Chapter 4 Mathematics of Finance - Creighton University · Chapter 4 Mathematics of Finance D. S....

Chapter 4Mathematics of Finance

D. S. MalikCreighton University, Omaha, NE

D. S. Malik Creighton University, Omaha, NE () Chapter 4 Mathematics of Finance 1 / 22

Interest

Two types of Interests

SimpleCompound


Simple Interest

Interest is �xed for the duration of the deposit or investment.

ExampleLisa wants to open a new account and deposit $1000. The bank agrees topay her the simple interest at the rate of 8% per year. Suppose no moredeposits are made in the account. Also suppose I denotes the interestamount at the end of the period.

What is the interest amount at the end of one year? How muchmoney will Lisa have at the end of one year?

Because the interest is simple, the bank pays 8% of the amount in theaccount. So the interest the bank pays is:

I = 1000 � 0.08 = 80.

The total amount at the end of one year is 1000+ 80 = $1080.


ExampleWhat is the interest amount at the end of �ve years? How muchmoney will Lisa have at the end of �ve years?

As in previous part, the interest is simple. Here we need to calculatethe interest for �ve years. Thus,

I = 1000 � (0.08) � 5 = 400.

The total interest at the end of �ve years is $400. The total amountat the end of �ve years is $1000+ $400 = $1400.


RemarkWhen the interest rate is speci�ed, it is usually speci�ed as an annualinterest rate, i.e., an interest rate per year.

For example, in the preceding example, the interest rate is 8% peryear.

Note that we multiplied 1000 with 0.08 rather than 8.

This is for the following reasons:

The interest rate is 8%. This is read as 8 percent, which means 8 forevery 100.This is equivalent to 8

100 = 0.08 for every 1.This can be interpreted as 0.08 of 1.


Formula to Compute Simple Interest

Suppose that the amount P is invested at the simple interest rate ofr percent per year for t years.

Let I denote the total interest and A be the total amount after tyears. Then

I = Prt,

andA = P + I = P + Prt = P(1+ rt).

P is called the principal amountr is called the interest rate per year or annual interest ratet is the number of years the amount P is invested.


RemarkIn a problem, we typically specify interest rate as a percent, such as7.25%.

During calculations, we convert a percent into a decimal number.

For example, 7.25% is converted into 0.0725.

We therefore say that as a percent the interest rate is 7.25 and as adecimal number the interest rate is 0.0725.

When we give the description of a formula, we typically say that theinterest rate is r percent per year or r% per year.

When we write the formula, r is expressed as a decimal number.


ExampleAn amount of $1500 is invested at the simple interest rate of 7.5%annually.

1 What is the total amount of interest after 12 year? What is the totalmoney at the end of 12 year.Here P = 1500, r = 0.075, and t = 1

2 . Let I denote the interestamount and A be the total amount at the end of 12 year. Then

I = Prt = 1500 � (0.075) � 12= 56.25.

Also,A = P + Prt = 1500+ 56.25 = 1556.25.

Hence, the interest is $56.25 and the total amount is $1, 556.25.


Example2. What is the total amount of interest after 6 years? What is the totalmoney at the end of 6 years?Here P = 1500, r = 0.075, and t = 6. Let I denote the interestamount and A be the total amount at the end of 6 years. Then

I = Prt = 1500 � (0.075) � 6 = 675.

Also,A = P + Prt = 1500+ 675 = 2175.

Hence, the interest is $675 and the total amount is $2, 175.


Compound Interest

De�nitionSuppose that P dollars is invested at the interest rate of r percent peryear for t years and the interest is calculated m times a year.

Suppose that the interest is added to the balance at the end of eachperiod and the interest for the next period is calculated on thebalance at the end of the previous period.

In this case, we say that the interest is compounded periodically.


Suppose that the interest is compounded m times a year. Then

If the interest is compounded monthly, then m = 12.

If the interest is compounded quarterly, then m = 4.

If the interest is compounded semiannually, then m = 2.

If the interest is compounded daily, then m = 365 or 366, dependingon whether the year is a leap year.

If the interest is compounded annually, then m = 1.

If the interest is compounded weekly, then m = 52.

If the interest is compounded biweekly (twice a month), then m = 26.

If the interest is compounded bimonthly (every other month), thenm = 6.


In order to calculate the compound interest, the �rst thing that weneed to do is to �nd the interest rate per period.

Note that the interest rate r is for the year, not necessarily for eachperiod.

Suppose that i denotes the interest per period.If the interest is calculated m times a year, then

i =rm.

Moreover, if the interest is calculated m times a year for t years, thenthe number of times the interest is calculated is mt.

Let n denote the number of interest periods, then

n = mt.


ExampleSuppose that the interest is compounded monthly at the rate of 8.4%per year for 5 years.

Then r = 0.084, m = 12, and t = 5. Hence,

i =rm=0.08412

= 0.007

andn = mt = 12 � 5 = 60.


TheoremSuppose that P dollars is invested at the interest rate of r percent per yearfor t years. Suppose that the interest is compounded m times a year andA denotes the total amount at the end of t years. Then

A = P(1+ i)n,

wherei = r

m , is the interest per period, andn = mt is the number of interest periods.


ExampleSuppose $1, 500 is deposited in an account that pays 6% interest peryear compounded quarterly.

Let us determine the total amount at the end of the sixth year.

Here P = 1500, r = 0.06, m = 4, and t = 6.

Thus, i = rm =

0.064 = 0.015 and n = mt = 4 � 6 = 24.

Therefore, the total amount, A, at the end of the sixth year is

A = P(1+ i)n = 1500(1+ 0.015)24 = 1500 � 1.01524 = 2144.25.

Hence, the total amount is $2, 144.25.


ExampleRavi won a jackpot worth $100, 000. He wants to buy a home after 8 yearsand would like to make a down payment of $80, 000. His bank o¤ers anaccount paying 7.25% interest per year compounded monthly. Ravi wantsto know how much money, out of $100, 000, should be deposited in thataccount so that he will have $80, 000 at the end of 8 years.Note that Ravi wants $80, 000 at the end of 8 years. That is, the totalamount, A, to be accumulated is given. So we want to determine theamount, P, that is needed to be deposited now.


Example

Here A = 80, 000, r = 0.0725, m = 12, and t = 8. Thus, i = rm =

0.072512

and n = mt = 12 � 8 = 96. We want to �nd P. Now

A = P(1+ i)n.

This implies that

P =A

(1+ i)n=

80000�1+ 0.0725

12

�96 = 44870.10.Hence, Ravi needs to deposit $44, 870.10 in the account to accumulate$80, 000 at the end of 8 years.


De�nition(Present Value) The present value of an amount A which is to be paidor accumulated after n periods of time at the interest rate of i per periodis the principal amount, P, which, when invested at the interest rate of iper period for n periods, grows to A.

TheoremLet P be the present value of an amount A which is to be paid oraccumulated after n periods of time at the interest rate of i per period.Then

P =A

(1+ i)n.


E¤ective Rate (Annual Percent Yield)

De�nitionSuppose that the interest is r percent per year compounded m timesa year.

Let Ir denote the interest earned at the end of the year with thisinterest rate.

Suppose that the interest rate is re percent per year compoundedannually and let Ie be the interest earned at the end of the year withthis interest rate.

Then re is called the e¤ective rate or annual percent yieldcorresponding to the interest rate r if

Ir = Ie .

Also, r is called the nominal rate.


Theorem(E¤ective Rate or Annual Percent Yield)(i) The e¤ective rate or the annual percent yield, re , corresponding to agiven interest rate, r , compounded m periods per year is given by

re =�1+

rm

�m� 1.

(ii) Suppose that P dollars increases to an amount of A dollars in t years.Then the e¤ective rate or annual percent yield re is

re =�AP

� 1t

� 1.


ExampleWe �nd the e¤ective rate for an account in which the interest iscompounded monthly at the rate of 7.5% per year.Here r = 0.075 and m = 12. Thus,

re =�1+

rm

�m� 1 =

�1+

0.07512

�12� 1 = 0.07763.

Hence, the e¤ective rate is 7.763%.


Exercise: What is the e¤ective rate for an account in which interest iscompounded quarterly at the rate of 8% per year?

Solution: Here r = 0.08 and m = 4. Thus,

re =�1+

rm

�m� 1 =

�1+

0.084

�4� 1 � 0.0824.

Hence, the e¤ective rate is 8.24%.


Chapter 4 Mathematics of Finance - Creighton University · Chapter 4 Mathematics of Finance D. S....

Documents

Transcript of Chapter 4 Mathematics of Finance - Creighton University · Chapter 4 Mathematics of Finance D. S....