## 06 Nov Complete Finance exercises 1, 3, 5, 11, 16, 19, and 23.? Submit your work below as a Word document or PDF.?Finance.pdf

Complete Finance exercises 1, 3, 5, 11, 16, 19, and 23. Submit your work below as a Word document or PDF.

Finance 197

© David Lippman Creative Commons BY-SA

Finance We have to work with money every day. While balancing your checkbook or calculating

your monthly expenditures on espresso requires only arithmetic, when we start saving,

planning for retirement, or need a loan, we need more mathematics.

Simple Interest Discussing interest starts with the principal, or amount your account starts with. This could

be a starting investment, or the starting amount of a loan. Interest, in its most simple form, is

calculated as a percent of the principal. For example, if you borrowed $100 from a friend

and agree to repay it with 5% interest, then the amount of interest you would pay would just

be 5% of 100: $100(0.05) = $5. The total amount you would repay would be $105, the

original principal plus the interest.

Simple One-time Interest

0I P r=

( )0 0 0 0 1A P I P P r P r= + = + = +

I is the interest

A is the end amount: principal plus interest

P0 is the principal (starting amount)

r is the interest rate (in decimal form. Example: 5% = 0.05)

Example 1

A friend asks to borrow $300 and agrees to repay it in 30 days with 3% interest. How much

interest will you earn?

P0 = $300 the principal

r = 0.03 3% rate

I = $300(0.03) = $9. You will earn $9 interest.

One-time simple interest is only common for extremely short-term loans. For longer term

loans, it is common for interest to be paid on a daily, monthly, quarterly, or annual basis. In

that case, interest would be earned regularly. For example, bonds are essentially a loan made

to the bond issuer (a company or government) by you, the bond holder. In return for the

loan, the issuer agrees to pay interest, often annually. Bonds have a maturity date, at which

time the issuer pays back the original bond value.

Example 2

Suppose your city is building a new park, and issues bonds to raise the money to build it.

You obtain a $1,000 bond that pays 5% interest annually that matures in 5 years. How much

interest will you earn?

198

Each year, you would earn 5% interest: $1000(0.05) = $50 in interest. So over the course of

five years, you would earn a total of $250 in interest. When the bond matures, you would

receive back the $1,000 you originally paid, leaving you with a total of $1,250.

We can generalize this idea of simple interest over time.

Simple Interest over Time

0I P rt=

( )0 0 0 0 1A P I P P rt P rt= + = + = +

I is the interest

A is the end amount: principal plus interest

P0 is the principal (starting amount)

r is the interest rate in decimal form

t is time

The units of measurement (years, months, etc.) for the time should match the time

period for the interest rate.

APR – Annual Percentage Rate

Interest rates are usually given as an annual percentage rate (APR) – the total interest

that will be paid in the year. If the interest is paid in smaller time increments, the APR

will be divided up.

For example, a 6% APR paid monthly would be divided into twelve 0.5% payments.

A 4% annual rate paid quarterly would be divided into four 1% payments.

Example 3

Treasury Notes (T-notes) are bonds issued by the federal government to cover its expenses.

Suppose you obtain a $1,000 T-note with a 4% annual rate, paid semi-annually, with a

maturity in 4 years. How much interest will you earn?

Since interest is being paid semi-annually (twice a year), the 4% interest will be divided into

two 2% payments.

P0 = $1000 the principal

r = 0.02 2% rate per half-year

t = 8 4 years = 8 half-years

I = $1000(0.02)(8) = $160. You will earn $160 interest total over the four years.

Try it Now 1

A loan company charges $30 interest for a one month loan of $500. Find the annual interest

rate they are charging.

Finance 199

Compound Interest With simple interest, we were assuming that we pocketed the interest when we received it.

In a standard bank account, any interest we earn is automatically added to our balance, and

we earn interest on that interest in future years. This reinvestment of interest is called

compounding.

Suppose that we deposit $1000 in a bank account offering 3% interest, compounded monthly.

How will our money grow?

The 3% interest is an annual percentage rate (APR) – the total interest to be paid during the

year. Since interest is being paid monthly, each month, we will earn 3%

12 = 0.25% per month.

In the first month,

P0 = $1000

r = 0.0025 (0.25%)

I = $1000 (0.0025) = $2.50

A = $1000 + $2.50 = $1002.50

In the first month, we will earn $2.50 in interest, raising our account balance to $1002.50.

In the second month,

P0 = $1002.50

I = $1002.50 (0.0025) = $2.51 (rounded)

A = $1002.50 + $2.51 = $1005.01

Notice that in the second month we earned more interest than we did in the first month. This

is because we earned interest not only on the original $1000 we deposited, but we also earned

interest on the $2.50 of interest we earned the first month. This is the key advantage that

compounding of interest gives us.

Calculating out a few more months:

Month Starting balance Interest earned Ending Balance

1 1000.00 2.50 1002.50

2 1002.50 2.51 1005.01

3 1005.01 2.51 1007.52

4 1007.52 2.52 1010.04

5 1010.04 2.53 1012.57

6 1012.57 2.53 1015.10

7 1015.10 2.54 1017.64

8 1017.64 2.54 1020.18

9 1020.18 2.55 1022.73

10 1022.73 2.56 1025.29

11 1025.29 2.56 1027.85

12 1027.85 2.57 1030.42

200

To find an equation to represent this, if Pm represents the amount of money after m months,

then we could write the recursive equation:

P0 = $1000

Pm = (1+0.0025)Pm-1

You probably recognize this as the recursive form of exponential growth. If not, we could go

through the steps to build an explicit equation for the growth:

P0 = $1000

P1 = 1.0025P0 = 1.0025 (1000)

P2 = 1.0025P1 = 1.0025 (1.0025 (1000)) = 1.0025 2(1000)

P3 = 1.0025P2 = 1.0025 (1.00252(1000)) = 1.00253(1000)

P4 = 1.0025P3 = 1.0025 (1.00253(1000)) = 1.00254(1000)

Observing a pattern, we could conclude

Pm = (1.0025)m($1000)

Notice that the $1000 in the equation was P0, the starting amount. We found 1.0025 by

adding one to the growth rate divided by 12, since we were compounding 12 times per year.

Generalizing our result, we could write

0 1

m

m

r P P

k

= +

In this formula:

m is the number of compounding periods (months in our example)

r is the annual interest rate

k is the number of compounds per year.

While this formula works fine, it is more common to use a formula that involves the number

of years, rather than the number of compounding periods. If N is the number of years, then

m = N k. Making this change gives us the standard formula for compound interest.

Compound Interest

0 1

Nk

N

r P P

k

= +

PN is the balance in the account after N years.

P0 is the starting balance of the account (also called initial deposit, or principal)

r is the annual interest rate in decimal form

k is the number of compounding periods in one year.

If the compounding is done annually (once a year), k = 1.

If the compounding is done quarterly, k = 4.

If the compounding is done monthly, k = 12.

If the compounding is done daily, k = 365.

Finance 201

The most important thing to remember about using this formula is that it assumes that we put

money in the account once and let it sit there earning interest.

Example 4

A certificate of deposit (CD) is a savings instrument that many banks offer. It usually gives a

higher interest rate, but you cannot access your investment for a specified length of time.

Suppose you deposit $3000 in a CD paying 6% interest, compounded monthly. How much

will you have in the account after 20 years?

In this example,

P0 = $3000 the initial deposit

r = 0.06 6% annual rate

k = 12 12 months in 1 year

N = 20 since we’re looking for how much we’ll have after 20 years

So

20 12

20

0.06 3000 1 $9930.61

12 P

= + =

(round your answer to the nearest penny)

Let us compare the amount of money earned from compounding against the amount you

would earn from simple interest

Years Simple Interest

($15 per month)

6% compounded

monthly = 0.5%

each month.

5 $3900 $4046.55

10 $4800 $5458.19

15 $5700 $7362.28

20 $6600 $9930.61

25 $7500 $13394.91

30 $8400 $18067.73

35 $9300 $24370.65

As you can see, over a long period of time, compounding makes a large difference in the

account balance. You may recognize this as the difference between linear growth and

exponential growth.

0

5000

10000

15000

20000

25000

0 5 10 15 20 25 30 35

A c c o

u n

t B

a la

n c

e (

$ )

Years

202

Evaluating exponents on the calculator

When we need to calculate something like 53 it is easy enough to just multiply

5⋅5⋅5=125. But when we need to calculate something like 1.005240 , it would be very

tedious to calculate this by multiplying 1.005 by itself 240 times! So to make things

easier, we can harness the power of our scientific calculators.

Most scientific calculators have a button for exponents. It is typically either labeled

like:

^ , yx , or xy .

To evaluate 1.005240 we'd type 1.005 ^ 240, or 1.005 yx 240. Try it out – you should

get something around 3.3102044758.

Example 5

You know that you will need $40,000 for your child’s education in 18 years. If your account

earns 4% compounded quarterly, how much would you need to deposit now to reach your

goal?

In this example,

We’re looking for P0.

r = 0.04 4%

k = 4 4 quarters in 1 year

N = 18 Since we know the balance in 18 years

P18 = $40,000 The amount we have in 18 years

In this case, we’re going to have to set up the equation, and solve for P0. 18 4

0

0.04 40000 1

4 P

= +

( )040000 2.0471P=

0

40000 $19539.84

2.0471 P = =

So you would need to deposit $19,539.84 now to have $40,000 in 18 years.

Rounding

It is important to be very careful about rounding when calculating things with

exponents. In general, you want to keep as many decimals during calculations as you

can. Be sure to keep at least 3 significant digits (numbers after any leading zeros).

Rounding 0.00012345 to 0.000123 will usually give you a “close enough” answer, but

keeping more digits is always better.

Finance 203

Example 6

To see why not over-rounding is so important, suppose you were investing $1000 at 5%

interest compounded monthly for 30 years.

P0 = $1000 the initial deposit

r = 0.05 5%

k = 12 12 months in 1 year

N = 30 since we’re looking for the amount after 30 years

If we first compute r/k, we find 0.05/12 = 0.00416666666667

Here is the effect of rounding this to different values:

If you’re working in a bank, of course you wouldn’t round at all. For our purposes, the

answer we got by rounding to 0.00417, three significant digits, is close enough – $5 off of

$4500 isn’t too bad. Certainly keeping that fourth decimal place wouldn’t have hurt.

Using your calculator

In many cases, you can avoid rounding completely by how you enter things in your

calculator. For example, in the example above, we needed to calculate 12 30

30

0.05 1000 1

12 P

= +

We can quickly calculate 12×30 = 360, giving

360

30

0.05 1000 1

12 P

= +

.

Now we can use the calculator.

r/k rounded to:

Gives P30 to be: Error

0.004 $4208.59 $259.15

0.0042 $4521.45 $53.71

0.00417 $4473.09 $5.35

0.004167 $4468.28 $0.54

0.0041667 $4467.80 $0.06

no rounding $4467.74

Type this Calculator shows

0.05 ÷ 12 = . 0.00416666666667

+ 1 = . 1.00416666666667

yx 360 = . 4.46774431400613

× 1000 = . 4467.74431400613

204

Using your calculator continued

The previous steps were assuming you have a “one operation at a time” calculator; a

more advanced calculator will often allow you to type in the entire expression to be

evaluated. If you have a calculator like this, you will probably just need to enter:

1000 × ( 1 + 0.05 ÷ 12 ) yx 360 = .

Annuities For most of us, we aren’t able to put a large sum of money in the bank today. Instead, we

save for the future by depositing a smaller amount of money from each paycheck into the

bank. This idea is called a savings annuity. Most retirement plans like 401k plans or IRA

plans are examples of savings annuities.

An annuity can be described recursively in a fairly simple way. Recall that basic compound

interest follows from the relationship

11m m

r P P

k −

= +

For a savings annuity, we simply need to add a deposit, d, to the account with each

compounding period:

11m m

r P P d

k −

= + +

Taking this equation from recursive form to explicit form is a bit trickier than with

compound interest. It will be easiest to see by working with an example rather than working

in general.

Suppose we will deposit $100 each month into an account paying 6% interest. We assume

that the account is compounded with the same frequency as we make deposits unless stated

otherwise. In this example:

r = 0.06 (6%)

k = 12 (12 compounds/deposits per year)

d = $100 (our deposit per month)

Writing out the recursive equation gives

( )1 1

0.06 1 100 1.005 100

12 m m mP P P− −

= + + = +

Assuming we start with an empty account, we can begin using this relationship:

Finance 205

0 0P = ( )1 01.005 100 100P P= + =

( ) ( )( ) ( )2 11.005 100 1.005 100 100 100 1.005 100P P= + = + = +

( ) ( ) ( )( ) ( ) ( ) 2

3 21.005 100 1.005 100 1.005 100 100 100 1.005 100 1.005 100P P= + = + + = + +

Continuing this pattern, after m deposits, we’d have saved:

( ) ( ) ( ) 1 2

100 1.005 100 1.005 100 1.005 100 m m

mP − −

= + + + +

In other words, after m months, the first deposit will have earned compound interest for m-1

months. The second deposit will have earned interest for m-2 months. Last months deposit

would have earned only one month worth of interest. The most recent deposit will have

earned no interest yet.

This equation leaves a lot to be desired, though – it doesn’t make calculating the ending

balance any easier! To simplify things, multiply both sides of the equation by 1.005:

( ) ( ) ( )( )1 2 1.005 1.005 100 1.005 100 1.005 100 1.005 100

m m

mP − −

= + + + +

Distributing on the right side of the equation gives

( ) ( ) ( ) 1 21.005 100 1.005 100 1.005 100(1.005) 100 1.005

m m

mP −

= + + + +

Now we’ll line this up with like terms from our original equation, and subtract each side

( ) ( ) ( )

( ) ( )

1

1

1.005 100 1.005 100 1.005 100 1.005

100 1.005 100 1.005 100

m m

m

m

m

P

P

−

−

= + + +

= + + +

Almost all the terms cancel on the right hand side when we subtract, leaving

( )1.005 100 1.005 100 m

m mP P− = −

Solving for Pm

( )( )0.005 100 1.005 1 m

mP = −

( )( )100 1.005 1

0.005

m

mP −

=

Replacing m months with 12N, where N is measured in years, gives

206

( )( )12 100 1.005 1

0.005

N

NP −

=

Recall 0.005 was r/k and 100 was the deposit d. 12 was k, the number of deposit each year.

Generalizing this result, we get the saving annuity formula.

Annuity Formula

1 1

Nk

N

r d

k P

r

k

+ − =

PN is the balance in the account after N years.

d is the regular deposit (the amount you deposit each year, each month, etc.)

r is the annual interest rate in decimal form.

k is the number of compounding periods in one year.

If the compounding frequency is not explicitly stated, assume there are the same

number of compounds in a year as there are deposits made in a year.

For example, if the compounding frequency isn’t stated:

If you make your deposits every month, use monthly compounding, k = 12.

If you make your deposits every year, use yearly compounding, k = 1.

If you make your deposits every quarter, use quarterly compounding, k = 4.

Etc.

When do you use this

Annuities assume that you put money in the account on a regular schedule (every

month, year, quarter, etc.) and let it sit there earning interest.

Compound interest assumes that you put money in the account once and let it sit there

earning interest.

Compound interest: One deposit

Annuity: Many deposits.

Finance 207

Example 7

A traditional individual retirement account (IRA) is a special type of retirement account in

which the money you invest is exempt from income taxes until you withdraw it. If you

deposit $100 each month into an IRA earning 6% interest, how much will you have in the

account after 20 years?

In this example,

d = $100 the monthly deposit

r = 0.06 6% annual rate

k = 12 since we’re doing monthly deposits, we’ll compound monthly

N = 20 we want the amount after 20 years

Putting this into the equation:

( )20 12

20

0.06 100 1 1

12

0.06

12

P

+ − =

( )( ) ( )

240

20

100 1.005 1

0.005 P

− =

( )

( )20

100 3.310 1

0.005 P

− =

( )

( )20

100 2.310 $46200

0.005 P = =

The account will grow to $46,200 after 20 years.

Notice that you deposited into the account a total of $24,000 ($100 a month for 240 months).

The difference between what you end up with and how much you put in is the interest

earned. In this case it is $46,200 – $24,000 = $22,200.

Example 8

You want to have $200,000 in your account when you retire in 30 years. Your retirement

account earns 8% interest. How much do you need to deposit each month to meet your

retirement goal?

In this example,

We’re looking for d.

r = 0.08 8% annual rate

k = 12 since we’re depositing monthly

N = 30 30 years

P30 = $200,000 The amount we want to have in 30 years

208

In this case, we’re going to have to set up the equation, and solve for d. ( )30 12

0.08 1 1

12 200,000

0.08

12

d

+ &

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