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Chapter 5 Mathematics of Finance

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**Chapter Outline Compound Interest Present Value**

Chapter 5: Mathematics of Finance Chapter Outline Compound Interest Present Value Interest Compounded Continuously Annuities

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**Chapter 5: Mathematics of Finance**

Compound Interest Example 1 – Compound Interest Compound amount S at the end of n interest periods at the periodic rate of r is as Suppose 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.

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**There are 2 × 3 = 6 interest periods.**

Chapter 5: Mathematics of Finance 5.1 Compound Interest Example 1 – Compound Interest Solution: There are 2 × 3 = 6 interest periods. The semiannual rate was 2.75%, so the nominal rate was 5.5 % compounded semiannually.

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**The periodic rate is r = 0.06/4 = 0.015.**

Chapter 5: Mathematics of Finance 5.1 Compound Interest Example 3 – Compound Interest How 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

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**Effective 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

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**Respective effective rates of interest are**

Chapter 5: Mathematics of Finance 5.1 Compound Interest Example 7 – Comparing Interest Rates If 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 are The 2nd choice gives a higher effective rate.

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Exercises Southern 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 ?

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**Money NOW is worth more than money LATER!**

The Time Value of Money Money NOW is worth more than money LATER!

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To 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 value Future value of an annuity Present value of an annuity Perpetuities and growing perpetuities

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**A sum of money today is called a present value.**

We designate it mathematically with a subscript, as occurring in time period 0 For example: P0 = 1,000 refers to $1,000 today

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**A 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.

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**As already noted, the number of time periods in a time value problem is designated by n.**

n may be a number of years n may be a number of months n may be a number of quarters n may be a number of any defined time periods

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**The 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.

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**Annuities Sequences and Geometric Series**

Chapter 5: Mathematics of Finance Annuities Example 1 – Geometric Sequences Sequences and Geometric Series A geometric sequence with first term a and common ratio r is defined as a. The geometric sequence with a = 3, common ratio 1/2 , and n = 5 is

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**b. Geometric sequence with a = 1, r = 0.1, and n = 4. **

Chapter 5: Mathematics of Finance 5.4 Annuities Example 1 – Geometric Sequences b. Geometric sequence with a = 1, r = 0.1, and n = 4. c. Geometric sequence with a = Pe−kI , r = e−kI , n = d. Sum of Geometric Series The sum of a geometric series of n terms, with first term a, is given by

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**Find the sum of the geometric series: **

Chapter 5: Mathematics of Finance 5.4 Annuities Example 3 – Sum of Geometric Series Find the sum of the geometric series: Solution: For a = 1, r = 1/2, and n = 7 Present Value of an Annuity The present value of an annuity (A) is the sum of the present values of all the payments.

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**Chapter 5: Mathematics of Finance**

5.4 Annuities Example 5 – Present Value of Annuity Find 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) = 42 From Appendix A, . Hence,

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**Chapter 5: Mathematics of Finance**

5.4 Annuities Example 7 – Periodic Payment of Annuity If $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,

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The 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

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An 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)n Sn = 1,000(1.10)5 Sn = $1,610.51

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**Another 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?

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**One 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?

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**The 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.

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For 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.

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**A common mathematical symbol for an annuity amount is PMT**

A financial calculator usually has a key labeled PMT Time 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:

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**For the future value of an annuity:**

FV = PMT[(1+i)n - 1]/i

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**The amount S of ordinary annuity of R for n periods at r per period is**

Chapter 5: Mathematics of Finance 5.4 Annuities Example 9 – Amount of Annuity Amount of an Annuity The amount S of ordinary annuity of R for n periods at r per period is Find 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,

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**For the present value of an annuity:**

PV = PMT[(1+i)n -1]/[i(1+i)n]

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An 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]/i FV = 50[(1.01) ]/.01 FV = $49,463

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**Another 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?

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An 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

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**Chapter 5: Mathematics of Finance**

Present Value Example 1 – Present Value P that must be invested at r for n interest periods so that the present value, S is given by Find the present value of $1000 due after three years if the interest rate is 9% compounded monthly. Solution: For interest rate, Principle value is

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**a final payment at the end of five years. **

Chapter 5: Mathematics of Finance 5.2 Present Value Example 3 – Equation of Value A debt of $3000 due six years from now is instead to be paid off by three payments: $500 now, $1500 in three years, and a final payment at the end of five years. What would this payment be if an interest rate of 6% compounded annually is assumed?

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**The equation of value is**

Chapter 5: Mathematics of Finance 5.2 Present Value Solution: The equation of value is

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**Example 5 – Net Present Value**

Chapter 5: Mathematics of Finance 5.2 Present Value Example 5 – Net Present Value Net Present Value You 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. Year Cash Flow 2 $10,000 3 8000 5 6000

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**Solution: Chapter 5: Mathematics of Finance 5.2 Present Value**

Example 5 – Net Present Value Solution:

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**Interest Compounded Continuously**

Chapter 5: Mathematics of Finance Interest Compounded Continuously Example 1 – Compound Amount Compound Amount under Continuous Interest The compound amount S is defined as If $100 is invested at an annual rate of 5% compounded continuously, find the compound amount at the end of a. 1 year. b. 5 years.

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**Effective Rate under Continuous Interest **

Chapter 5: Mathematics of Finance 5.3 Interest Compounded Continuously Effective Rate under Continuous Interest Effective rate with annual r compounded continuously is Present Value under Continuous Interest Present value P at the end of t years at an annual r compounded continuously is .

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**We want the present value of $25,000 due in 20 years.**

Chapter 5: Mathematics of Finance 5.3 Interest Compounded Continuously Example 3 – Trust Fund A 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.

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Exercises Suppose 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 ?

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