2Chapter Outline Compound Interest Present Value Chapter 5: Mathematics of FinanceChapter OutlineCompound InterestPresent ValueInterest Compounded ContinuouslyAnnuities
3Chapter 5: Mathematics of Finance Compound InterestExample 1 – Compound InterestCompound amount S at the end of n interest periods at the periodic rate of r is asSuppose that $500 amounted to $ in a savings account after three years. If interest was compounded semiannually, find the nominal rate of interest compounded semiannually, that was earned by the money.
4There are 2 × 3 = 6 interest periods. Chapter 5: Mathematics of Finance5.1 Compound InterestExample 1 – Compound InterestSolution:There are 2 × 3 = 6 interest periods.The semiannual rate was 2.75%, so the nominal rate was 5.5 % compounded semiannually.
5The periodic rate is r = 0.06/4 = 0.015. Chapter 5: Mathematics of Finance5.1 Compound InterestExample 3 – Compound InterestHow long will it take for $600 to amount to $900 at an annual rate of 6% compounded quarterly?Solution:The periodic rate is r = 0.06/4 =It will take
6Effective Rate or Annual Percentage Yield (APY) If principal P is invested at the annual (nominal) rate r compounded m times a year, then the annual percentage yields is
7Respective effective rates of interest are Chapter 5: Mathematics of Finance5.1 Compound InterestExample 7 – Comparing Interest RatesIf an investor has a choice of investing money at 6% compounded daily or % compounded quarterly, which is the better choice?Solution:Respective effective rates of interest areThe 2nd choice gives a higher effective rate.
8ExercisesSouthern Pacific Bank recently offered a 1-year CD that paid 6.8% compounded daily and Washington Savings Bank offered one that paid 6.85% compounded quarterly. Find the APY (expressed as a percentage, correct to three decimal places) for each CD. Which has the higher return ?A savings and loan wants to offer a CD with a monthly compounding rate that has an effective rate of 7.5%. What annual nominal rate compounded monthly should they use ?
9Money NOW is worth more than money LATER! The Time Value of MoneyMoney NOWis worth more thanmoney LATER!
10To simplify this material as much as possible, you should understand that there are only a few basic types of problems, though each has several variations.Future value or present valueFuture value of an annuityPresent value of an annuityPerpetuities and growing perpetuities
11A sum of money today is called a present value. We designate it mathematically with a subscript, as occurring in time period 0For example: P0 = 1,000 refers to $1,000 today
12A sum of money at a future time is termed a future value We designate it mathematically with a subscript showing that it occurs in time period n.For example: Sn = 2,000 refers to $2,000 after n periods from now.
13As already noted, the number of time periods in a time value problem is designated by n. n may be a number of yearsn may be a number of monthsn may be a number of quartersn may be a number of any defined time periods
14The interest rate or growth rate in a time value problem is designated by i i must be expressed as the interest rate per period.For example if n is a number of years, i must be the interest rate per year.If n is a number of months, i must be the interest rate per month.
15Annuities Sequences and Geometric Series Chapter 5: Mathematics of FinanceAnnuitiesExample 1 – Geometric SequencesSequences and Geometric SeriesA geometric sequence with first term a and common ratio r is defined asa. The geometric sequence with a = 3, common ratio 1/2 , and n = 5 is
16b. Geometric sequence with a = 1, r = 0.1, and n = 4. Chapter 5: Mathematics of Finance5.4 AnnuitiesExample 1 – Geometric Sequencesb. Geometric sequence with a = 1, r = 0.1, andn = 4.c. Geometric sequence with a = Pe−kI , r = e−kI ,n = d.Sum of Geometric SeriesThe sum of a geometric series of n terms, with first term a, is given by
17Find the sum of the geometric series: Chapter 5: Mathematics of Finance5.4 AnnuitiesExample 3 – Sum of Geometric SeriesFind the sum of the geometric series:Solution: For a = 1, r = 1/2, and n = 7Present Value of an AnnuityThe present value of an annuity (A) is the sum of the present values of all the payments.
18Chapter 5: Mathematics of Finance 5.4 AnnuitiesExample 5 – Present Value of AnnuityFind the present value of an annuity of $100 per month for years at an interest rate of 6% compounded monthly.Solution:For R = 100, r = 0.06/12 = 0.005, n = ( )(12) = 42From Appendix A, .Hence,
19Chapter 5: Mathematics of Finance 5.4 AnnuitiesExample 7 – Periodic Payment of AnnuityIf $10,000 is used to purchase an annuity consisting of equal payments at the end of each year for the next four years and the interest rate is 6% compounded annually, find the amount of each payment.Solution:For A= $10,000, n = 4, r = 0.06,
20The first of the general type of time value problems is called future value and present value problems. The formula for these problems is:Sn = P0(1+i)n
21An example problem:If you invest $1,000 today at an interest rate of 10 percent, how much will it grow to be after 5 years?Sn = P0(1+i)nSn = 1,000(1.10)5Sn = $1,610.51
22Another example problem: Assume you will receive an inheritance of $100,000, six years from now. How much could you borrow from a bank today and spend now, such that the inheritance money will be exactly enough to pay off the loan plus interest when it is received? Assume the bank charges an interest rate of 12 percent?How long will it take for $10,000 to grow to $20,000 at an interest rate of 15% per year?
23One more example problem: If you invest $11,000 in a mutual fund today, and it grows to be $50,000 after 8 years, what compounded, annualized rate of return did you earn?
24The next two general types of time value problems involve annuities An annuity is an amount of money that occurs (received or paid) in equal amounts at equally spaced time intervals.These occur so frequently in business that special calculation methods are generally used.
25For example:If you make payments of $2,000 per year into a retirement fund, it is an annuity.If you receive pension checks of $1,500 per month, it is an annuity.If an investment provides you with a return of $20,000 per year, it is an annuity.
26A common mathematical symbol for an annuity amount is PMT A financial calculator usually has a key labeled PMTTime value tables for future value of annuities and for present value of annuities can also be used to simplify calculations.OR, the following formulas can be used:
27For the future value of an annuity: FV = PMT[(1+i)n - 1]/i
28The amount S of ordinary annuity of R for n periods at r per period is Chapter 5: Mathematics of Finance5.4 AnnuitiesExample 9 – Amount of AnnuityAmount of an AnnuityThe amount S of ordinary annuity of R for n periods at r per period isFind S consisting of payments of $50 at the end of every 3 months for 3 years at 6% compounded quarterly. Also, find the compound interest.Solution: For R=50, n=4(3)=12, r=0.06/4=0.015,
29For the present value of an annuity: PV = PMT[(1+i)n -1]/[i(1+i)n]
30An example problem:If you save $50 per month at 12 percent per annum, how much will you have at the end of 20 years?Note that since time periods are months, i = 12%/12 months = 1% per period, for 240 periods.FV = PMT[(1+i)n - 1]/iFV = 50[(1.01) ]/.01FV = $49,463
31Another example problem: If you want to save $500,000 for retirement after 30 years, and you earn 10 percent per annum, how much must you save each year?
32An example problem:If you borrow $100,000 today at 9 percent interest per annum, and repay it in equal annual payments over 10 years, how much are the payments?PV = PMT[(1+i)n -1]/[i(1+i)n]100,000 = PMT[(1+.09)10 -1]/[.09(1.09)10]PMT = $15,582 per year
33Chapter 5: Mathematics of Finance Present ValueExample 1 – Present ValueP that must be invested at r for n interest periods so that the present value, S is given byFind the present value of $1000 due after three years if the interest rate is 9% compounded monthly.Solution:For interest rate,Principle value is
34a final payment at the end of five years. Chapter 5: Mathematics of Finance5.2 Present ValueExample 3 – Equation of ValueA debt of $3000 due six years from now is instead to be paid off by three payments:$500 now,$1500 in three years, anda final payment at the end of five years.What would this payment be if an interest rate of 6% compounded annually is assumed?
35The equation of value is Chapter 5: Mathematics of Finance5.2 Present ValueSolution:The equation of value is
36Example 5 – Net Present Value Chapter 5: Mathematics of Finance5.2 Present ValueExample 5 – Net Present ValueNet Present ValueYou can invest $20,000 in a business that guarantees you cash flows at the end of years 2, 3, and 5 as indicated in the table.Assume an interest rate of 7% compounded annually and find the net present value of the cash flows.YearCash Flow2$10,0003800056000
37Solution: Chapter 5: Mathematics of Finance 5.2 Present Value Example 5 – Net Present ValueSolution:
38Interest Compounded Continuously Chapter 5: Mathematics of FinanceInterest Compounded ContinuouslyExample 1 – Compound AmountCompound Amount under Continuous InterestThe compound amount S is defined asIf $100 is invested at an annual rate of 5% compounded continuously, find the compound amount at the end ofa. 1 year.b. 5 years.
39Effective Rate under Continuous Interest Chapter 5: Mathematics of Finance5.3 Interest Compounded ContinuouslyEffective Rate under Continuous InterestEffective rate with annual r compounded continuously isPresent Value under Continuous InterestPresent value P at the end of t years at an annual r compounded continuously is .
40We want the present value of $25,000 due in 20 years. Chapter 5: Mathematics of Finance5.3 Interest Compounded ContinuouslyExample 3 – Trust FundA trust fund is being set up by a single payment so that at the end of 20 years there will be $25,000 in the fund. If interest compounded continuously at an annual rate of 7%, how much money should be paid into the fund initially?Solution:We want the present value of $25,000 due in 20 years.
41ExercisesSuppose you decide to deposit $100 every 6 months into an account that pays 6% compounded semiannually. If you make six deposits, one at the end of each interest payment period, over 3 years, how much money will be in the account after the last deposit is made ?How much should you deposit in an account paying 6% compounded semiannually in order to be able to withdraw $1,000 every 6 months for the next 3 years ?