and K.S. Coombs, Prentice Hall, Toronto, Canada. (2018)The following instructions for a financial calculator is for the Texas Instruments BAII+. You can solve these questions on your calculator to check your work. The following is a written reminder about how to use it:Press 2nd, “CLR TVM” before every question.P/Y = payments per year.C/Y = compounding periods per year.I/Y = interest per year (nominal interest rate in %)N = number of compounding periodsPV = Present value / principal.FV = Future value.To solve for FV, enter in everything, press “CPT”, then “FV”. Likewise for “PV”New Formulas:Calculating future value (compound interest):FV=PV1+inFinding i in the compound interest formula:i=jm Calculating present value (compound interest): PV=FV1+inEqual payments formula (compound interest: PV=FV1+in1+ FV1+in2… FV1+innNote: Solve it for FVEach of the variables have the following meaning:FV = Future ValuePV = Present ValueI = Interest rateN = Number of Compounding periodsJ = Nominal (annual) rate of interestM = Number of compounding periods per yearIntroducing the Future Value formula.Formula: FV=PV1+inWhat is the Future Value of a $500 loan at 6% p.a. compounded quarterly for 3.5 years?A deposit of $5,000 earns interest at 7% p.a. compounded quarterly for four years, six months. What is the future value?Introducing Present ValueFormula: PV=FV1+inI want $10,000 5 years from now. If interest is 5% compounded monthly, what would I need to deposit into the bank today?Find the principal that will amount to $10,000 in 6 years at 4% p.a. compounded semi-annually.Find the proceeds (Present Value) of a non-interest bearing note for $3,000 discounted 2 years before maturity. The rate money is worth is 9% compounded bining it all together.Debt payments of $500 due three months ago, $1,000 due today, and $2,000 due in fifteen months are to be combined into one payment due six months from today at 12% p.a. compounded monthly. Find the size of that payment.Equal payments and compound interestFormula: PV=FV1+in1+ FV1+in2… FV1+innNote: You’ll be solving for FV.Craig is due to make a payment of $2,500 now. Instead, he has negotiated to make two equal payments one year and two years from now. Determine the size of the equal payments if money is worth 7.5% compounded quarterly.TipsA common mistake is to write an incorrect number for “n.” Double check that you are calculating it correctly.Part 4 – Further Topics in Compound InterestIntroductionThis chapter introduces logarithms. You will be solving these on your calculator. In summary, when you are trying to solve for a variable that happens to be an exponent (as we are doing with the variable “n” to find the number of compounding periods), you will use logarithms. “ln” stands for “natural logarithm.” It is a logarithm with a base of “e,” which is a mathematical constant that approximately equals 2.718.New Formulas:Calculating n (compound interest): n=lnFVPVln1+iCalculating i (compound interest);i=FVPV1n-1Compound interest formula and nIntroducing the formula n=lnFVPVln1+iHow many years will it take for $10,000, invested at 7.5% p.a compounded quarterly to grow to $20,000?In how many years will $100 to double at 8% compounded semi-annually?A payment of $1000 is due today and another payment of $3000 is due in 5 years. Find the date at which a payment of $3800 will settle the debt if interest is set at 4% p.a. compounded annually.Find the date at which payments of $1,000 due six months ago and $500 due today could be settled by a payment of $2,000, if interest is 9% compounded monthly.Finding i in compound interestIntroducing the formula i=FVPV1n-1Find the nominal rate of interest compounded semi-annually if $3,000 accumulates to $5,201.96 in 8 years.TipsRemember that “n” is equal to the number of compounding periods – NOT the number of years. Therefore, at the end of the question you will need to divide by the number of compounding periods to find the answer in years.Remember to multiply “i” by the number of compounding periods to find the nominal rate of interest.Part 5 – Ordinary AnnuitiesIntroductionAnnuities refer to a series of payments being made on a regular basis. For example, your rental payments, mortgage payments, car payments, or any other kind of regular payments is an annuity.An ordinary annuity is when payments are made at the end of each payment interval. For example, you might pay for a mortgage or your car at the end of every month.A simple annuity refers to the compounding periods and the payment intervals being the same. For example, if your car payments are due monthly, the interest is also compounded monthly.It is possible to use the formulas we’ve learned to this point to solve the questions we are about to do. However, the formulas would be highly tedious. Take the following as an example:What is the future value of a loan at the end of year 5 if the interest rate is 6% compounded annually, and there are five payments of $1,000 at the end of every year?We would solve this using the current formulas like this:-523876278130Year 1NowYear 2Year 3Year 4Year 51000100010001000100000Year 1NowYear 2Year 3Year 4Year 510001000100010001000Source: Mathematics of Finance, with Canadian Applications, 11th Edition, S.A. Hummelbrunner and K.S. Coombs, Prentice Hall, Toronto, Canada. (2018)Instead of doing this, we will use a new formula:FVn=PMT1+in-1iWhere:PMT: the payment being made at regular intervals ($1,000)i: interest per compounding period (compounding annually, so 0.06 / 1 = 0.06)n: number of payments (5 payments)Note: N is always the number of payments in annuities problems.(Math done in question 1 of this part of the workbook)When annuities have the same compounding and payment periods (for example: payments semi-annually and compounding semi-annually), problems the math is straightforward. However, what if the compounding and payment periods are different?We are also introducing a new set of formulas called “general annuities”General Annuities (compounding periods and payment periods are different)c=THE?NUMBER?OF?INTEREST?COMPOUNDING?PERIODS?PER?YEARTHE?NUMBER?OF?PAYMENTS?PER?YEAR138112513970p=1+ic-1p=1+ic-11190625189230FVg=PMT1+pn-1pFVg=PMT1+pn-1pFuture value:1219200169545PVg=PMT1-1+p-npPVg=PMT1-1+p-np Present value: These are better explained by doing examples, as you can find later in this workbook. P is the effective interest rate per payment, and c is a ratio for compounding periods to payments per year. A chart shows some example ratios.First, you find c, followed by p, and then you can safely input the numbers into the formulas. Below is a chart of some sample ratios for C.Source: Mathematics of Finance, with Canadian Applications, 11th Edition, S.A. Hummelbrunner and K.S. Coombs, Prentice Hall, Toronto, Canada. (2018)Before doing any question, you need to ask yourself: are payment periods and compounding periods the same or different? If they are the same, you use the ordinary annuities formulas. If they are different, you use the general annuities formulas, which are both listed for you below.Last, it is essential that you remember that n is equal to the total number of payments.New FormulasOrdinary Annuities (compounding periods and payment periods are the same)Future Value:FVn=PMT1+in-1i-266700112395PVn=PMT1-1+i-ni00PVn=PMT1-1+i-niPresent Value:General Annuities (compounding periods and payment periods are different)c=THE?NUMBER?OF?INTEREST?COMPOUNDING?PERIODS?PER?YEARTHE?NUMBER?OF?PAYMENTS?PER?YEAR138112513970p=1+ic-1p=1+ic-11190625189230FVg=PMT1+pn-1pFVg=PMT1+pn-1pFuture value:1219200169545PVg=PMT1-1+p-npPVg=PMT1-1+p-np Present value: Ordinary Annuities – Future ValueIntroducing FVn=PMT1+in-1iPayments of $1000 each are made at the end of each year for 5 years to a savings account. Find the future value of the account at the end of 5 years. The interest rate is 6% p.a. compounded annually.Paige deposits $200 into her savings account at the end of each quarter for seven years at a rate of 5% p.a. compounded quarterly. How much will be in her account immediately after she makes her last deposit?Ordinary Annuities – Present ValueIntroducing PVn=PMT1-1+i-niPayments of $1000 are made at the end of every year for five years. The nominal interest rate is 6% p.a. compounded annually. What is the value of the annuity today?A house is bought for $5000 down and payments of $1000 at the end of each six months for 15 years. The interest rate is 12% p.a. compounded semi-annually. What is the cash value (present value) of the home?General Annuities – Future Value FVg=PMT1+pn-1p-80010203200 p=1+ic-1 p=1+ic-1 c=THE?NUMBER?OF?INTEREST?COMPOUNDING?PERIODS?PER?YEARTHE?NUMBER?OF?PAYMENTS?PER?YEARNote in this problem how the compounding periods and payment periods are different.To attend school, Sam deposits $2,000 at the end of every six months for four and a half years. What is the future value of the deposits if interest is 6% p.a. compounded quarterly?What is the future value of deposits of $100 made at the end of each year for four years if interest is 4% compounded quarterly?General Annuities – Present ValueIntroducing PVg=PMT1-1+p-npA 30-year mortgage on a condominium requires payments of $1,200 at the end of each month. If interest is 6% compounded semi-annually, what was the mortgage principal [Present Value]?What is the present value of $2,500 paid at the end of each year for 9 years if interest is 8% compounded quarterly?TipsYou should start to get in the habit of asking “are payments at the end of the period or the beginning of the period,” and “are compounding periods the same or different than the payments in a year?” This will inform the formulas you use in the future.The following is a chart about how you solve questions on your financial calculator. The example below is searching for “i” which we have not done in this workbook.Mathematics of Finance, with Canadian Applications, 11th Edition, S.A. Hummelbrunner and K.S. Coombs, Prentice Hall, Toronto, Canada. (2018)Part 6 – Ordinary Annuities: Payment Amount and Number of PaymentsIntroductionThis part is a continuation of the last chapter on annuities. Instead of searching for Present Value and Future Value, you are now searching for the payment amount or N.At the beginning of every question, you should be asking yourself “are compounding periods and payment periods the same or different?” That will bring you to the correct set of formulas.Then, you need to know what the question is asking you. Are you searching for Present Value, Future Value, the Payment Amount, or N? You’ll need to carefully read the question to identify this.Finally, if you’re searching for Payment or N, you’ll know either the present value or future value. You should correctly identify which one to pull the correct formula.There are numerous formulas in this section of Mathematics of Finance. You need to be methodical about how you approach questions to ensure you arrive at the correct one. At the end of this workbook, all the formulas are compiled into a sheet for your convenience.New Formulas:Simple Annuities formulas (compounding periods and payment periods are the same)Payment (Future Value known): PMT=FVni1+in-1Payment (Present Value known): PMT=PVni1-1+i-nN (Future Value known): n=lnFVn×iPMT+1ln1+iN (Present Value known): n=ln1-PVn×iPMT-ln1+iGeneral Annuities formulas (compounding periods and payment periods are different)229552547625PMT=FVgp1+pn-1PMT=FVgp1+pn-1Payment (Future Value known): 2371725-142875PMT=PVgp1-1+p-nPMT=PVgp1-1+p-nPayment (Present Value known): 203835017780n=lnFVgpPMT+1ln1+pn=lnFVgpPMT+1ln1+pN (Future Value known): 221932526670n=ln1-PVgpPMT-ln1+pn=ln1-PVgpPMT-ln1+pN (Present Value known)Calculating payment (ordinary annuity - future value known)When you are trying to calculate the amount of the payment, you will either know the future value or the present value. When you know the future value, you will use the following formula:PMT=FVni1+in-1What deposit made at the end of each month will accumulate to $20,000 in five years at 12% p.a. compounded monthly?Calculating payment (ordinary annuity – present value known)When you are looking for the payment and know the present value, you use this formula: PMT=PVni1-1+i-nCorey bought a car worth $25,000. He put a down payment of $3,000 on the car and financed the remainder over four years at 6% p.a. compounded monthly. How much must Corey pay at the end of each month?Calculating N (ordinary annuities)When searching for N, you use the following formulas (note: N gives you the number of payments, so you will have to divide by the number of payments in a year to find the answer in years):N (Future Value known): n=lnFVn×iPMT+1ln1+iN (Present Value known): n=ln1-PVn×iPMT-ln1+iHow much time will it take for $100 deposited at the end of each quarter to amount to $2,500 at 6% p.a. compounded quarterly?How many years would it take you to repay $5000 by making payments of $75 at the end of every month at an interest rate of 6% p.a. compounded monthly?Calculating payment (general annuities – Future Value known)Recall that you will use the general annuities formulas when payment periods and compounding periods are different.Introducing PMT=FVgp1+pn-1Macy wants to save $30,000 over the next four years to buy a new car. What monthly payment must she make over that time if interest is 3% compounded semi-annually?Calculating payment (general annuities – Present Value known)Introducing PMT=PVgp1-1+p-nAnn received a $50,000 loan at 4% compounded semi-annually. What monthly payment will repay the loan in 9 years?Calculating n (general annuities)Introducing n=lnFVgpPMT+1ln1+pAnd n=ln1-PVgpPMT-ln1+pHow many months will it take for monthly deposits of $500 to accumulate to $100,000 at 6.5% compounded semi-annually?How many months will it take you to repay $20,000 if you repay $1,500 a month and interest is 7% compounded quarterly? TipsN is equal to the number of payments.When using your financial calculator, you must set the “P/Y” and “C/Y” functions. The following instructions will assist you with this (note: this assumes a TI BAII+ calculator):Mathematics of Finance, with Canadian Applications, 11th Edition, S.A. Hummelbrunner and K.S. Coombs, Prentice Hall, Toronto, Canada. (2018)Part 7 – Annuities DueIntroductionAn annuity due means that payments are due at the beginning of the period. An excellent example of this is your rental payments for housing. You typically pay rent in advance of the month you intend to live in a unit. This typically changes your timeline as follows:Source: Mathematics of Finance, with Canadian Applications, 11th Edition, S.A. Hummelbrunner and K.S. Coombs, Prentice Hall, Toronto, Canada. (2018)The new formula for simple annuities due is FVndue=PMT1+in-1i1+i. The only difference between this and the ordinary annuities formulas is the addition of the (1+i) at the end. In the case of general annuities (compounding periods and payments per year are different), the formula becomes (1+p).The primary difficulty students have is in ensuring they have correctly identified that payments are due at the beginning of the period so you can make a correct timeline.These formulas function the exact same way as the previous annuities formulas we have done to this point with minor variations.Note: At the beginning of every question, you MUST follow these steps to ensure you get the correct formula:Are payments at the beginning or end of the period? Are payment intervals and compounding periods the same or different?What am I trying to solve for? (FV, PV, PMT, N, etc.)New Formulas:Simple Annuities Due(Payment at beginning of period, compounding periods and the payment periods are the same):131445056515FVndue=PMT1+in-1i1+iFVndue=PMT1+in-1i1+iFuture value:1209675160655PVndue=PMT1-1+i-ni1+iPVndue=PMT1-1+i-ni1+iPresent value:19431008890PMT=FVndue1+in-1i1+iPMT=FVndue1+in-1i1+iPayment (Future Value known): 2524125131445PMT=PVndue1-1+i-ni1+iPMT=PVndue1-1+i-ni1+iPayment (Present Value known)General Annuities Due(Payment at beginning of period, compounding periods and the payment periods are different):center198120FVgdue=PMT1+pn-1p1+pFVgdue=PMT1+pn-1p1+pFuture value:342900131445PVgdue=PMT1-1+p-np1+pPVgdue=PMT1-1+p-np1+pPresent value: 226695083820PMT=FVgdue1+pn-1p1+pPMT=FVgdue1+pn-1p1+pPayment (future value known):22098008255PMT=PVgdue1-1+p-np1+pPMT=PVgdue1-1+p-np1+pPayment (present value known): Ordinary Annuities DueNote: You should start to get into the habit of identifying the correct formula you need to use. You do this by following three steps:Are payments at the beginning or end of the period? Are payment intervals and compounding periods the same or different?What am I trying to solve for?This will enable you to use the formula sheet to find the correct formula. You should start with these questions.Find the FV of $100 payments at 6% compounded annually made at beginning of the year for four yearsFind the present value of eight payments of $2000, each made at the beginning of every six months, if money is worth 8% compounded semi-annually.General Annuities DueIntroducing general annuities due (payments at beginning of period, compounding periods per year and payment intervals are different).What deposit made at the beginning of each year will accumulate to $25,000 at 8% compounded quarterly at the end of eight years?TipsYou should be in the habit of identifying the correct formula to use. Once again, you should follow these steps:Are payments at the beginning or end of the period? Are payment intervals and compounding periods the same or different?What am I trying to solve for?This will enable you to use the formula sheet to find the correct formula.If you are proficient with your calculator, you will be able to check your work quickly with it.Part 8 – Annuities: Special Situations (Deferred Annuities, Perpetuities, and Constant-Growth)IntroductionA deferred annuity does not require any new formulas. Rather, it just means that the first payment is delayed until some point in the future (which is why they are called deferred annuities). You will be using the compound interest formulas to assist in solving these questions. These questions are difficult to solve without reading the question carefully and constructing a timeline.A perpetuity refers to payments being made forever. For example, a university endowment fund, pension plans, dividends, or scholarship funds, are all examples of questions which use perpetuities.Present value when paymentsPer year and compounding periodsPV=PMTiPer year are the same.Payments at end of period.27813007620PV=PMTpPV=PMTpPresent value when paymentsPer year and compounding periodsPer year are differentPayments at end of period.2867025184785PVdue=PMT+PMTiPVdue=PMT+PMTiPresent value when paymentsPer year and compounding periodsPer year are the samePayments at beginning of periodPresent value when payments28765507620PVdue=PMT+PMTpPVdue=PMT+PMTpPer year and compounding periodsPer year are differentPayments at beginning of period.Constant-growth annuities are annuities where the payments are growing at a constant rate. For example, pension payments may grow with time. To find any individual payment, we use the formula PMT(1 + k)n – 1An example chart follows:1st payment = PMT2nd payment = PMT(1 + k)3rd payment = PMT(1 + k)24th payment = PMT(1 + k)3……10th payment = PMT(1 + k)9Size of the nth payment = PMT(1 + k)n – 1---K is the percentage growth for every subsequent payment and N is the payment number. When finding the sum of the payments (which does not include interest), you use the formula PMT1+kn-1k. When looking for the future value, you will use the formula FV=PMT1+in-1+kni-k.Constant-growth annuities:K is equal to the growth in paymentsEvery period.Size of an individual payment(N equals the payment number PMT(1 + k)n – 1PMT equals the first payment)260985035560PMT1+kn-1kPMT1+kn-1kSum of the payments: 226695036195FV=PMT1+in-1+kni-kFV=PMT1+in-1+kni-kFuture value:20383508255PV=PMT1-1+kn1+i-ni-kPV=PMT1-1+kn1+i-ni-kPresent value:Deferred AnnuitiesFind the size of the payment required at the end of every month to repay a 2-year loan of $5,000 if the payments are deferred for two years and interest is 6% compounded monthly.(note: in these questions, the term of the loan starts after the deferment period. In this case, it is 4 years total, 2 years deferment then 2 years of repayments).Find the size of the payment required at the end of every three months to repay a 4-year loan of $9,000 if the payments are deferred for one year and interest is 6% compounded quarterly.Ordinary PerpetuitiesFind the amount of money invested today at 5% compounded annually which will provide a scholarship of $2,000 at the end of every year in perpetuity.Perpetuities DueA tract of land is leased in perpetuity at $2,000 due at the beginning of each month. If money is worth 12% compounded monthly, what is the present value of the lease?General PerpetuitiesWhat sum of money invested today at 6% compounded quarterly will provide a scholarship of $5,000 at the end of each year in perpetuity?Constant-Growth AnnuitiesSeven deposits increasing at a constant rate of 2% are made at the end of each year. The size of the first deposit is $10,000 and the fund earns interest at 8% compounded annually.What is the size of the last deposit?What is the accumulated value of the deposits [Future Value]?TipsDeferred annuities require both reading a question carefully to ensure your timeline is correct, followed by carefully choosing your formulas. They require no new math, but do require attention to detail.Perpetuities formulas are used when payments are made indefinitely. This is also how a lot of property is valued. For example, a good approximation for the market value of a rental property is the amount of rent it earns monthly divided by the market interest rate compounded monthly (for example, let’s say the rental payment is $15,000 annually and interest is 5.5% compounded annually, the property is quickly estimated to be worth $272,727.27. (15,000 / .055)Perpetuities are also used in share purchase questions. Shares of companies are commonly valued by taking the value of the dividend (payment by the company to shareholders) and dividing it by the market interest rate.Part 9 – Loan Amortization / Loan Repayment SchedulesLoan amortization refers to repayment of interest-bearing debts by equal payments at equal periods in time. It is an annuity problem. Debts are typically repaid in regular instalments.These calculations for amortization schedules are frequently done for residential mortgages in Canada. A residential mortgage is purchasing a home with debt, but using the home as security for the loan. Equity is the difference between the fair market value of the property and the total of the debt against it. You can get either a fixed-rate (where the interest is fixed for a term) or a variable-rate mortgage (where the interest rate changes when market conditions change).Loans repaid in fixed instalments include both principal and interest payments. This is known as a blended payment. Banks regularly construct a loan repayment schedule to show current balances.New Formulas:Amortization Loan Schedule Template1234562 x i or p3 - 42 - 5Payment NumberPrevious BalanceAmount PaidInterest PaidPrincipal repaidOutstandingprincipal balanceNote: The second column (previous balance) is optional.Amortization ScheduleTo buy a car, Evan borrowed $3,000 at 8% compounded quarterly. The loan is to be repaid by equal quarterly payments over a one-year term.What is the size of the monthly payment?Construct an amortization schedule.James borrowed $1,000 at 10% p.a. and will be repaying the loan in semi-annual installments of $300. Construct a complete repayment schedule.Part 10 - BondsIntroductionBonds are debt that is issued to a large group of investors by corporations and governments. They have their own set of vocabulary, which includes some of the following:Face value (par value): The amount owed to the holder when it matures.Bond rate (coupon rate): The interest rate paid to the bond-holder at regular intervals.Maturity date (redemption date): The date on which the bond is to be repaid.Investors regularly receive interest while holding onto the bond. These payments are known as coupons. Most bonds are designed to be bought and sold on a market. The bonds may sell above their face value or below their face value, which is known as selling at a premium or a discount.It is essential that you understand the difference between the “interest rate (bond rate)” and the “rate money is worth (market rate).” The “interest rate” is the amount written on the bond itself. The “rate money is worth” is the prevailing market rate today. Interest rates in the market change on a daily basis, but the amount written on the loan itself stays constant. This gets reflected in the price of the bond itself.As an example, if a car dealership is offering 0% financing for a new car, then that is the “interest rate” they are offering. However, 0% is not the prevailing market rate. If you were to go to anyone else in the market and ask for a loan, they would demand the prevailing market rate instead. In our case, let’s say it’s7%. That 7% is the “rate money is worth.”Bringing this back to bonds, let’s suppose you purchased a bond with a face value of $10,000. The coupon rate on the bond is 5%. However, market conditions have changed and the prevailing interest rate in the market for debt of similar risk would now be 8%. You want to sell your bond in the market. You would now be forced to sell your bond for less than $10,000 (at a discount) because investors are not willing to give you full price to only receive 5% when the prevailing rate is 8%. A table below illustrates this:Mathematics of Finance, with Canadian Applications, 11th Edition, S.A. Hummelbrunner and K.S. Coombs, Prentice Hall, Toronto, Canada. (2018)Finally, bond premiums and discounts are amortized for accounting purposes. These schedules are done separately in an accounting course.The chapter on bonds does not involve any new math. Instead, it’s an application of all the math you have already learned to this point. You will be required to calculate some days between dates, but you can use either the functionality on your BA 2 plus calculator, or you can use this free website here:(Note: site functionality last checked August 2020).New Formulas:Cash Price = Market Price* + Accrued Interest*Market price (purchase price of bond on interest payment date) = PV of redemption price + PV of interest PMTsBond Pricing - Market Rate Equals Coupon RateA $20,000, three-year bond bearing interest at 6% payable semi-annually is purchased at a market price of $20,000 two years and 9 months before maturity. The market rate is also 6%. Determine the cash price of the bond.Bond Pricing on Interest Payment DateA $10,000 bond bearing interest at 8% payable semi-annually matures in ten years. If it is bought to yield 4% compounded semi-annually, what is the purchase price of the bond?Bond Pricing between Interest Payment DateTo solve these questions, you will have to find the market price of the bond on the interest date immediately preceding the purchase date, then you will use a compound interest calculation to find the purchase price on the desired date.A savings bond with a face value of $10,000 bearing interest at 5% payable semi-annually is issued on December 1, 2020 and matures on December 1, 2040. What is the purchase price of the bond on August 3, 2024, if the market rate is 7% compounded semi-annually?TipsWhen doing bonds questions, you should first ask if you expect to pay a premium or a discount. If your final answer does not give you what you expected, this will let you know you need to check your work.You should understand the difference between the interest rate and the rate money is worth. You may set an interest rate for 5 years, but the rest of the market isn’t staying stationary for the 5 years of your loan. The market is moving up and down while your loan’s interest rate is staying constant. The market’s movements affect how much you can resell your loan for.Part 11 – Net Present ValueIntroductionNet present value tells you the value of a prospective investment. It is calculated by taking all incoming cash flows subtracted by all outgoing cash flows. This calculation is regularly used to make financial decisions.For example, let’s suppose you have won a scholarship and you are given two options: immediately receiving a payment of $10,000 or receiving a payment of $3,500 at the end of each year for 3 years. Which should you accept? If money is worth 6% compounded annually, you should accept the $10,000 immediately because the present value of the payments in the second option adds only to $9,355.54. In other words, the net present value of the second option is -$644.46.You’ll be doing similar calculation in these questions. If the NPV is greater than 0, you should accept the project. If it is less than 0, you should decline the project.The internal rate of return, also known as return on investment (ROI) is the percentage increase in the value of an investment over time. Another way to think of it is the interest rate where NPV would equal 0. It cannot be calculated by hand without trial and error.Finally, you do not do any new math, but instead have an application of the math you’ve already done.New Formulas:Net?Present?Value(NPV)=Present?Value?of?Inflows-Present?Value?ofoutlaysCalculating Net Present ValueA study of a potential project estimates the following cash flows:Initial outlay: -$1,500,000Years 1-4 outlays: -$200,000 per yearYears 5-10 inflows: $500,000 per yearYears 11-20 inflows: $700,000 per year.Residual [leftover] value after 20 years: $400,000Should the venture be undertaken if the required return on investment is 15% compounded annually?Question 1 ContinuedThe Jund Corporation is considering developing a new product. The cash flow estimates are as follows:Initial outlay: -$100,000Years 1 – 2 outlays: -$100,000Years 4 – 15 inflows: $65,000Residual value after 15 years: $30,000If the corporation requires a return on investment of 14% compounded annually, should it develop the new product?Question 2 ContinuedTipsThese questions require a timeline to effectively complete.Midterm 1 Review QuestionsIn how many days will $3,100 grow to $3,195.72 at 5.75%?What is the present value of $3,780 due in nine months if interest is 5%?A loan of $5,000 is to be repaid in three equal instalments due 60, 120, and 180 days after the date of the loan. If interest is 6.9 p.a., calculate the amount of the instalments.Marshall borrowed $15,000 on a demand note with an interest rate of 9% per annum. Payments were made of $2,800 42 days after issue, and $2,400 36 days later. How much is the final payment 45 days after that?Debt payments of $400 due today, $500 due in 18 months, and $900 due in 3 years are to be combined into a single payment due 2 years from now. What is the size of the single payment if interest is 8% p.a. compounded quarterly?In how many years will money double at 8% compounded monthly?Kapil owed $4,000 on his purchase of a new oven for his bakery business. He repaid $1,500 in 9 months, $2,000 in 18 months, and the remaining amount owed in 27 months. If interest is 10% compounded quarterly, what is the amount of the final payment?Find the present value of a $1,600 promissory note 6 years, 4 months before maturity discounted at 5% compounded quarterly.Midterm 1 Review Answers196 days$3,643.37$1,704.33$10,174.66$1,820.328.6932 years$1,102.13$1,168.20Midterm 2 Review QuestionsDeposits increasing at a constant rate of 4% are made quarterly for six years from an account earning 6% compounded quarterly. The first deposit is $1,500.What is the size of the last deposit?What is the future value of the deposits?Sabia needs to save $14,000 for tuition. How much would she have to pay into an account at the beginning of every three months over three years if interest is 7% compounded semi-annually?Jennifer rents a suite and pays $1,150 in monthly rent in advance. What is the cash value (purchase price) of the property if money is worth 6.6% compounded monthly?Cameron won $8000 as a business grant. The money was deposited into a savings account earning 4.2% compounded monthly. He intends to leave the money for 5.5 years, then withdraw amounts at the end of each month for the next 4 years while he studies to become an business person. What will be the size of each withdrawal?A mortgage of $95,000 is to be amortized by monthly payments over 25 years. If the payments are made at the end of each month and interest is 8.5% compounded semi-annually, what is the size of the monthly payments?What is the cash value [present value] of a lease requiring payments of $750 at the beginning of each month for three years if interest is 8% compounded quarterly?An annuity purchased for $9,000 makes month-end payments for seven years and earns interest at 5% compounded quarterly. If payments are deferred for three years, how much is each payment?Lamarche borrowed $14,000 at 6.5% compounded semi-annually. If the loan is to be repaid in equal semi-annual payments over 3 years and the first payment is due 4 years after the date of the loan, what is the size of the semi-annual payment?Ryan plans to invest in a property that after three years will give her $1,200 at the end of each month indefinitely. How much should Ryan be willing to pay if an alternative investment yields 9% compounded monthly? [Perpetuity and deferred annuity problem combined]On December 31, Sabia borrowed $2,400, agreeing to repay the loan with blended payments (interest and principle) of $292 per month, starting next month on January 31. Interest was charged at 7.8% compounded annually calculated on the monthly unpaid balance. Construct a repayment schedule.Midterm 2 Review AnswersA) 3,697.07 b) 111,422.29$1,041.42$210,240.91$228.38$755.60$24,111.08$147.55$3,259.68$122,263.83Payment NumberBalance before payment ($)Amount Paid ($)Interest Paid ($)Principal Repaid ($)Balance after payment ($)01,60011,600300122881,31221,3123009.84290.161,021.8431,021.843007.66292.34729.504729.503005.47294.53434.975434.973003.26296.74138.236138.23139.271.04138.230.00Totals1,639.2739.271,600Final Exam Review QuestionsCorey Safety is considering purchasing a machine with the following cash flow estimates:Initial outlay: -$140,000Years 1 – 10 inflows: $30,000Year 4 outflow: -$20,000Year 7 outflow: -$40,000Should Corey Safety invest in the machine if it requires a return of 12% compounded annually on its investments?Question 1 Continued A $15,000, 2.5% bond is purchased six years and six months before maturity to yield 3% semi-annually. If the bond interest is payable semi-annually, what is the purchase price of the bond?What is the purchase price of a $1,000, 7.5% bond with semi-annual payments maturing in 10 years of the bond is bought to yield 6% compounded semi-annually?ContinuedA debt can be paid by payments of $2000 scheduled today, $2000 scheduled in three years, and $2000 scheduled in six years. What single payment would settle the debt four years from now if money is worth 10% compounded semi-annually?The Thanasi Flip-Flop Company has to make a decision about expanding its production facilities. The cash flow estimates are:Initial outlay: -$60,000Years 1 – 5 inflows: $15,000Year 5 outflow: -$60,000Year 6 – 15 inflows: $10,000Should the expansion project be undertaken if the required rate of return is 12% compounded annually?Question 5 ContinuedA company is considering a project with the following cash flow estimates:Initial outlay: -$15,000Years 1 – 3 outlays: -$15,000Year 4 Inflow: $60,000Year 5 inflow: $30,000Should the project be undertaken if the required rate of return is 16% compounded annually?A $5,000, 4% bond with interest payable annually that matures in seven years is purchased to yield 4.75% compounded annually. Find the purchase price.What single payment 6 months from now would be equivalent to payments of $1,250 due (but not paid) four months ago and $1,550 due in 12 months? Assume money can earn 3.7% compounded monthly.Phoebe Adventures charged interest at 2% p.a. on overdue accounts. An invoice for $21,000, plus interest, was paid 35 days past the due date. How much interest was paid (hint: You’ll need the I = PRT formula for this question)?Final Exam Review Answers$5,142.00$14,560.07$1,115.80$6,805.31-$7,913-$1,268$4,781.03$2,810.71$40.27Formula SheetSimple and Compound Interest FormulasCalculating interest: I = PrtCalculating maturity value (simple interest):S = P(1 + rt)Calculating principal (simple interest):P=S(1+rt)Equal payments formula (simple interest):P=S(1+rt1) + S(1+rt2) … S(1+rtn)Note: Solve it for SCalculating future value (compound interest):FV=PV1+inFinding i in the compound interest formula:i=jm j is the nominal interest ratem is number of compounding periods per yearCalculating present value (compound interest): PV=FV1+inCalculating n (compound interest): n=lnFVPVln1+iCalculating i (compound interest):i=FVPV1n-1Calculating effective rate of interest:f=1+im-1Equal payments formula (compound interest): PV=FV1+in1+ FV1+in2… FV1+innNote: Solve it for FVAnnuities FormulasOrdinary Simple Annuities(payment at end of period, compounding periods and payment periods are the same)-276225182245FVn=PMT1+in-1i00FVn=PMT1+in-1iFuture Value:-266700112395PVn=PMT1-1+i-ni00PVn=PMT1-1+i-niPresent Value:Payment (Future Value known): PMT=FVni1+in-1Payment (Present Value known): PMT=PVni1-1+i-nN (Future Value known): n=lnFVn×iPMT+1ln1+iN (Present Value known): n=ln1-PVn×iPMT-ln1+iOrdinary General Annuities(payment at end of period, compounding periods and payment periods are different)c=THE?NUMBER?OF?INTEREST?COMPOUNDING?PERIODS?PER?YEARTHE?NUMBER?OF?PAYMENTS?PER?YEAR138112513970p=1+ic-1p=1+ic-11190625189230FVg=PMT1+pn-1pFVg=PMT1+pn-1pFuture value:1219200169545PVg=PMT1-1+p-npPVg=PMT1-1+p-np Present value: 2324100187325PMT=FVgp1+pn-1PMT=FVgp1+pn-1Payment (Future Value known): 2371725-142875PMT=PVgp1-1+p-nPMT=PVgp1-1+p-nPayment (Present Value known): 203835017780n=lnFVgpPMT+1ln1+pn=lnFVgpPMT+1ln1+pN (Future Value known): 221932526670n=ln1-PVgpPMT-ln1+pn=ln1-PVgpPMT-ln1+pN (Present Value known):Constant Growth AnnuityThe size of the payments grows with every period.K is equal to the growth in paymentsEvery period.Size of an individual payment(N equals the payment number PMT(1 + k)n – 1PMT equals the first payment)260985035560PMT1+kn-1kPMT1+kn-1kSum of the payments: 226695036195FV=PMT1+in-1+kni-kFV=PMT1+in-1+kni-kFuture value:20383508255PV=PMT1-1+kn1+i-ni-kPV=PMT1-1+kn1+i-ni-kPresent value:Simple Annuities Due(Payment at beginning of period, compounding periods and the payment periods are the same):131445056515FVndue=PMT1+in-1i1+iFVndue=PMT1+in-1i1+iFuture value:1209675160655PVndue=PMT1-1+i-ni1+iPVndue=PMT1-1+i-ni1+iPresent value:19431008890PMT=FVndue1+in-1i1+iPMT=FVndue1+in-1i1+iPayment (Future Value known): 2524125131445PMT=PVndue1-1+i-ni1+iPMT=PVndue1-1+i-ni1+iPayment (Present Value known)General Annuities Due(Payment at beginning of period, compounding periods and the payment periods are different):center198120FVgdue=PMT1+pn-1p1+pFVgdue=PMT1+pn-1p1+pFuture value:342900131445PVgdue=PMT1-1+p-np1+pPVgdue=PMT1-1+p-np1+pPresent value: 226695083820PMT=FVgdue1+pn-1p1+pPMT=FVgdue1+pn-1p1+pPayment (future value known):22098008255PMT=PVgdue1-1+p-np1+pPMT=PVgdue1-1+p-np1+pPayment (present value known): PerpetuitiesPayments last forever.Present value when paymentsPer year and compounding periodsPV=PMTiPer year are the same.Payments at end of period.27813007620PV=PMTpPV=PMTpPresent value when paymentsPer year and compounding periodsPer year are differentPayments at end of period.2867025184785PVdue=PMT+PMTiPVdue=PMT+PMTiPresent value when paymentsPer year and compounding periodsPer year are the samePayments at beginning of periodPresent value when payments28765507620PVdue=PMT+PMTpPVdue=PMT+PMTpPer year and compounding periodsPer year are differentPayments at beginning of Present ValueNet?Present?Value(NPV)=Present?Value?of?Inflows-Present?Value?ofoutlaysBondsCash Price = Market Price* + Accrued Interest*Purchase price of bond on interest payment date = PV of redemption price + PV of interest PMTs ................
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