Financial Math

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Topics Covered• Proportions and Percents• Taxes• Simple interest • Compound interest• Annuities and sinking funds• Amortized loans

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Proportions

Example: Hy-Vee is selling mangos 3 for $1. If you purchase 8 mangos, how much will you owe?

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Solution: We set up a proportion

x =83

≈ $2.67

3 mangos1 dollar

=8 mangosx dollars

Percentages

Literally, percent means “per hundred”. So, for example, 4% is 4 per hundred, or 4/100. In decimal form, 4% is 0.04. In general, to convert from a percent to a decimal, we move the decimal two places to the left. To convert from a decimal to a percent, we move the decimal two places to the right.

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Example: A computer is advertised as 20% off. If the computer normally sells for $749.99, what is the sale price?

Solution: We need to find 20% of $749.99. First convert 20% to decimal form, then multiply by $749.99.

749.99 − 149.998 = 599.992

(0.20)(749.99) = 149.998(subtract from the full price)

So, the computer is on sale for $599.99.If the sales tax rate is 8.7%, what is your total cost to purchase the computer?

PercentagesExample: Joan sees an ad for a new bicycle for $300, down from the original price of $379. She also has a coupon for 5% off any purchase, before taxes. Determine the final purchase price of the bicycle and the overall percentage saved.

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Solution: The purchase price will be

300 − 0.05(300) = $285Joan saved $379 - $285 = $94. This is called the absolute change.She saved , or 24.8%, of the original purchase price.

There was a 24.8% decrease in the price. This is called the relative change in the price.

94379

= 0.248

Absolute and Relative Change

Given a starting quantity and ending quantity,

Relative change gives a percent change. The starting quantity is called the base.

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Relative change =absolute changestarting quantity

Absolute change = |starting quantity − ending quantity |

Example: Suppose a stock drops in value by 60% one week, then increases in value the next week by 75%. Is the value higher or lower than where it started?

�7Taken from http://www.opentextbookstore.com/mathinsociety/2.4/ProblemSolving.pdf

Solution: Let’s suppose the stock starts out worth $100. Then a decrease by 60% is$100 - (0.6)($100) = $40.

Now an increase by 75% is$40 + (0.75)($40) = $70.So, after the two weeks, the stock has reduced in value by $30.The relative change in the value was 30/100 = 0.3, or 30% lower than the initial worth of the stock.Will the relative change in value change if the initial worth of the stock changes?

Example: David’s credit card offers 5% cash back on gas and groceries, 2% cash back on restaurants and 1% cash back on all other purchases. His purchases over the last month are outlined in the table. Determine the amount of cash back he will receive. What is the overall percent back for purchases this month?

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Date Store Cost

05/01 Price Chopper

35.77

05/01 Phillips 66 - Platte City

10.20

05/02 Redbox 1.8505/08 QuikTrip 29.97

05/12 Gap 41.38

05/12 ProFlowers 35.00

05/17 HyVee 25.92

05/18 Panera Bread

10.80

05/20 Electric company

95.03

05/28 Jack Stack BBQ

48.00

First, determine the totals spent in each category: Gas and Groceries: 35.77 + 10.20 + 29.97 + 25.92 = 101.86Restaurants: 10.80 + 48.00 = 58.80Other: 1.85 + 41.38 + 35.00 + 95.03 = 173.26So, his total cash back is 0.05(101.86) + 0.02(58.80) + 0.01(173.26) = $8.00.His percent back is

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So, overall, he received 2.4% cash back.

amount backtotal spent

=8.00

333.92≈ 0.02396

Note this is not the same as taking the average of 5%, 2%, and 1%!

Example: Ramona went out to eat on Kids Eat Free night with her two children and husband. When she got the bill, the total was $21.40 (not including the kids’ meals). She knows that she should tip at least 15% of the total cost including kids meals. If the kids meals cost $5.50 each, what’s the minimum should she leave as a tip?

Example: Would it be better for the value of a stock to go up 40% on Monday and then fall 60% on Tuesday or for it to fall 60% on Monday and go up 40% on Tuesday?

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TaxesAccording to the IRS, “Taxes under the Federal Insurance Contributions Act (FICA) are composed of the old-age, survivors, and disability insurance taxes, also known as social security taxes, and the hospital insurance tax, also known as Medicare taxes. Different rates apply for these taxes. The current tax rate for social security is 6.2% for the employer and 6.2% for the employee, or 12.4% total. The current rate for Medicare is 1.45% for the employer and 1.45% for the employee, or 2.9% total.” If your gross annual income is $44,000, how much will you pay in social security and medicare taxes?

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Solution: Social security taxes: 0.062(44000) = $2728.Medicare taxes: 0.0145(44000) = $638So, you will pay a total of $2728 + $638 = $3366. Your employer will pay the same.

Income TaxesIncome taxes are computed using your taxable income. First, you must choose your filing status. For purposes of this class, we will stick with “single” or “married, filing jointly.” Your taxable income is computed as either Taxable income = gross income - standard deduction, orTaxable income = gross income - itemized deductions

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Standard Deductions for 2018

Income TaxesExample: Suppose your gross annual income is $30000 and your filing status is single. Determine your taxable income.Solution: taxable income = 30000 - 12000 = $18000

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Standard Deductions for 2018

Example: Suppose your gross annual income is $30000 and your filing status is single. Your taxable income is $18000. Determine the tax owed in 2018. Solution: According to the table, the tax owed is $952.50 + 12% of the amount over $9,525.

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Standard Deductions for 2018

Example: Suppose your gross annual income is $30000 and your filing status is single. Your taxable income is $18000. Determine the tax owed in 2018. Solution: According to the table, the tax owed is $952.50 + 12% of the amount over $9,525.

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First, find the amount over $9525: $18000 - 9525 = $8475. Now, find 12% of $8475: 0.12(8475) = $1017.Finally, the taxes owed is $952.50 + $1017 = $1969.50Your effective tax rate is the ratio

total tax expensegross income

=1969.5030000

≈ 0.06565

Here, your effective rate is 6.57%.

Simple Interest- a fixed percent of your principal

whereP = principalr = interest rate per year (in decimal form)t = time in yearsI = interest earned after t yearsA = amount you have in the bank after t years

Simple interest = I= Prt

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

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Simple InterestWe could ask four questions: 1. How much money will be in the account after t years?2. How much should you invest for t years if you want a specific amount after t years?3. What interest rate will give you the amount A after t years?4. How long will it take for the account to rise to the amount A?

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Simple InterestExample : Suppose you invest $2000 in an account that earns an interest rate of 1.5% per year. How much money will be in the account after 5 years?

Solution: P = 2000, r = 0.015, t = 5I = Prt = (2000)(0.015)(5) = 150

so, A = 2000 + 150 = 2150

There will be $2150 in the account after 5 years.

How much will be in the account after 8 years?�18

Simple InterestExample: Tanner wants to have $3000 in 5 years to help pay for his college education. If he invests in an account earning a simple interest rate of 5% per year, how much should he invest?

Solution: t = 5, r = 0.05, A=3000, P=?. Use A=P(1+rt) to solve for P.

Tanner should invest $2400.�19

Simple InterestExample : Kelsey invested $200 in a bank account earning simple interest. After three years, her account had $255. What was the interest rate? Solution: P=200, t = 3, A = 255. Use A = P(1+rt):

So, she invested at the rate of 9.17%. �20

Simple Interest Example: Sarah is saving for a new car. If she places $2000 in a bank account that earns 2% simple interest a year, how long should she leave it in the account to reach $3000 in the account? Would this account be a good choice to save for her car?

Solution: P = 2000, r = 0.02, t = ?. We know we want A=3000 (or I = 3000-2000=1000). Solve for t:

So, t = 25. It would take 25 years for Sarah to get $3000.

How long would it take for the amount in the account to rise to $2300? �21

Compound Interest

• Interest that is paid on principal plus previously earned interest is called compound interest.

• If the interest is added yearly, we say that the interest is compounded annually.

• If the interest is added every three months (4 times a year), we say the interest is compounded quarterly.

We can also discuss interest compounded semi-annually, daily, monthly, or continuously.

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Example: $3000 is deposited for 3 years in a bank account that pays 10% annual interest, compounded annually. How much will be in the account at the end of 3 years?

Solution: At the end of the first year, the account will be worth $3000(1 + (0.10)(1)) = $3300.

At the end of the second year, interest is calculated using the $3300 as your principal: A = $3300(1+0.10) = $3630

We can continue in this way.�23

The table below shows the amount in the account after 1, 2 , and 3 years:

So, after 3 years, there will be $3993 in the account.

If the account had earned a simple interest rate of 10%, after 3 years, the account would have had $3000(1+(0.10)(3))=$3900.

Year Principal (Beginning of the Year)

Future Value (Amount at the end of the year.)

1 $3000 $3300

2 $3300 $3630

3 $3630 $3993

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Compound Interest

If P dollars are deposited in an account paying an annual rate of interest r compounded (paid) m times per year, then after t years the account will contain A dollars, where

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Compound InterestWe can ask the same four questions about accounts earning compound interest:

1. How much money will be in the account after t years?2. How much should you invest for t years if you want a specific amount after t years?3. What interest rate will give you the amount A after t years? 4. How long will it take for the account to rise to the amount A? (Requires understanding of logarithms.)

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Example: Suppose $1000 is deposited into an account paying 4% interest per year compounded quarterly (four times per year). (a) Find the amount in the account after 10 years with no

withdrawals.(b) How much interest is earned over the 10-year period?Solution: (a) P=1000, r =0.04, m = 4, t = 10

So the account will have $1488.86 after 10 years.(b) The interest is I = A – P = $1488.86 - $1000 = $488.86

Compound Interest

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Compound InterestExample: How much money should you invest in an account earning 1.25% interest compounded monthly if you would like to have $4000 in the account after 3 years?

Solution: P = ?, r = 0.0125, t = 3, m=12, A = 4000. Solve for P:

4000 = P

✓1 +

0.0125

12

◆(12)(3)

4000✓1 +

0.0125

12

◆(12)(3)= P

3852.85 = P

So, you would need to invest $3852.82 to have $4000 in the account after 3 years. How much would you need to invest if the account earned 2% compounded monthly?

Example: Trenton wants to invest $10,000 in an account earning interest compounded quarterly (4 times per year). If at the end of 6 years he has $11,500 in the account, what interest rate did the account have?

Solution: P = 10,000, r = ?, t = 6, A = 11,500, m = 4. Solve for r:

Compound Interest

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Divide both sides by 10,000:

We want to get rid of the exponent. We do this by taking both sides to the power 1/24:

Solution, cont:

So, he invested his money in an account earning approximately 2.34% interest.

✓11500

10000

◆1/24

� 1 =r

4

4 · ✓

11500

10000

◆1/24

� 1

!= r

0.0234 = r

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Example: Bob’s Discount Furniture offers financing with no down payment required and no interest for 6 months. For new accounts, the APR for purchases is 28.99% (compounded monthly). You’ve found a sofa and loveseat set for $1598. What interest will accumulate on your purchase over the next 6 months if you make no payments?

Solution: We’ll first find the value of the loan in 6 months. P = 1598, m = 12, t = 0.5, r = 0.2899.

A = 1598

✓1 +

0.2899

12

◆(12)(0.5)

= 1844.08

So, in 6 months your loan will be worth $1844.08. The interest accumulated will be $1844.08 – $1598 = $246.08.

Suppose you choose to put $500 down and then make no payments for 6 months. If the dealer still charges an interest rate of 28.99% compounded monthly, how much interest will have accumulated after 6 months?

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Compound Interest

If P dollars are deposited in an account paying an annual rate of interest r compounded continuously, then after t years the account will contain A dollars, where

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Example: Joe has $25,000 to invest for 10 years. He is considering two accounts. The first account has a simple interest rate of 2.5%. The second account has an interest rate of 2.3% compounded continuously. Which account would be the better choice?

Solution: We will determine the amount at the end of ten years for both accounts. For the first account, P = 25,000, r = 0.025, and t = 10.

For the second account, interest is compounded continuously and r = 0.023 and t = 10.

The better investment would be the second account.

A = 25000(1 + (.025)(10)) = $31250

A = 25000e(0.023)(10) = $31465

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Recap

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(1)Kristen loans Jenn $400 at the annual simple interest rate of 4%. If Jenn waits a year to start paying her back, how much interest will have accumulated?

(2)Determine the amount in an account after 5 years if $10,000 is invested in an account earning

a) 5% interest compounded monthlyb) 4.75% interest compounded continuously

(3)Matt invested $1000 in an account for 15 years, earning interest compounded quarterly. At what rate should he invest if he wants to have $3000 at the end of the 15 years? �35

A review of exponents and logarithms

A function of the form f(x) = ax where a is a real number is called an exponential function.

Example: Let f(x) = 3x. Then f(0) = 30 = 1f(1) = 31 = 3What’s f(-1)? f(5)?

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The number eConsider

Fill in the table with the function values.

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The number e

As x gets larger and larger,

is getting closer and closer to the number

Find e on your calculator. What’s e2? e10?�38

The number e

e is used often in applications of exponential growth or decay, such as compound interest or in carbon-dating.

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The Natural Logarithm

The function

is called the natural log function. It has the property that

if and only if

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The Natural Logarithm

What is ln e2? What is ln e-3? What is ln 1 ?

Use your calculator to find ln 2.

Example:

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A Useful Property of the Natural Log

For any number M > 0 and n a real number,

lnMn = n lnM

Example: Rewrite the following expressions. Assume all variables represent positive real numbers.

ln ex

ln(1 + .0512 )

12tln(1.10)tln e0.025x

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Solving Equations involving exponentials and logarithms

Example: Solve the equation below for x

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Solving Equations involving exponentials and logarithms

Example: Solve the equation below for x

Solution: Take the natural log of both sides:

Use the exponent property of the natural log.

Solve for x by dividing by ln 5 on both sides.

You try solving �44

Using Natural Log to solve Compound Interest ProblemsExample: Kyle’s parents invested $4200 in an account earning 3.25% interest compounded annually to help him save for college. How long will it take for them to double their money?Solution: To double their money, they want A =2(4200)=8400. We know r = 0.0325, m = 1, P = 4200, and t = ?.

It will take approximately 21.672 years to double their money.

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Example: John invests $60,000 in a pension plan that offers 4.75% interest compounded continuously. How long will it take for his investment to grow to $80,000?Solution: r = 0.0475, P = 60,000, t=?, we want A = 80,000:

Using Natural Log to solve Compound Interest Problems

ln

✓4

3

◆= .0475t

So, it will take approximately 6.056 years for the investment to grow to $80,000.

ln (4/3)

.0475= t

6.056 ⇡ t

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(1) You have $4500 to invest towards putting hardwood floors in your house. If you invest it in an account earning 3.75% interest, compounded quarterly, how long will it take for the amount in the account to grow to $6000?

(2) Joe puts $8000 in an account earning 4.3% interest compounded continuously when his new baby is born. How old will the baby be when the account has $20,000?

Annuities

• An annuity is an interest-bearing account into which we make a series of payments of the same size.

• If one payment is made at the end of every compounding period, the annuity is called an ordinary annuity.

• The future value of an annuity is the amount in the account, including interest, after making all payments.

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Annuities

Assume we are making n=mt regular payments, R, into an ordinary annuity. The interest is being compounded m times a year and deposits are made at the end of each compounding period. The future value (or amount), A, of this annuity at the end of the n periods is given by the equation.

A = R

"�1 + r

m

�mt � 1rm

#

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Example: A payment of $50 is made at the end of each month into an account paying a 6% annual interest rate, compounded monthly. How much will be in that account after 3 years?

Solution: R = $50, r = 0.06, m = 12, t = 3, n = 3 x 12 = 36.

Plug everything into your calculator at once, to find A = $1966.81

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Example: A payment of $200 is made each quarter into an account with an annual interest rate of 3%, compounded quarterly. How much will be in the account at the end of 8 years?

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Sinking Funds

When you want to save regularly in an annuity to have a fixed amount available in the future, the account is called a sinking fund.

Because a sinking fund is a special type of annuity, it is not necessary to find a new formula. We can use the formula for calculating the future value of an ordinary annuity that we found earlier. In this case, we will know the value of A and we will want to find R.

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Example: Assume that you wish to save $1,800 in a sinking fund in 2 years. The account pays 6% compounded quarterly and you will also make payments quarterly. What should be your quarterly payment?

Solution: A = 1800, r = 0.06, m = 4, t = 2, n = 2 x 4 = 8

So, your quarterly payments should be $213.45.

1800 · 0.064�

1 + 0.064

�8 � 1= R

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Example: Suppose you have decided to retire as soon as you have saved $1,000,000. Your plan is to put $200 each month into an ordinary annuity that pays an annual interest rate of 8%, compounded monthly. In how many years will you be able to retire? Solution: A = 1,000,000, R = 200, m = 12, r=0.08, n = ?

1000000 = 200

"�1 + 0.08

12

�n � 10.0812

#

1000000 · 0.0812

200=

✓1 +

0.08

12

◆n

� 1

5000 · 0.0812

+ 1 =

✓1 +

0.08

12

◆n

ln

✓5000 · 0.08

12+ 1

◆= ln

✓1 +

0.08

12

◆n

ln

✓5000 · 0.08

12+ 1

◆= n ln

✓1 +

0.08

12

◆

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Example, cont

So, n = 532.1836. Now, n = 12 x t, where t is the number of years that we invest in the annuity. So 532.1836 = 12t, and t = 44.3486. You will need to invest for about 44.3486 years until your retirement.

ln�5000 · 0.08

12 + 1�

ln�1 + 0.08

12

� = n

532.1836 = n

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You Try: (1) Suppose you want to save $500,000 for retirement and you make monthly payments into a sinking fund that earns 7.5% compounded monthly. If you make payments for 30 years, what should your monthly payment be?(2) If you put $150 per month into an annuity that pays an annual interest rate of 9%, compounded monthly, how long will it take the annuity to have a value of $100,000?

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Amortization

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YouBank

Compound Interest Annuity=

You need to borrow P dollars from the bank. From the bank’s perspective, they could invest that P dollars for the term of your loan at a compound interest rate. You need to pay them back what they would get from that investment.

AmortizationThe process of paying off a loan (plus interest) by making a series of regular, equal payments is called amortization, and such a loan is called an amortized loan.

Assume that you borrow an amount P, which you will repay by taking out an amortized loan. You will make m periodic payments per year for mt = n total payments and the annual interest rate is r. Then, you can make your payment by solving for R in the equation

P⇣1 +

r

m

⌘mt= R

"�1 + r

m

�mt � 1rm

#

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Amortization - Present ValueSolving for P in the equation below, we have

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P⇣1 +

r

m

⌘mt= R

"�1 + r

m

�mt � 1rm

#

We thus have the following formula for the present value of an annuity:

Example: An amortized loan of $10,000 is made to pay off a car in 4 years. If the yearly interest rate is 18%, what is your monthly payment?

Solution: We know P = 10,000, m= 12, t = 4, n = 4 x 12 = 48, r = 0.18

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Example: Chart the history of an amortized loan of $1000 for 3 months at 12% interest with monthly payments of $340.

Solution: Month 1: Interest owed: I = Prt = (1000)(0.12)(1/12) = $10, so $10 of the first month’s payment goes toward interest and the rest goes toward reducing the balance:

Month Payment Interest Payment to

Principal

New Balance

1 $340 $10 $330 $670

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Month Payment InterestPayment

to Principal

New Balance

1 $340 $10 $330 $6702 $340 $6.70 $333.30 $336.70

Month Payment InterestPayment

to Principal

New Balance

1 $340 $10 $330 $6702 $340 $6.70 $333.30 $336.703 $340.07 $3.37 $336.70 $0

Month 3: Interest owed: I = (336.70)(0.12)(1/12) = $3.37. Our last payment must cover the balance and the interest, so it must be $336.70+$3.37 = $340.07.

Month 2: Interest owed: I = (670)(0.12)(1/12) = $6.70, so $6.70 of our payment goes toward interest and the rest goes toward the principal:

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Amortization Schedules

The table that we found in the previous example is called an amortization schedule. It is a table detailing each periodic payment on an amortized loan.

Month Payment InterestPayment

to Principal

New Balance

1 $340 $10 $330 $6702 $340 $6.70 $333.30 $336.703 $340.07 $3.37 $336.70 $0

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Example: You wish to borrow $120,000 to buy a house. A bank offers a 30-year mortgage at an annual rate of 7%. (a) Calculate the monthly payment. (b) Construct an amortization schedule for the first three payments on this loan.

Solution: (a) P = 120,000, r = 0.07, m = 12, t = 30, n = 30 x 12 = 360.

R ⇡ 798.36�64

(b) Month 1: Interest = Prt = 120,000(0.07)(1/12) = $700.

Month 2: Interest = Prt = (119,901.64)(0.07)(1/12) = $699.43

Month 3: Interest = Prt = (119,802.71)(0.07)(1/12) = $698.85

Month Payment Interest Payment to Principal

New Balance

1 $798.36 $700 $98.36 $119,901.64

Month Payment Interest Payment to Principal

New Balance

1 $798.36 $700 $98.36 $119,901.64

2 $798.36 $699.43 $98.93 $119,802.71

Month Payment Interest Payment to Principal

New Balance

1 $798.36 $700 $98.36 $119,901.63

2 $798.36 $699.43 $98.93 $119,802.71

3 $798.36 $698.85 $99.51 $119,703.20 �65

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0

100

200

300

400

500

600

700

800

900

1 12 23 34 45 56 67 78 89 100

111

122

133

144

155

166

177

188

199

210

221

232

243

254

265

276

287

298

309

320

331

342

353

Interest Payment to Principal

Example, cont. If you make the monthly payment of $798.36 for 30 years, how much will you have paid in interest on your $120,000 loan?

Solution: We will have paid $798.36*360 = $287,409.60

total to pay off the loan. The interest paid is $287,409.60 -$120,000 = $167,409.60

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Buying a House• A sample listing from Zillow • Mortgage Rates from Zillow• Amortization Schedule • What are the monthly payments? • How much will you pay in interest over the

course of the 30 years?• What is the difference between a fixed rate loan

and an ARM?• What if you took a 15 year loan at a rate of

3.5%?• Paying ahead – make sure your extra payments

go toward the principal, and not future interest�68

Paying Ahead

• Paying ahead – make sure your extra payments go toward the principal, and not future interest

• Accelerating Your Payments• Why should you make extra payments?

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Student Loans

• Information on government subsidized and unsubsidized student loans versus private loans can be found here.

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The Present Value of an AnnuityIf we know the monthly payment, the interest rate, and the number of payments, then the amount that we can borrow is called the present value of the annuity.

Assume that you are making m periodic payments per year for n=mt total payments into an annuity with an annual interest rate of r. Also assume each of your payments is R. Then to find the present value of your annuity, solve for P in the equation

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Example: If you can afford to pay $250 a month on car payments and the bank offers you a 4-year car loan with an annual rate of 12%, what is the present value of this annuity? (i.e., how expensive a car can you afford?)

Solution: We have R = 250, r = 0.12, m = 12, t = 4, n=4 x 12 = 48. We want to solve for P below:

P = 9493.49

So, you can afford to borrow $9493.49.

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Example: If Addison can afford car payments of $275 per month for 4 years, what is the price of a car that she can afford now? Assume an interest rate of 10.8%.

Example: Suppose you purchase a new home for $195,500. If you put 20% down and finance the rest in a 30 year mortgage at 4.125% interest, what will your monthly payment be?

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Finding the Unpaid Balance of a Loan

During times when interest rates are high, people are forced to borrow money at those high rates if they want to buy a house or car on credit. When rates lower, one can refinance the loan by taking out a second loan at a lower interest rate on the remaining debt from the original loan.

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Example: (a) Suppose you have a 30-year mortgage for $120,000 at an annual interest rate of 6%. After 10 years, you refinance. How much remains to be paid on your mortgage?

(b) The remaining 20 years is financed at an annual interest rate of 4.2%. What are the monthly payments?

(c) How much will you save in interest in 20 years by paying the lower rate?

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(a) We first find the monthly payment for the 30 year mortgage. We know P = $120,000, r = 0.06, m = 12, n = 30x12=360, r/m = 0.005.

$719.46 ⇡ R

Multiplying both sides by the reciprocal, we have

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(a), cont. Let N be the number of payments that are left to be made on your loan. The unpaid balance U on the loan is the present value of the loan if you were to make N more payments of $R at the current rate r:

Therefore, you still owe $100,422.78 on this mortgage.

Here, P = $120,000, r = 0.06, m = 12, r/m = 0.005, N = 20 x 12 = 240 (since you’ve been paying on the loan for 10 years, there are 30 - 10 = 20 years left) and R = $719.46, so

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(b) If you can refinance your mortgage for the remaining 20 years at an annual interest rate of 4.2%, what will your monthly payments be?

Solution: We have P = $100,422.78, n = 12x20 = 240, r = 0.042, and m = 12. We need to find R.

Your new monthly payment is $619.18.

$619.18 ⇡ R

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(c) How much will you save in interest by paying the lower rate?

Payment reduction: $719.46 – $619.18 = $100.28 per month.Total amount paid over 20 years at the old interest rate: 240 × $719.46 = $172,670.40. Total amount paid over 20 years at the new interest rate: 240 × $619.18 = $148,603.20. Amount saved: $172,670.40 – $148,603.20 = $24,067.20.

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Refinancing a Mortgage

Often, when refinancing you have to pay a refinancing fee, which is usually a percentage of the remaining balance on your mortgage. For example, suppose you had to pay a 2% refinancing fee in the previous example. Two percent of $100,422.78 is $2008.46. You would gain this fee back in 21 months with the reduced payments on the loan (21 x $100.28 = $2105.88).

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Example: Suppose you take out a 30 year loan for $150,000 at 5% interest. The monthly payment on this loan is $805.23. (a) How soon could you pay off the loan if you pay an

extra $500 each month? (b) How much money would you save in interest by doing

this? (c) Suppose that, after paying off your loan early, you

continue to make your original monthly payments into an annuity at 5% interest compounded monthly. At the original end of your loan, how much would the annuity be worth?

Paying Extra

Example: Suppose you take out a 30 year loan for $150,000 at 5% interest. The monthly payment on this loan is $805.23. (a) How soon could you pay off the loan if you pay an extra $500

each month?

We have P = $150,000, r = 0.05, R = $1305.23, and m=12. We want to plug this into the mortgage loan formula to solve for n.

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So, it will take approximately 156.7345 payments to payoff the loan. Since you make 12 payments a year, we divide this number by 12 to find that the loan will be paid off in a little over 13 years if we add an extra $500 to our monthly payments.

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Example: Suppose you take out a 30 year loan for $150,000 at 5% interest. The monthly payment on this loan is $805.23. (b) How much money would you save in interest by doing this?

If we just made the minimum monthly payments for 30 years, we would pay a total of

$805.23 x 360 = $289,882.80 to the bank. By paying $500 extra each month, we will pay a total of

$1305.23 x 156.7345 = $204,574.57 to the bank.This saves you a total of

$289,882.80 - 204,574.57 = $85,308.23.

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Example: Suppose you take out a 30 year loan for $150,000 at 5% interest. The monthly payment on this loan is $805.23. (c) Suppose that, after paying off your loan early, you continue to make monthly payments into an annuity at 5% interest compounded monthly. At the original end of your loan, how much would the annuity be worth?

We need to plug into the annuity formula with R = $805.23, m = 12, t = 17, n = 12 x 17 = 204, and r = 0.05.

A = 805.23

(1 + 0.05

12 )204 � 1

0.05/12

�⇡ $258, 095.96

Formulas

Annuities and Sinking Funds:

Present Value (Amortized Loans):

R = amount of each payment, r = interest rate, m = number of times interest is compounded per year, n = mt = total number of payments

A = R

"�1 + r

m

�mt � 1rm

#

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Example: After graduating from college, your student loans total $29,000 at an interest rate of 7.21%. The term of the loan is 10 years. The monthly payment for your loan is $339.86. If you pay an extra 30% every month, how soon will you pay off the loan? How much will you save in interest by paying ahead?