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Chapter 9: Mathematics of

Finance

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9-1: Time Value of Money Problems Whenever we take out loans, make investments, or

deal with money over time, common questions arise. How much will this be worth in 10 years? How will inflation eat into my retirement savings over

the next 30 years? How much do I need to save each month to have

$1,000,000 at age 60? How much do I need to pay each month on a 30 year

mortgage of $400,000 if the interest rate is 5.25% These are called Time Value of Money (TVM)

problems.

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Common Notations P = Present Value: the current value of an

investment or loan or sum of money. F = Future Value: The value of an

investment, loan or sum of money in the future.

r = Annual Percentage Rate m = Number of periods per year. Example: If

you make monthly loan payments, them m = 12

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Common Notations…cont’d

i = Interest Rate per Period: Example: The annual interest rate is 6% and there are 12 payments per year. Then i = .06/12 = .005, or 0.5% per month.

i = r/m t = Time, measured in years n = Total number of periods. Example: You have a

10 year loan that is paid monthly. Then you have n = 10*12=120 total periods. n = m*t

R = The amount of a Rent. This is the regular payment made on a loan or into an investment.

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Example Suppose you deposit $1000 into an account that

compounds interest quarterly. The annual rate of interest is 2.3% and you are going to keep it in the account for 4.5 years. At the end of this time, the account will be worth $1108.72

P = $1000 F = $1108.72 r = .023 m = 4 i = .023/4 = .00575 t = 4.5 n = 18

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9-2: Percent Increase/Decrease

If a quantity increases by some percent, we can create a multiplier that helps us convert a beginning value to an ending value.

To create the appropriate multiplier: Percent Increase: 1 + i Percent Decrease: 1 - i

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Example

This year, the SCCC student population is 11,350.

The administration estimates that will increase by 2% next year. How many students can we expect next year? The multiplier = 1+0.02 = 1.02 New student population = 11,350(1.02) = 11577

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Example

The current balance of my retirement account is $244,350.

If the value of the account drops by 5.2% over the next year, what will be the new value? Multiplier = 1 - .052 = .948 New Value = $244,350(.948) = $231,643

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9-3: Compound Interest

When we invest money, interest may be applied to the account on a regular basis. For example, if we invest in an account that pays interest monthly, we say the interest compounds monthly. Anything in the account at the time of compounding gets interest added to it. In the monthly case, we have 12 compoundings per year, with each compounding representing 1/12 of the total annual interest rate.

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Example

We invest $1000 in an account that compounds monthly. The annual interest rate is 3.6%. If we keep it in the account for 5 years, adding or removing nothing, how much will be in the account at the end of 5 years?

First, we need to note that the interest per period is i = .036/12 = .003.

Let’s begin by building a table…

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Example

Period #

Previous Balance New Balance

0 $0 $1000

1 $1000 $1000(1.003)

2 $1000(1.003) $1000(1.003)(1.003)

3 $1000(1.003)(1.003) $1000(1.003) (1.003) (1.003)

…

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Example

Period #

New Balance

0 $1000 =$1000(1.003)0

1 $1000(1.003) =$1000(1.003)1

2 $1000(1.003)(1.003) =$1000(1.003)2

3 $1000(1.003) (1.003) (1.003)

=$1000(1.003)3

…

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Example

Period # New Balance

3 $1000(1.003) (1.003) (1.003)

=$1000(1.003)3

4 … =$1000(1.003)4

5 … =$1000(1.003)5

After 5 years, or 60 periods, we have…

60 … =$1000(1.003)60

After n total periods, we have…

n … =$1000(1.003)n

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Compound Interest Formula

If P dollars earn an annual interest rate of i per period for n periods, with no additional principal added or removed, then the future value (F) is given by:

F = P(1+i)n

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Example

A bank account is opened with $4,000 in the account. It earns 6% annual interest. If it earns interest quarterly (four times per year), then what is in the account after 10 years?

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Example

Suppose you invest $500 today at an annual rate of 1.5%, compounded daily. How long before the balance doubles?

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9-4: Rule of 72

Given some investment that grows at an annual interest rate, r, (not expressed as a decimal), then the amount of time in years it takes for the investment to double is approximately:

r

72

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Example

How long will it take for an investment to double if it earns 5% annual interest?

Note that estimating the doubling time does not require that we know how much is originally invested!

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Example

If you want your investment to double in 30 months, what annual interest rate do you need to secure?

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9-5: Yield

Because each compounding acts on the original balance and any interest that has been previously earned, the net interest earned will not be the same as the annual interest rate at the end of the investment. The “true” interest rate earned is called the Yield.

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Example

Invest $100 for 1 years, compounded monthly at an annual rate of 12%.

F = 100(1+.01)12 = $112.68 This represents a yield of 12.68%, which is higher than

the original 12% stated above. The yield is often called the Annual Percentage Yield (APY). Always ask what this is when you take out a loan…time, compounding and bank fees can substantially increase your rate of interest and therefore your total payments due!

APR = 12% APY = 12.68%

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Yield Formula

The formula for yield in the t = 1 year case is:

11 niy

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Where did the formula come from?

1)1(

1)1(

)1(

n

n

n

i

P

iP

P

PiP

old

oldnewy

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Why is it important to know the APY?

The APY, or yield, is helpful since it simplifies calculations. If we know the APY, then it does not matter how many times we compound per year because the APR will give us actual percentage increase at the end of a year.

APR = Annual Percentage Rate or nominal rate or rate

APY = Annual Percentage Yield or effective rate or yield

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9-6: Annuities

When additional payments or deposits (rents) are made at regular intervals into an investment, then we call these annuities:

Ordinary Annuity: Payment is due at the END of each period.

Annuity Due: Payment is due at the START of each period.

This will complicate our Future Value calculations.

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Example

We invest in an account that compounds annually. The annual interest rate is 3%. If we add $800 at the end of each year (an ordinary annuity), how much will be in the account at the end of 5 years total?

Let’s look at a picture of what is going on here.

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Example

Year 1 Year 2 Year 3 Year 4 Year 5

$800 $800 $800 $800 $800

Start

Each of these $800 investments earns interest for a different period of time. Hence, the value of each of these deposits is different at the end of the 5-year period.

Investment Period

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Example

End Year 1 End Year 2 End Year 3 End Year 4 End Year 5

$800 $800 $800 $800 $800

Start

The End

This one is worth

$800(1+.03)4 at the End

$900.41

This one is worth

$800(1+.03)3 at the End

$874.18

This one is worth $800 at

the End

$800

This one is worth

$800(1+.03)1 at the End

$824

This one is worth

$800(1+.03)2 at the End

$848.72

Investment Period

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Example

We can add all of these up: $800(1.03)4 +

$800(1.03)3 + $800(1.03)2 +

$800(1.03)1 +$800

=$900.41 + $874.18 + $848.72 + $824 + $800 =$4247.31

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A General Formula

Now imagine if the monthly payments were deposited and monthly interest credited. We would then have 5*12 = 60 different deposits to find the values for so we can add them up.

To avoid this inefficiency, we instead use the following formula, which is equivalent to going through that process.

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A General Formula

If R dollars are paid at the end of each period, with an interest rate of I per period, then the Future Value of the Annuity is:

i

iRF

n 11

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Where did the formula come from? We can generalize the example before and

think about adding R + R(1+i) + R(1+i)2 + R(1+i)3 + … + R(1+i)n-1. Let us call this sum S.

Hence, (1+i)S - S = R(1+i)n - R

S (1+i - 1) = R(1+i)n - R

Hence, the sum S is

(R(1+i)n - R)/i

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Check

We invest in an account that compounds annually. The annual interest rate is 3%. If we add $800 at the end of each year (an ordinary annuity), how much will be in the account at the end of 5 years total?

F 800(1 .03)5 1

.03

800.159275

.03

800 5.309136 4247.31

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Example

What is the future value if you invest $95 per month for 7 years at an annual rate of 3.75%?

R = 95 i = .0375/12 n = 12*7 = 84

19.9109

8862111.9595

003125.

2996441.95

12/0375.

112/0375.195

84

F

Note that I try to keep as many decimal places as possible until the end

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FV for Annuities Due

When payments or deposits are made at the beginning of a period (rather than at the end as in the previous examples), an adjustment is needed.

We can view each payment as if it were made at the end of the preceding period. This would require one more payment (n+1 total) than usual and would require that we subtract the last payment so we don’t overpay.

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FV for Annuities Due

R

i

iRF

n

11 1

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Example

If $300 payments are made at the beginning of the month for 18 years (a college savings fund), what is the Future Value if the annual interest rate is 5.5%

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9-7: Future Value (FV) on Excel

The FV command will do these computations for us automatically.

Command Format: =FV(rate, nper, pmt, [pv], [type])

This is i, the rate

per period

This is n, the total

# of periods

This is R, the

amount of rent

This is P, the

Present Value

Blank for ordinary

annuity, 1 for annuity due

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Excel Examples

If $300 payments are made at the beginning of the month for 18 years (a college savings fund), what is the Future Value if the annual interest rate is 5.5%

What is the future value if you invest $95 per month (paid at the end of the month) for 7 years at an annual rate of 3.75%?

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9-8: PV of Annuities

Suppose you have an ordinary 20-year annuity that you pay $500 into at the end of each quarter. The annual interest rate is 7%.

What is the lump-sum of money which should be deposited at the start of the annuity that would produce the exact same amount of money at the end of the period, without any additional payments?

This is know as the Present Value of the Annuity

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Present Value of an Ordinary Annuity

i

iRP

n11

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Example

Suppose you set up an ordinary annuity account which is to last 10 years and earn 4% annual interest rate. If your rent payment is $150 per month, how much do you need to deposit as a lump sum up front to achieve the same end result without any regular payments?

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9-9: Present Value (PV) on Excel

The PV command will do these computations for us automatically.

Command Format: =PV(rate, nper, pmt, [fv], [type])

This is i, the rate

per period

This is n, the total

# of periods

This is R, the

amount of rent

This is the FV you

want after the last pmt…

Optional

Blank for ordinary

annuity, 1 for annuity due

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Can you figure out how this comes from the formula for the Future Value of an Ordinary Annuity?

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Loan Payment Formula

R

ii

P

ii

ii

R

ii

P

i

iRP

n

n

n

n

n

11

11

11

11

11 Start here with the original formula

Divide both sides by [ ] to get R alone

Here is the formula for the loan payment.

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Loan Payment Formula

Using basic algebra, we can rewrite this as:

n

n

i

Pi

i

PiR

1

1111

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Example

Suppose you want to buy a home and take out a 30-year mortgage for $240,000. The annual interest rate is 5.75%.

a) What is the monthly payment?

b) What total amount of money do you pay over the life of the loan (assuming all regular payments are made)?

c) How much of your total payments is interest?

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Example (a)

We use the formula to get $1400.57 per month .

n

n

i

Pi

i

PiR

1

1111

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Example (b) and (c)

The total amount of money we pay is:

360x$1400.57 = $504,205.20

Hence, the amount of interest paid is:

$504,205.20 - $240,000 = $264,205.20

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#69

You borrowed $150,000, which you agree to pay back with monthly payments at the end of each month for the next 10 years. At 6.25% interest, how much is each payment?

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9-10: Loan Payments (PMT) on Excel

The PMT command will do these computations for us automatically.

Command Format: =PMT(rate, nper, pv, [fv], [type])

This is i, the rate

per period

This is n, the total

# of periods

This is the

present value of the loan

This is the FV you

want after the last pmt…

Default=0

Blank for ordinary

annuity, 1 for annuity due

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9-12: Adjusting for Inflation

Inflation can seriously devalue a loan or asset over time.

For example, if the average inflation rate is 3.5%, how much will $50 be worth in 5 years (in terms of today’s dollars)? In other words, in five years how much can I buy with a $50 bill compared to what I can buy today?

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Example

F = P(1+i)n

50 = P(1+.035)5

50 = P(1.187686306) 50/(1.187686306) = P $42.10 = P

Important: Notice that we substituted $50 for F since that is what we know we will have in the future. We solve for P since we want to know what the Future $50 is worth in Present dollars.

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Terms

Nominal Dollars are those that have not been adjusted for inflation.

Real Dollars = Present Dollars are those that have been adjusted for inflation and therefore reflect the spending power of some future amount of money in terms of today’s dollars.

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We’ll disregard amortization tables.