Chapter 5 Mathematics of Finance

INTRODUCTORY MATHEMATICAL INTRODUCTORY MATHEMATICAL ANALYSISANALYSISFor Business, Economics, and the Life and Social Sciences

2007 Pearson Education Asia

Chapter 5 Chapter 5 Mathematics of Finance Mathematics of Finance


Compound Interest

Present Value

Interest Compounded Continuously

Annuities

Chapter 5: Mathematics of Finance

Chapter OutlineChapter Outline



Compound InterestCompound Interest

Example 1 – Compound Interest

• Compound amount S at the end of n interest periods at the periodic rate of r is as

nrPS 1

Suppose that $500 amounted to $588.38 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.


Solution:

There are 2 × 3 = 6 interest periods.

The semiannual rate was 2.75%, so the nominal rate was 5.5 % compounded semiannually.


5.1 Compound Interest


0275.01500

38.588

500

38.5881

500

38.5881

38.5881500

6

6

6

6

r

r

r

r


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 = 0.015.

It will take .




233.27015.1ln

5.1ln

5.1ln015.1ln

5.1ln015.1ln

5.1015.1

015.1600900

n

n

n

n

n

months 9 years,68083.6 21

4233.27


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

11

m

e m

rrAPY

11

n

e n

rr




Example 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 are

The 2nd choice gives a higher effective rate.

%27.614

06125.01

and %18.61365

06.01

4

365

e

e

r

r

8

16


Exercises

1. 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 ?

2. 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 ?


The Time Value of Money

Money NOW

is worth more than

money LATER!


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


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


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.


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


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.



AnnuitiesAnnuities

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

0 where,...,,,, 132 aarararara n

432

2

13 ,

2

13 ,

2

13 ,

2

13 ,3



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.

001.0 ,01.0 ,1.0 ,1

dkIkIkI PePePe ,...,, 2

Sum of Geometric Series

• The sum of a geometric series of n terms, with first term a, is given by

1r for

1

11

0

r

raars

nn

i

i



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

62

2

1...

2

1

2

11

64

127

21

1

21

11

1

1

21

128127

7

r

ras

n

Present Value of an Annuity

• The present value of an annuity (A) is the sum of the present values of all the payments.

nrRrRrRA 1...11 21



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,

005.042__100aA

798300.37005.042

__ a

83.3779$798300.37100 A

2

13

2

13



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,

91.2885$465106.3

000,10000,10

000,10

06.04

06.04

____

__

aa

AR

Ra

rn


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


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


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?


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?


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.


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.


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:


For the future value of an annuity:

• FV = PMT[(1+i)n - 1]/i



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,

r

rRS

n 11

06.652$041211.135050015.012

__ S

06.52$501206.652 Interest Compund


For the present value of an annuity:

• PV = PMT[(1+i)n -1]/[i(1+i)n]


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)240 - 1]/.01

• FV = $49,463


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?


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



Present ValuePresent 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 .

nrSP 1

15.764$0075.11000 36 P

0075.012/09.0 r



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?



5.2 Present Value

Solution:

The equation of value is

27.1257$

02.160002.11000 208

x



5.2 Present Value

Example 5 – 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.

Net Present Value

investment Initial - values present of Sum NPV ValuePresent Net

Year Cash Flow

2 $10,000

3 8000

5 6000



5.2 Present Value

Example 5 – Net Present Value

Solution:

31.457$

000,2007.1600007.1800007.1000,10NPV 532



Interest Compounded ContinuouslyInterest 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 ofa. 1 year.

b. 5 years.

kt

k

rPS

1

13.105$100 105.0 ePeS rt

40.128$100100 25.0505.0 eeS



5.3 Interest Compounded Continuously

Effective Rate under Continuous Interest

• Effective rate with annual r compounded continuously is .1 r

e er

Present Value under Continuous Interest

• Present value P at the end of t years at an annual r compounded continuously is .rtSeP



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.

6165$000,25

000,254.1

2007.0

e

eSeP rt


Exercises

1. 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 ?

2. 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 ?

Chapter 5 Mathematics of Finance

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