Financial Mathematics For the next few weeks we will study the mathematics of finance. Apart from basic arithmetic, financial mathematics is probably the most practical math you will learn. practical in the every-day sort of sense) After you complete this chapter you should be able to determine how much money you need to save for major purchases like cars, houses, college savings accounts for children, retirement. You should also be able to determine the size of payments on loans. This can help you make more informed decisions when financing a car or a home. If time permits, we will also look into credit card billing and credit ratings. So many people get so deeply in debt simply due to a misunderstanding of how credit cards work. Nearly everything in finance is measured in terms of percent changes, so we will begin reviewing percent increase/ percent decreases. Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 1 / 17
Percent Increase and Percent Decrease Cent means. For example, cents in a dollar, or years in a century, etc. Percent means per hundred, or out of a hundred. 78 percent just means 78. 78 percent of 200 means 78 200 which is 156. This can also be thought of as In general, P % of N equals A means 78 = 156 200 P N = A OR P = A N In applications, any two of P, A, N could be given. The above formula can be used to find the remaining variable. Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 2 / 17
Percents At a certain high school, 72% of entering freshman earn a diploma. Suppose 306 students finish with a diploma. How large was the entering class? so 72 = 306 N so 72 N = 306 N = The entering class had 425 students. 306 72 = 425 Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 3 / 17
Percents Tuition for resident students at a certain university which will remain nameless) was $3, 735 in 2001 and is currently $9, 128. Express the current tuition as a percentage of the 2001 tuition. so P = 9128 3735 P = 9128 = 244.39% 3735 Thus, the current tuition is 244.39% of the 2001 tuition. Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 4 / 17
Percents Tuition for resident students at a certain university was $3, 735 in 2001 and is currently $9, 128. Express the 2001 tuition as a percentage of the current tuition. so P = 3735 9128 P = 3735 = 40.92... % 9128 Thus, the 2001 tuition is 40.92% of the current tuition. Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 5 / 17
Setting up the correct percentage The previous two examples used the same data, but a subtle change in wording switched the role of N and A. How can we check if we set up the fractions correctly? The current tuition is more than the 2001 tuition, thus, if the current tuition is expressed as a percentage of the 2001, this percentage had better be greater than % Likewise, the 2001 tuition is less than the current tuition, thus, if the 2001 tuition is expresses as a percentage of the current tuition, this percentage had better be less than %. Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 6 / 17
Percent Increase / Percent Decrease In applications, it is very common to express only the percent change between two quantities, not the absolute percentages. The current tuition is 244.39% of the 2001 tuition, which could be expressed by saying Tuition increased by 144.39% If the population of a town was 0 people and the population increased by 25%, what is the new populaton? 25 0 = 250 The population increased by 250 people, therefore the total population is 0 + 250 = 1250 Original Population + Increase = New Population) Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 7 / 17
Percent Increase / Percent Decrease You buy a new computer. The list price is $1200. You have to pay 6% sales tax. How much do you actually pay? METHOD 1: The amount of tax is 6 1200 = 72. Thus, you pay 1200 + 72 = $1272. METHOD 2: You pay % for the computer plus 6% for the tax, thus the actual price is + 6) = 106% of the list price. Thus, you pay 106 1200 = $1272 Notice 106 = + 6 = 1 + 6 thus, the final price could also be expressed as 1200 1 + 6 ) = $1272 Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 8 / 17
The Growth Factor of a Percentage Change In the previous example, we saw that adding 6% to the price corresponded to multiplying the price by 1 + 6 ) = 1.06) I will refer to this as the growth factor corresponding to the 6% increase. For a percent increase, the growth factor will be greater than 1: A 34% increase corresponds to the growth factor 1 + 34 = 1 + 0.34 = 1.34. A 271% increase corresponds to the growth factor 1 + 271 = 1 + 2.71 = 3.71. For a percent decrease, the growth factor will be bewteen 0 and 1: A 28% decrease corresponds to the growth factor 1 28 = 1 0.28 = 0.72. Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 9 / 17
The Growth Factor of a Percentage Change To recover the percent change from a growth factor, first subtract 1, then multiply by i.e., move decimal two places to the right). The percent change is the absolute value of the number you just found. If the number is postive, then the change is a percent increase. If the number if negative, the change is a percent decrease. Find the percent change corresponding to the growth factor 1.74: Subtract 1: 1.74 1 = 0.74 Multiply : 0.74 = 74% Since this is positive, this is a 74% increase. Find the percent change corresponding to the growth factor 0.62: Subtract 1: 0.62 1 = 0.38 Multiply : 0.38 = 38%. Since this is negative, this is a 38% decrease. When applying a percent change, we can approach the problem in two ways: a) Add the percent increase or subtract the percent decrease b) Multiply by the growth factor The former is simpler in small examples. We ll see that the latter method greatly simplifies longer problems. Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 10 / 17
Percent Increase / Percent Decrease Your textbook gives these two formulas: If you start with a quantity Q and increase it by x% then you end up with the quantity I, I = 1 + x ) Q If you start with a quantity Q and decrease it by x% then you end up with the quantity D, D = 1 x ) Q You may find it easier to only remember the first quantity and to think of decreases as negative increases. Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 11 / 17
Percent Increase / Percent Decrease On Monday morning, the price of gold was $1845.00 per ounce. On Tuesday morning the price of gold was $1841.00 per ounce. Find the percent increase / percent decrease in the price of gold. Using the percent increase formula with Q = 1845 and I = 1841 we find 1841 = 1 + x ) 1845 1 + x ) = 1841 1845 = 0.997831... x = 0.997831 1 = 0.0021680... x = 0.0021680...) = 0.21680 Thus, the price of gold decreased by 0.21680%. Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 12 / 17
Back to where we started? An item typically sells for $.00 but is on sale, at 7% off. The sales tax is 7%. After the sale and the tax, do you pay $.00 for this item? Why or why not? First, deduct 7% from the original price. The sale price is therefore 1 7 ) = $93.00 The price after tax is obtained by adding 7% of the sale price, so the final price is 93 1 + 7 ) = $99.51 Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 13 / 17
How did this happen? We subtracted 7% then added 7%. Why isn t this 0%? Shouldn t 7% 7% = 0? We need to pay special attention to the bases of the percentages. In particular, 7% of 7% of IS equal to 0 BUT our problem is concerned with 7% of 93 7% of = 0.49 Thus, the combined effect of the two percent changes was to lower the price by $0.49 Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 14 / 17
Using Growth Factors When percent changes are combined, we can add the changes together, provided we remember to attach each percentage to its appropriate base. Alternatively, we can convert each percent change to its corresponding growth factor, then MULTIPLY all of the growth factors together, then convert back to a percent change. This latter method is preferable, since we don t need to keep track of the bases of each of the percentages. In the previous example, 7% off is a 7% decrease, so the corresponding growth factor is 1 0.07 = 0.93. 7% sales tax is a 7% increase, so the corresponding growth factor is 1 + 0.07 = 1.07. The net effect of applying the discount and the sales tax is to multiply: 0.93 1.07 = 0.9951. Subtracting 1 from 0.9951 and multiplying by gives the growth factor of 0.49%. The combined effect is a 0.49% decrease in the price. The final price of the item is 0.9951 $.00 = $99.51 so the two percent changes combined to lower the price by $0.49. Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 15 / 17
A more complex example The Dow Jones Industrial average started the week at 11260 points. On Monday it increased 1.9%. On Tuesday it decreased 0.7%. On Wednesday it decreased 3.5%. On Thursday it decreased 2.7%. On Friday it increased 1.4%. What was the total percent change over the week? Is this a percent increase or a percent decrease? The growth factors are 1 + 1.9 ) = 1.019 add since increase 1 0.7 ) = 0.993 subtract since decrease 1 3.5 1 2.7 ) = 0.965 subtract since decrease ) = 0.973 subtract since decrease ) = 1.014 add since increase 1 + 1.4 Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 16 / 17
A more complex example The Dow Jones Industrial average started the week at 11260 points. On Monday it increased 1.9%. On Tuesday it decreased 0.7%. On Wednesday it decreased 3.5%. On Thursday it decreased 2.7%. On Friday it increased 1.4%. What was the total percent change over the week? Is this a percent increase or a percent decrease? For the whole week, the growth factor is the product 1.019 0.993 0.965 0.973 1.014 = 0.9634 Now, if the percent change over the entire week is x% then 1 + x = 0.9634 x = 0.9634 1 = 0.0366 x = 0.0366 = 3.66% The percent change is 3.66% and this is a decrease since x is negative. Paul Koester ) MA 111, Percent Increase / Percent Decrease February 13 2012 17 / 17