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Transcript of F&S Presentation
A= P (1+r/n)^nt Planning for the Future with an Annuity:
P= A (r/n) __________ [(1+r/n)^nt-1] Calculating Annuities Value of an Annuity: Interest Compounded Once a Year
A= P (1+r)^t Methods Two methods an Annuity payout would be, annuitization method and systematic withdrawal schedule. An annuitization method would give you monthly income for a period of time. There are many options you could choose from with this method is: Life option, joint-life option, period certain, and life with guaranteed term.
Systematic withdrawal schedule, you could set an amount of money you would want to receive each month. With this you can also choose a lump-sum payment. Example A= 2,000 (1+.05/12)^12x1 You deposit $2,000 in an account that pays 5% interest compounded monthly. =2,000(1.00416666667)^12 = 2,000 (1.05116189788) = 2102.32 Formula: A= P (1+r/n)^nt Income Tax Tax calculations depend on filing status which consist of four categories: 1. Single - unmarried, divorced, or legally separated.
2. Married filing jointly - married, you and your spouse file a single tax return.
3. Head of Household - unmarried and paying more than half the cost of supporting a dependent child or parent.
4. Exemptions & Deductions - subtracted from your adjusted gross income Exemptions: amounts fixed for each person, also for each of your dependents. Deductions: one person to another
-Standard Deduction: depends on your filing status
-Itemized Deduction: sum of all individual deductions to which you are entitled. Calculating Income Tax:
1. Adjusted gross income = gross income - adjustments
2. taxable income=
adjusted gross income - (exemptions + deductions)
3. Income tax: tax computation - tax credits Example Single woman with no dependents, whose gross income, adjustments, deductions, and credits are given. Gross income: $52,100
Tax credit: $200
1500: property taxes
2400: charitable contributions
2300: medical expenses not covered the insurance Example adjusted gross income= 51,000 - 3,000 taxable income = 48,000 - (3,500 + 6,500) = $38,000 income tax = 5,843.75 - 200 = 5,643.75 = 48,000 Installment Loans loan with payments to payoff each time period. P (r/n) 1- (1+r/n) -nt ] [ ______________ formula: example You decide to borrow $15,000 to buy a new car. Loan A: 3 year loan at 5%
Loan B: 5 year loan at 7% First you would have to find the monthly payments and total interest for Loan A.
Secondly, find the monthly payments and total interest for Loan B. And for the third step, compute the monthly payments and total interest loans. 1. PMT = 15000 (0.05/12) [1-(1+0.05/12) -(12)(3) _____________ = 62.5 0.13902375534 _______ = 449.56 = $450 Simple and Compound Interest Simple interest- interest paid only on the original principal and NOT on any interest added at later dates. Compound Interest- is interest paid on both the original principal and on all interest that has been added to the original principal. Interest- the dollar amount that we yet paid for lending money or pay for borrowing money. Rate- which is given as a percent, the amount of interest. Principal- in financial formulas is the balance upon which interest is paid. Formulas Simple Interest: PrincipalxRatexTime I= Prt Example: P=100, r=8% (0.08), t=1yr I= 100x0.08x1 I=8 Future Value Formula for Simple Interest: A=P(1+rt) Compound Interest Formula: A=P(1+r) t Example Continued Symbols A(future value)= accumulated balance after T years. P=starting Principal r or APR= interest rate (written as a decimal n= number of compounding periods per yr t= number of years semi-annually= 2 periods monthly= 12 periods quarters in a year= 4 annually= 1 Compound Formula:
Monthly- A=P(1+r/n) t rt nt Example 1. Show how quarterly compounding affects a $1000 investment at 8% per year.
= 1082.43 Sampling Random Sampling: is a sample obtained in such a way that every element in the population has an equal chance of being selected. Simple Random Sampling: choose a sample of items in such a way that every sample of a given size has an equal chance of being selected. Systematic Sampling: we use a sample to choose the sample, such as selecting every 10th of every 50th number of population. Sampling Continued Convenience Sampling: we use sample that is convenient to select, such as people who happen to be in the same classroom. Stratified Sampling: we use this method of sampling when we are concerned about differences among the subgroups, or strata, within a population. First would have to identify the subgroups and then draw a simple random sample within each group The total sample consist of all the samples from the individual subgroups. Representative Sampling: is a sample in which the relevant characteristics of the sample members match those of the population. Methodolgy:
-Identify each element in the population
-assign numbers to each element in the population
-randomly select numbers
-assign the element in the populations who have those numbers to the sample set. Basic Steps of Statistical Study 1. State the goal of your study precisely. 2. Choose a representative sample from the population. 3. Collect data from the sample and summarize these data by finding sample statistics of interest. 4. Use the sample statistics to infer the population parameters. (use math here) 5. Draw conclusions. Start Population Sample Sample Statistics Population
Parameters 1. Identify goal 2. Draw from population 3. Collect raw data
and summarize 4. make inferences
about population 5. draw
conculsion Frequent Distribution Frequency Tables have 2 columns. The first column is categories, the second column lists the frequency of each category, which is the number of times each category appears in the data set. Relative Frequency: expressed as a fraction or percentage of the total. Cumulative Frequency; total of frequencies for the given category. A bar chart shows using frequency or relative. A pie chart is used when relative is a total of 100%. Frequency Distribution Mean = average
Median = middle (least to greatest)
Mode = most repeated number to find the mean you divide sum of all values by total number of values. median = n+1/2 midrange = lowest data value+highest data value/2 xf x f 1
46.9 mean= 217.7/117.2 = 1.857 mode= 6.7 median= 46.9+1/2 = 23.95 f= frequency of data value x= each data value Measures of Dispersion Measures of Dispersion are used to describe the spread of data items in a data set. There are two most common measures of dispersion are range and standard deviation. Range = highest value - lowest value Deviation from Mean = data item - mean Computing The Standard Deviation for a Data Set 1. Find the mean of data items 2. Find deviation of each data item from the mean:
data item - mean 3. Square each deviation;
(data item - mean)^2 4. Sum the squared deviations:
E(data item - mean)^2 5. Divide the sum in step 4 by n-1, where n represents the number of data items:
E(data item - mean)^2/n-1 6. Take the square root of quotient in steps 5. This value is the standard deviation for data set.
standard deviation = E(data item - mean)^2/n-1 Example Data Item x-x - (x-x)^2 - 17
3^2=9 E(x-x)=0 E(x-x)^2=28 - - 28/7-1=28/6=4.7 SD= (sqr) of 4.7=2.17 = = Normal Distribution Normal Distribution - is a bell shaped and symmetric about a vertical line through the center. For mean, median, and mode it would be equal on both sides. The shape of the normal distribution depends on the mean. There are some graphs with the same mean but different standard deviation. Condition for Normal Distribution 1. Most data value are bunched up near the mean and gives the distribution a nice peak. 2. Data values are spread evenly around the mean, makes the distribution symmetric. 3.Large deviation from the mean are rare. 4. Individual data values result from a combination of many different factors. 68-95-99 Rule for Normal Distribution 1. About 68% of the data items fall within 1 sd of mean. 2. About 95% of data items fall within 2 sd of mean 3. about 99.7% of data items fall within 3 sd of mean. Scatter Plot, Correlation, and Regression Lines Scatter Plot- collection of data points, one data point per person or object. - can be used to determine if 2 quantities are related Correlation- clear relationships between 2 quantities. - used to determine if relationship between 2 variables and if so the strength and direction of that relationship, Correlation Coefficient- measure that is used to see the strength and direction of a relationship between variables. Examples of plots and correlations. Positive Correlation- tend to increase or decrease together. Negative Correlation- on variable tends to decrease Perfect Correlation- which all points lie precisely on the regression line that rises from the left to right. Perfect Negative Correlation- which all points in the scatter point lie precisely on regression line that falls left to right . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . ... .. . . . . . . . . . . . . . . . . ... . . . .. .. . .. . . . . . ... . . . .. . . . . . . Permutation Permutation- is an ordered arrangement of item that occurs when: -No item is used more than once
-The order of arrangement make a difference Example Four friends are walking. How many different ways could they arrange themselves in this side by side pattern? 4!= 4*3*2*1= 24 ways Notation of Permutation- nPr= n!/(n-r)! != Factorial Factorial notation- n! Permutation with Repetition- n!/a!b!c! Combination Combination- items occur when: -Items are selected from the same group
-No item is used more than once
-The order of items make no difference nCr= n!/(n-r)! Example In poker a person is dealt 5 cards from a standard 52 card deck. How many different 5-card poker hands are possible? 52C5= 52!/(52-5)!= 52!?47!5!= 52*51*50*49*48/5*4*3*2*1= 2,598,960 Probability With Permutation & Combination Example: A group consist of four men and five women. Three people are selected to attend a conference. A. In how many ways can three people be selected from the five women? B. In how many ways can three women be select from the five women? 9C3= 9!/(9-3)!3!= 9*8*7*6!/6!*3*2*1=84 5C3= 5!/(5-3)!3!= 5*4*3!/2*1*3!=10 C. Find the probability that the selected group will consist of all women 10/84= 5/42 Events Involving Not and Or; Odds Complements Rules of Probability P(not E)= 1-P(E) P(E)=1-P(E) Example: If you are dealt on card from a standard 52 card deck, find the probability that you are not dealt a queen. P(not E)= 1-P(E) P(not a queen)=1-P(queen) Then The Probability of being dealt a queen is 4/52=1/13 P(not a queen)=1-P(queen)=1-1/13=13/13-1/13=13 The probability that you are dealt a queen is 12/13. Expected Value Expected Value is a mathematical way to use probabilities to determine what to expect in various situation over the long run. Example: Find the expected value for the outcome of the roll of a fair die. Die have 6 sides, each with a probability of 1/6. E=1*1/6*2*1/6*3*1/6*4*1/6*5*1/6*6*1/6= 1+2+3+4+5+6/6 =21/6 =3.5 The expected value of the roll of a fair die is 3.5 Exponential Functions Exponential Functions can be used to model this explosive growth, typically associated with populations, epidemics, and interest-bearing bank accounts. The exponential function with base b is defined by y=bx or f(x)=bx
Where b is positive constant other than 1 (b>0 and b=1) Example Solve for y when x=5 y=1.5^5 that means we need to plug in x (x=5) Y=1.5^5 = 7.59375= 7.6 Log Function For x>0 and b>0, b=1 y=log10 x is equivalent to b^y=x The function f(x)= logb^x is the log function with base b. Example for f(x)= 2^x -- table value x f(x)=2^x -2 -1 0 1 2 3 1/4 1/2 1 2 4 8 Population Growth and Decay At the start of an experiment there are 100 bacteria. If the bacteria follow an exponential growth pattern with rate k = 0.02, what will be the population after 5 hours? Formula- P(t)=P(o)e^kt P(o)=100 ; K= 0.02 : t=5 Therefore- P(5)=100e^0.02*5= 110.517 How long will it take for the population to double? 2-100=100e^0.02t - 2=e^0.02t LOG on both sides - ln2= lne^0.02t= 0.02tlne= 0.02t t=ln2/0.02= 34.65= 35 hours