MATH104– Chapter 8 Math of Finance 8.1: Introduction Percents – Write 30% as a decimal – What is the definition of percent

Percents, decimals, fractions PercentDecimalFraction 7% 3.5% 2 ¼ % 3/5.8 ½ % 2

Applications For a $60,000 house, find a 20% down payment On a $40,000 salary, calculate a 8% raise Calculate your final pay after the 8% raise

Salary… If you received a 2% salary increase, and now make $32,640, what was your previous salary?

Percent change Percent change = Amount change/base amt Find percent change from: 2001 to to 2003

Stock ex If the stock market went down 50% today, what percent increase would you need tomorrow to return to the previous level? Percent change from Jan 1 to Jan 2 From Jan 2 to Jan 3

Sales Tax On a $8000 car, calculate a 6% sales tax. What is the total paid?

FICA tax Calculate FICA (Social Security and Medicare) on a $140,000 income if you: – Are not self employed – you pay 7.65% FICA on the first $102,000 – And you pay 1.45% on the income exceeding $102,000

Federal Income Tax– pg AGI=Gross income – Adjustments= 2.Taxable income = AGI – (Exempt + Deduct)= 3.Tax computation (see table on p. 448) Income tax= tax computation – tax credit

Do p. 447 Marital status ____ Number of kids ____ Gross income ____ Adjustments Deductions Tax credit ____ 1.AGI=Gross income – Adjustments= 2.Taxable income = AGI – (Ex +Ded)= 3.Tax computation (see table on p. 448) Income tax= tax computation – tax credit

Fed Tax Ex Marital status ____ Number of kids ____ Gross income ____ Adjustments Deductions Tax credit ____ 1.AGI=Gross income – Adjustments= 2.Taxable income = AGI – (Ex +Ded)= 3.Tax computation (see table on p. 448) Income tax= tax computation – tax credit

8.2: Simple Interest Ex: If you invest $2000 at r=10% for 1 year, what amount of interest will you earn? Ex: If you invest $2000 at r=10% for 3 years, what amount of interest will you earn? Ex: If you invest $2000 at r=10% for 6 months, what amount of interest will you earn? Simple Interest formula: I = _____

Examples If P=$5200, r=7%, t=4 years, find I and A If P=$4500, r=7%, t=9 months, find I and A If P=$3500, r=2% per month, t=6 months, find I and A

Example If P=$2300, r=2 ¼ %, t=10 years, find I and A

Solve for another variable If simple interest is calculated to be I=$400, where r = 8%, and t = 2 years, find P

Solve for Principal Since I=Prt and A=P+I, then A=P+Prt=P(1+rt) If you borrow money at r=7% for 3 years and you pay back $6050, how much money did you borrow?

Amortization tables If you take a car loan for $8000 at r=8% for 5 years, your monthly payment will be $ (Trust me for now. We’ll find out how to calculate those later.) Use an amortization table to describe where your monthly payment goes…

Ex #1: $8000, 8%, 5 years MonthPaymentNew balance $8000 1$

Ex #2: $8000, 5%, 3 years: MonthPaymentNew balance $8000 1$

8.3: Compound Interest Definition— Example: Consider borrowing $1000 at r=5%, compounded annually. How much will you have at the end of each year…

Compound- example P=$1000, r=5%, compounded annually YearStarting balanceAmount at year’s end 11000A=P+I =1000+(1000*.05) 2 3

Compound- example P=$1000, r=5%, compounded annually YearStarting balanceAmount at year’s end 11000A=P+I =1000+(1000*.05) = = $1050 =1000 ( ) = $ A= (1050*.05) =1050(1+.05)= $ =1000(1+0.05)(1+0.05) = 1000(1+.05) 2 = $ A= ( *.05)= $ = (1+.05) =1050(1+.05) (1+.05) =1000(1+.05)(1+.05)(1+.05) = 1000(1+.05) 3 = $

$1000 Compounded quarterly, at 8% QuarterStart.balanceAmount at quarter’s end 11000A=P+I = (1000*.08 *1/4) =1000+(1000*.08/4) =1000 (1 +.08/4) = A= (1020*.08*1/4) = 1020 (1+.08/4) =1000 (1+.08/4)(1+.08/4) =1000 (1 +.08/4) 2 = A= (1040.4*.08*1/4) = (1+.08/4) =1000 (1+.08/4) 3 = A= ( *.08*1/4) = (1+.08/4) =1000 (1+.08/4) 4 =

Compound interest P = ______, r = ________, n = ______, t= _____ A =

Compound I P = ______, r = ________, n = ______, t= _____ A =

Compounded continuously A=Pe rt

Present Value A = So solve for P=

Present value You wish to have $100,000 in 20 years. If you can earn 8%, compounded monthly, how much should you invest today? P=

Effective yield Use A =

8.4: Intro to Annuities If you invest $100 at the end of every year, at r=5%, you’ll have: End of 1 st year:100 End of 2 nd year: (1.05)*100 End of 3 rd year: End of 4 th year:

8.4 Annuities Formula Derivation: Value after 1 year is: P After 2 years: P + P(1+r) After 3 years: P+P(1+r) +P(1+r) 2 Using a summation formula = We get A = =

Annuity compounded once a year A =

Ex # 1: Annuity, compounded quarterly We plan to deposit _____ each quarter for ___ years, at a rate of ___, compounded quarterly. Find the total amount, A, in the account at the end. P = ___, r = ___, n =____, t = ____ A =

Ex #2: Annuity compounded monthly We plan to deposit _____ each _____ for ___ years, at a rate of ___, compounded monthly. Find the total amount, A, in the account at the end. P = ___, r = ___, n =____, t = ____ A =

Calculate the interest In the previous example, A= ___________ and P = __________ Find the amount you invested to contribute to the final amount Find the amount of interest that contributed to the total

Ex #3: Annuity compounded We plan to deposit ___ each _____ for ___ years, at a rate of ___, compounded _______. Find the total amount, A, in the account at the end. P = ___, r = ___, n =____, t = ____ A = Also, find amount invested Find interest earned

Solve for P to calculate the periodic payment A = P =

Ex #1: Find P, the periodic payment We plan to have __________in ___ years. If we can earn a rate of ___, compounded _______, how much should we invest each ____. (In other words, find the periodic payment P). A = ___, r = ___, n =____, t = ____ P = Also, find amount invested Find interest earned

Ex #2: Find P, the periodic payment We plan to have __________in ___ years. If we can earn a rate of ___, compounded _______, how much should we invest each ____. (Find periodic payment P). A = ___, r = ___, n =____, t = ____ P = Also, find amount invested Find interest earned

8.5 Car and Home Payments From sections 8.3 and 8.4, we know that A equals A== Solve for PMT PMT=

8.5 Car Loans Ex #1 You wish to buy a $8,000 car. You put $2000 down and find a 5 year loan at 9%. (Note: some hints are provided below that will NOT be provided on the test. You’ll need to know what formulas to use). Find: Purchase/ Cash Price= Down payment= Amount Financed/Loan= Cash Price-Down=

… Car Ex #1: $8,000 car. $2000 down, 5 yrs at 9%. Monthly payment = PMT == Total installment price/ total price paid= (monthly amount*no. months)+down= Total interest paid/ finance charge= total installment price – cash price=

Car Ex 1– Amortization MonthMonthly payment Towards I (I=Prt) Towards PEnding balance 1 2 3

Car Ex #2 You wish to buy a $10,000 car. You put $3000 down and find a __ year loan at __%. Down payment= Amt Financed/Loan= Cash Price - Down= Monthly payment PMT = =.

…Car Ex 2 Total installment price/ total price paid= (monthly amount*no. months)+down= Total interest paid/ finance charge= total installment price – cash price=

Car Ex 2 Amortization MonthMonthly payment Towards I (I=Prt) Towards PEnding balance 1 2 3

8.5 Home Mortgages You wish to buy a $110,000 house with 20% down. Loan: 8.5% for 30 years, with 3 points. Insurance: $50/ mn; property taxes: $150/mn Down payment= Amount Financed/Loan= Cash Price-Down *Points (paid at closing)= percentage of loan=.

PMT Monthly payment to cover principal and interest PMT =

Home Ex 1 *Entire Monthly payment, including taxes and insurance = Total installment price/ total paid using PMT (note: this covers principal and interest, not tax or insurance) =(PMT*no. months)+down = Total interest paid/ finance charge = total installment price – cash price=.

Home Ex 1 Amortization MonthMonthly payment Towards I (I=Prt) Towards PEnding balance 1 2

Saving money on a home What are some ways to save money on the final total amount paid? ______________ _________ ___________ _______________ _____________ Give an example of two specific changes you would make to save money on this home _______________. Next, we’ll redo the above problem..

Redo Home Ex 1 Redo above example: Purchase Price stays the same Down payment= Amt Financed/Loan= Cash Price-Down = *Points (paid at closing)= percentage of loan= Monthly paymt= PMT = = *Entire Monthly payment, including taxes and insurance =

Home Ex 1--revised Total installment price/ total paid using PMT (note: principal and interest, not tax or insurance) =(PMT*no. months)+down = Total interest paid/ finance charge = total installment price – cash price= MonthMonthly payment Towards I (I=Prt) Towards PEnding balance 1

Home Ex #2 You wish to buy a $150,000 house with 20% down. Loan: 5 ¼ % for 20 years, with 2 points. Insurance: $60/ mn; property taxes: $170/mn Down payment= Amount Financed/Loan= Cash Price-Down *Points (paid at closing)= percentage of loan= Monthly payment to cover principal and interest PMT =.

Home Ex 2 *Entire Monthly payment, including taxes and insurance = Total installment price/ total paid using PMT (note: this covers principal and interest, not tax or insurance) =(PMT*no. months)+down = Total interest paid/ finance charge = total installment price – cash price=

Home Ex 2 Amortization MonthMonthly payment Towards I (I=Prt) Towards PEnding balance 1

Home Ex #3 You wish to buy a $________ house with __% down. Loan: ___% for __ years, with __ points. Insurance: $__/ mn; property taxes: $__/mn Down payment= Amount Financed/Loan= Cash Price-Down *Points (paid at closing)= percentage of loan= Monthly payment to cover principal and interest PMT =

Home Ex 3 *Entire Monthly payment, including taxes and insurance = Total installment price/ total paid using PMT (note: this covers principal and interest, not tax or insurance) =(PMT*no. months)+down = Total interest paid/ finance charge = total installment price – cash price=

Home Ex 3 Amortization MonthMonthly payment Towards I (I=Prt) Towards PEnding balance 1

Credit Card Amortization problem Consider a $10,000 debt, paid back over 20 years at R=21%. Calculate monthly payment: PMT= Calculate first month interest and principal payments: I= MonthMonthly paymentTowards I (I=Prt) Towards PEnding balance 1$177.76