Chapter 2: Mathematical Computations

Creating a Basic Formula

FormulasUsed to calculate a variety of mathematical outputs in Excel and can be used to create virtually any custom calculation required for your objective. are used to calculate a variety of mathematical outputs in Excel and can be used to create virtually any custom calculation required for your objective. Furthermore, when constructing a formula in Excel, you use cell locations that, when added to a formula, become cell referencesCell addresses used in formulas. Excel references the values entered into the cell locations used in formulas to produce outputs. When the values are changed in a cell referenced by a formula, Excel automatically produces new outputs. When a cell reference is used by itself in a cell location (cell address preceded by an equal sign [=]), Excel displays the value entered into that cell location.. This means that Excel uses, or references, the number entered into the cell location when calculating a mathematical output. As a result, when the numbers in the cell references are changed, Excel automatically produces a new output. This is what gives Excel the ability to create a variety of what-if scenarios, which will be explained later in the chapter.

To demonstrate the construction of a basic formula, we will begin working on the Budget Detail worksheet in the Personal Budget workbook, which is shown in Figure 2.2 “Budget Detail Worksheet”. To complete this worksheet, we will add several formulas and functions. Table 2.1 “Spend Category Definitions” provides definitions for each of the spend categories listed in the range A3:A11. When you develop a personal budget, these categories are defined on the basis of how you spend your money. It is likely that every person could have different categories or define the same categories differently. Therefore, it is important to review the definitions in Table 2.1 “Spend Category Definitions” to understand how we are defining these categories before proceeding.

The first formula that we will add to the Budget Detail worksheet will calculate the Monthly Spend values. The formula will be constructed so that it takes the values in the Annual Spend column and divides them by 12. This will show how much money will be spent per month for each of the categories listed in Column A. The following explains how this formula is created:

1. Click the Budget Detail worksheet tab to open the worksheet.
2. Click cell C3.
3. Type an equal sign (=). When the first character entered into a cell location is an equal sign, it signals Excel to perform a calculation or produce a logical output.
4. Type D3. This adds D3 to the formula, which is now a cell reference. Excel will use whatever value is entered into cell D3 to produce an output.
5. Type the slash symbol (/). This is the symbol for division in Excel. As shown in Table 2.2 “Excel Mathematical Operators”, the mathematical operators in Excel are slightly different from those found on a typical calculator.
6. Type the number 12. This divides the value in cell D3 by 12. In this formula, a number, or constantNumerical value used in the construction of formulas. Constants should have some universal quality that does not change, such as days in a week or minutes in an hour., is used instead of a cell reference because it will not change. In other words, there will always be 12 months in a year.
7. Press the ENTER key.

Figure 2.3 “Adding a Formula to a Worksheet” shows how the formula appears in cell C3 before you press the ENTER key. Figure 2.4 “Formula Output for Monthly Spend” shows the output of the formula after you press the ENTER key. The monthly spend for Household Utilities is $250 because the formula is taking the Annual Spend in cell D3 and dividing it by 12. If the value in cell D3 is changed, the formula automatically produces a new output. We are calculating the spend per month for each category because people often get paid and are billed for these items on a monthly basis. This formula allows you to compare your monthly income to your monthly bills to determine whether you have enough income to pay these expenses.

Relative References (Copying and Pasting Formulas)

Once a formula is typed into a worksheet, it can be copied and pasted to other cell locations. For example, Figure 2.4 “Formula Output for Monthly Spend” shows the output of the formula that was entered into cell C3. However, this calculation needs to be performed for the rest of the cell locations in Column C. Since we used the D3 cell reference in the formula, Excel automatically adjusts that cell reference when the formula is copied and pasted into the rest of the cell locations in the column. This is called relative referencingExcel automatically adjusts a cell reference used in a formula or function relative to its original location when it is pasted into new cell locations. and is demonstrated as follows:

1. Click cell C3.
2. Click the Copy button in the Home tab of the Ribbon.
3. Highlight the range C4:C11.
4. Click the Paste button in the Home tab of the Ribbon.
5. Double click cell C6. Notice that the cell reference in the formula is automatically changed to D6.
6. Press the ENTER key.

Figure 2.5 “Relative Reference Example” shows the outputs added to the rest of the cell locations in the Monthly Spend column. For each row, the formula takes the value in the Annual Spend column and divides it by 12. You will also see that cell D6 has been double clicked to show the formula. Notice that Excel automatically changed the original cell reference of D3 to D6. This is the result of relative referencing, which means Excel automatically adjusts a cell reference relative to its original location when it is pasted into new cell locations. In this example, the formula was pasted into eight cell locations below the original cell location. As a result, Excel increased the row number of the original cell reference by a value of one for each row it was pasted into.

Creating Complex Formulas (Controlling the Order of Operations)

The next formula to be added to the Personal Budget workbook is the percent change over last year. This formula determines the difference between the values in the LY (Last Year) Spend column and shows the difference in terms of a percentage. This requires that the order of mathematical operations be controlled to get an accurate result. Table 2.3 “Standard Order of Mathematical Operations” shows the standard order of operations for a typical formula. To change the order of operations shown in the table, we use parentheses to process certain mathematical calculations first. This formula is added to the worksheet as follows:

1. Click cell F3 in the Budget Detail worksheet.
2. Type an equal sign (=).
3. Type an open parenthesis (().
4. Click cell D3. This will add a cell reference to cell D3 to the formula. When building formulas, you can click cell locations instead of typing them.
5. Type a minus sign (−).
6. Click cell E3 to add this cell reference to the formula.
7. Type a closing parenthesis ()).
8. Type the slash (/) symbol for division.
9. Click cell E3. This completes the formula that will calculate the percent change of last year’s actual spent dollars vs. this year’s budgeted spend dollars (see Figure 2.6 “Adding the Percent Change Formula”).
10. Press the ENTER key.
11. Click cell F3 to activate it.
12. Place the mouse pointer over the Auto Fill Handle.
13. When the mouse pointer turns from a white block plus sign to a black plus sign, click and drag down to cell F11. This pastes the formula into the range F4:F11.

Figure 2.6 “Adding the Percent Change Formula” shows the formula that was added to the Budget Detail worksheet to calculate the percent change in spending. The parentheses were added to this formula to control the order of operations. Any mathematical computations placed in parentheses are executed first before the standard order of mathematical operations (see Table 2.3 “Standard Order of Mathematical Operations”). In this case, if parentheses were not used, Excel would produce an erroneous result for this worksheet.

Figure 2.7 “Removing the Parentheses from the Percent Change Formula” shows the result of the percent change formula if the parentheses are removed. The formula produces a result of a 299900% increase. Since there is no change between the LY spend and the budget Annual Spend, the result should be 0%. However, without the parentheses, Excel is following the standard order of operations. This means the value in cell E3 will be divided by E3 first (3,000/3,000), which is 1. Then, the value of 1 will be subtracted from the value in cell D3 (3,000−1), which is 2,999. Since cell F3 is formatted as a percentage, Excel expresses the output as an increase of 299900%.

Auditing Formulas

Excel provides a few tools that you can use to review the formulas entered into a worksheet. For example, instead of showing the outputs for the formulas used in a worksheet, you can have Excel show the formula as it was entered in the cell locations. This is demonstrated as follows:

1. With the Budget Detail worksheet open, click the Formulas tab of the Ribbon.
2. Click the Show Formulas button in the Formula Auditing group of commands. This displays the formulas in the worksheet instead of showing the mathematical outputs.
3. Click the Show Formulas button again. The worksheet returns to showing the output of the formulas.

Figure 2.8 “Show Formulas Command” shows the Budget Detail worksheet after activating the Show Values command in the Formulas tab of the Ribbon. As shown in the figure, this command allows you to view and check all the formulas in a worksheet without having to click each cell individually. After activating this command, the column widths in your worksheet increase significantly. The column widths were adjusted for the worksheet shown in Figure 2.8 “Show Formulas Command” so all columns can be seen. The column widths return to their previous width when the Show Formulas command is deactivated.

Two other tools in the Formula Auditing group of commands are the Trace Precedents and Trace Dependents commands. These commands are used to trace the cell references used in a formula. The Trace Dependents command shows where any given cell is referenced in a formula. The Trace Precedents command shows what cells have been referenced in a formula that exists in an activated cell. The following is a demonstration of these commands:

1. Click cell D3 in the Budget Detail worksheet.
2. Click the Trace Dependents button in the Formula Auditing group of commands in the Formulas tab of the Ribbon. A double blue arrow appears, pointing to cell locations C3 and F3 (see Figure 2.9 “Trace Dependents Example”). This indicates that cell D3 is referenced in formulas that are entered in cells C3 and F3.
3. Click the Remove Arrows command in the Formula Auditing group of commands in the Formulas tab of the Ribbon. This removes the Trace Dependents arrow.
4. Click cell F3 in the Budget Detail worksheet.
5. Click the Trace Precedents button in the Formula Auditing group of commands in the Formulas tab of the Ribbon. A blue arrow running through cells D3 and E3 and pointing to cell F3 appears (see Figure 2.10 “Trace Precedents Example”). This indicates that cells D3 and E3 are references in a formula entered in cell F3.
6. Click the Remove Arrows command in the Formula Auditing group of commands in the Formulas tab of the Ribbon. This removes the Trace Precedents arrow.

Figure 2.9 “Trace Dependents Example” shows the Trace Dependents arrow on the Budget Detail worksheet. The blue dot represents the activated cell. The arrows indicate where the cell is referenced in formulas.

Figure 2.10 “Trace Precedents Example” shows the Trace Precedents arrow on the Budget Detail worksheet. The blue dots on this arrow indicate the cells that are referenced in the formula contained in the activated cell. The arrow is pointing to the activated cell location that contains the formula.

The SUM Function

The SUM function is used when you need to calculate totals for a range of cells or a group of selected cells on a worksheet. With regard to the Budget Detail worksheet, we will use the SUM function to calculate the totals in row 12. It is important to note that there are several methods for adding a function to a worksheet, which will be demonstrated throughout the remainder of this chapter. The following illustrates how a function can be added to a worksheet by typing it into a cell location:

1. Click the Budget Detail worksheet tab to open the worksheet.
2. Click cell C12.
3. Type an equal sign (=).
4. Type the function name SUM.
5. Type an open parenthesis (().
6. Click cell C3 and drag down to cell C11. This places the range C3:C11 into the function.
7. Type a closing parenthesis ()).
8. Press the ENTER key. The function calculates the total for the Monthly Spend column, which is $1,496.

Figure 2.11 “Adding the SUM Function to the Budget Detail Worksheet” shows the appearance of the SUM function added to the Budget Detail worksheet before pressing the ENTER key.

As shown in Figure 2.11 “Adding the SUM Function to the Budget Detail Worksheet”, the SUM function was added to cell C12. However, this function is also needed to calculate the totals in the Annual Spend and LY Spend columns. The function can be copied and pasted into these cell locations because of relative referencing. Relative referencing serves the same purpose for functions as it does for formulas. The following demonstrates how the total row is completed:

1. Click cell C12 in the Budget Detail worksheet.
2. Click the Copy button in the Home tab of the Ribbon.
3. Highlight cells D12 and E12.
4. Click the Paste button in the Home tab of the Ribbon. This pastes the SUM function into cells D12 and E12 and calculates the totals for these columns.
5. Click cell F11.
6. Click the Copy button in the Home tab of the Ribbon.
7. Click cell F12, then click the Paste button in the Home tab of the Ribbon. Since we now have totals in row 12, we can paste the percent change formula into this row.

Figure 2.12 “Results of the SUM Function in the Budget Detail Worksheet” shows the output of the SUM function that was added to cells C12, D12, and E12. In addition, the percent change formula was copied and pasted into cell F12. Notice that this version of the budget is planning a 1.7% decrease in spending compared to last year.

Absolute References (Calculating Percent of Totals)

Since totals were added to row 12 of the Budget Detail worksheet, a percent of total calculation can be added to Column B beginning in cell B3. The percent of total calculation shows the percentage for each value in the Annual Spend column with respect to the total in cell D12. However, after the formula is created, it will be necessary to turn off Excel’s relative referencing feature before copying and pasting the formula to the rest of the cell locations in the column. Turning off Excel’s relative referencing feature is accomplished through an absolute reference. The following steps explain how this is done:

1. Click cell B3 in the Budget Detail worksheet.
2. Type an equal sign (=).
3. Click cell D3.
4. Type a forward slash (/).
5. Click cell D12.
6. Press the ENTER key. You will see that Household Utilities represents 16.7% of the Annual Spend budget (see Figure 2.14 “Adding a Formula to Calculate the Percent of Total”).

Figure 2.14 “Adding a Formula to Calculate the Percent of Total” shows the completed formula that is calculating the percentage that Household Utilities Annual Spend represents to the total Annual Spend for the budget (see cell B3). Normally, we would copy this formula and paste it into the range B4:B11. However, because of relative referencing, both cell references will increase by one row as the formula is pasted into the cells below B3. This is fine for the first cell reference in the formula (D3) but not for the second cell reference (D12). Figure 2.15 “#DIV/0 Error from Relative Referencing” illustrates what happens if we paste the formula into the range B4:B12 in its current state. Notice that Excel produces the #DIV/0Error code displayed in a cell containing a formula or function that is trying to divide by zero. error code. This means that Excel is trying to divide a number by zero, which is impossible. Looking at the formula in cell B4, you see that the first cell reference was changed from D3 to D4. This is fine because we now want to divide the Annual Spend for Insurance by the total Annual Spend in cell D12. However, Excel has also changed the D12 cell reference to D13. Because cell location D13 is blank, the formula produces the #DIV/0 error code.

To eliminate the divide-by-zero error shown in Figure 2.15 “#DIV/0 Error from Relative Referencing”, we must add an absolute reference to cell D12 in the formula. An absolute referencePrevents Excel from using relative referencing to change a cell reference used in a formula or function; an absolute reference is applied to a cell reference by typing a dollar sign ($) in front of the column letter and row number. prevents relative referencing from changing a cell reference in a formula. This is also referred to as locking a cell. The following explains how this is accomplished:

1. Double click cell B3.
2. Place the mouse pointer in front of D12 and click. The blinking cursor should be in front of the D in the cell reference D12.
3. Press the F4 key. You will see a dollar sign ($) added in front of the column letter D and the row number 12. You can also type the dollar signs in front of the column letter and row number.
4. Press the ENTER key.
5. Click cell B3.
6. Click the Copy button in the Home tab of the Ribbon.
7. Highlight the range B4:B11.
8. Click the Paste button in the Home tab of the Ribbon.

Figure 2.16 “Adding an Absolute Reference to a Cell Reference in a Formula” shows the percent of total formula with an absolute reference added to D12. Notice that in cell B4, the cell reference remains D12 instead of changing to D13 as shown in Figure 2.15 “#DIV/0 Error from Relative Referencing”. Also, you will see that the percentages are being calculated in the rest of the cells in the column, and the divide-by-zero error is now eliminated.

The COUNT Function

The next function that we will add to the Budget Detail worksheet is the COUNT function. The COUNT function is used to determine how many cells in a range contain a numeric entry. For the Budget Detail worksheet, we will use the COUNT function to count the number of items that are planned in the Annual Spend column (Column D). The following explains how the COUNT function is added to the worksheet by using the function list:

1. Click cell D13 in the Budget Detail worksheet.
2. Type an equal sign (=).
3. Type the letter C.
4. Click the down arrow on the scroll bar of the function list (see Figure 2.17 “Using the Function List to Add the COUNT Function”) and find the word COUNT.
5. Double click the word COUNT from the function list.
6. Highlight the range D3:D11.
7. Type a closing parenthesis ()).
8. Press the ENTER key. The function produces an output of 9 since there are 9 items planned on the worksheet.

Figure 2.17 “Using the Function List to Add the COUNT Function” shows the function list box that appears after completing steps 2 and 3 for the COUNT function. The function list provides an alternative method for adding a function to a worksheet.

Figure 2.18 “Completed COUNT Function in the Budget Detail Worksheet” shows the output of the COUNT function after pressing the ENTER key. The function counts the number of cells in the range D3:D11 that contain a numeric value. The result of 9 indicates that there are 9 categories planned for this budget.

The AVERAGE Function

The next function we will add to the Budget Detail worksheet is the AVERAGE function. This function is used to calculate the arithmetic mean for a group of numbers. For the Budget Detail worksheet, we will use the function to calculate the average of the values in the Annual Spend column. We will add this to the worksheet by using the Function Library. The following steps explain how this is accomplished:

1. Click cell D14 in the Budget Detail worksheet.
2. Click the Formulas tab on the Ribbon.
3. Click the More Functions button in the Function Library group of commands.
4. Place the mouse pointer over the Statistical option from the drop-down list of options.
5. Click the AVERAGE function name from the list of functions that appear in the menu (see Figure 2.19 “Selecting the AVERAGE Function from the Function Library”). This opens the Function Arguments dialog box.
6. Click the Collapse Dialog button in the Function Arguments dialog box (see Figure 2.20 “Function Arguments Dialog Box”).
7. Highlight the range D3:D11.
8. Click the Expand Dialog button in the Function Arguments dialog box (see Figure 2.21 “Selecting a Range from the Function Arguments Dialog Box”). You can also press the ENTER key to get the same result.
9. Click the OK button on the Function Arguments dialog box. This adds the AVERAGE function to the worksheet.

Figure 2.19 “Selecting the AVERAGE Function from the Function Library” illustrates how a function is selected from the Function Library in the Formulas tab of the Ribbon.

Figure 2.20 “Function Arguments Dialog Box” shows the Function Arguments dialog box. This appears after a function is selected from the Function Library. The Collapse Dialog button is used to hide the dialog box so a range of cells can be highlighted on the worksheet and then added to the function.

Figure 2.21 “Selecting a Range from the Function Arguments Dialog Box” shows how a range of cells can be selected from the Function Arguments dialog box once it has been collapsed.

Figure 2.22 “Function Arguments Dialog Box after a Cell Range Is Defined for a Function” shows the Function Arguments dialog box after the cell range is defined for the AVERAGE function. The dialog box shows the result of the function before it is added to the cell location. This allows you to assess the function output to determine whether it makes sense before adding it to the worksheet.

Figure 2.23 “Completed AVERAGE Function” shows the completed AVERAGE function in the Budget Detail worksheet. The output of the function shows that on average we expect to spend $1,994 for each of the categories listed in Column A of the budget. This average spend calculation per category can be used as an indicator to determine which categories are costing more or less than the average budgeted spend dollars.

The MAX and MIN Functions

The final two statistical functions that we will add to the Budget Detail worksheet are the MAX and MIN functions. These functions identify the highest and lowest values in a range of cells. The following steps explain how to add these functions to the Budget Detail worksheet:

1. Click cell D15 in the Budget Detail worksheet.
2. Type an equal sign (=).
3. Type the word MIN.
4. Type an open parenthesis (().
5. Highlight the range D3:D11.
6. Type a closing parenthesis ()).
7. Press the ENTER key. The MIN function produces an output of $1,200, which is the lowest value in the Annual Spend column (see Figure 2.24 “MIN Function Added to the Budget Detail Worksheet”).
8. Click cell D16.
9. Type an equal sign (=).
10. Type the word MAX.
11. Type an open parenthesis (().
12. Highlight the range D3:D11.
13. Type a closing parenthesis ()).
14. Press the ENTER key. The MAX function produces an output of $3,500. This is the highest value in the Annual Spend column (see Figure 2.25 “MAX Function Added to the Budget Detail Worksheet”).

Copy and Paste Formulas (Pasting without Formats)

As shown in Figure 2.25 “MAX Function Added to the Budget Detail Worksheet”, the COUNT, AVERAGE, MIN, and MAX functions are summarizing the data in the Annual Spend column. You will also notice that there is space to copy and paste these functions under the LY Spend column. This allows us to compare what we spent last year and what we are planning to spend this year. Normally, we would simply copy and paste these functions into the range E13:E16. However, you may have noticed the double-line style border that was used around the perimeter of the range B13:E16. If we used the regular Paste command, the double line on the right side of the range E13:E16 would be replaced with a single line. Therefore, we are going to use one of the Paste Special commands to paste only the functions without any of the formatting treatments. This is accomplished through the following steps:

1. Highlight the range D13:D16 in the Budget Detail worksheet.
2. Click the Copy button in the Home tab of the Ribbon.
3. Click cell E13.
4. Click the down arrow below the Paste button in the Home tab of the Ribbon.
5. Click the Formulas option from the drop-down list of buttons (see Figure 2.26 “Paste Formulas Option”).

Figure 2.26 “Paste Formulas Option” shows the list of buttons that appear when you click the down arrow below the Paste button in the Home tab of the Ribbon. One thing to note about these options is that you can preview them before you make a selection by dragging the mouse pointer over the options. As shown in the figure, when the mouse pointer is placed over the Formulas button, you can see how the functions will appear before making a selection. Notice that the double-line border does not change when this option is previewed. That is why this selection is made instead of the regular Paste option.

Sorting Data (Multiple Levels)

The Budget Detail worksheet shown in Figure 2.26 “Paste Formulas Option” is now producing several mathematical outputs through formulas and functions. The outputs allow you to analyze the details and identify trends as to how money is being budgeted and spent. Before we draw some conclusions from this worksheet, we will sort the data based on the Percent of Total column. As mentioned in Chapter 1 “Fundamental Skills”, sorting is a powerful tool that enables you to analyze key trends in any data set. We demonstrated the process of executing a single-level sort in Chapter 1 “Fundamental Skills”. For the purposes of the Budget Detail worksheet, we need to set multiple levels for the sort order. This is accomplished through the following steps:

1. Highlight the range A2:F11 in the Budget Detail worksheet.
2. Click the Data tab in the Ribbon.

3. Click the Sort button in the Sort & Filter group of commands. This opens the Sort dialog box, as shown in Figure 2.27 “Sort Dialog Box”.

   Figure 2.27 Sort Dialog Box

   

4. Click the down arrow next to the “Sort by” box.
5. Click the Percent of Total option from the drop-down list.
6. Click the down arrow next to the sort Order box.
7. Click the Largest to Smallest option.
8. Click the Add Level button. This allows you to set a second level for any duplicate values in the Percent of Total column.
9. Click the down arrow next to the “Then by” box.
10. Select the LY Spend option.
11. Click the OK button at the bottom of the Sort dialog box.

Figure 2.28 “Budget Detail Worksheet after Sorting” shows the Budget Detail worksheet after it has been sorted. Notice that there are three identical values in the Percent of Total column. This is why a second sort level had to be created for this worksheet. The second sort level arranges the values of 8.4% based on the values in the LY Spend column in ascending order. Excel gives you the option to set as many sort levels as necessary for the data contained in a worksheet.

Now that the Budget Detail worksheet is sorted, a few key trends can be easily identified. The worksheet clearly shows that the top three categories as a percentage of total budgeted spending for the year are Taxes, Household Utilities, and Food. All three categories are necessities (or realities) of life and typically require a significant amount of income for most households. Looking at the Percent Change column, we can see how our planned spending is expected to change from last year. This is perhaps the most import column on the worksheet because it allows you to assess whether your plan is realistic. You will see that there are no changes planned for Taxes and Household Utilities. While Taxes can change from year to year, it is not too difficult to predict what they will be. In this case, we are assuming that there are no changes to the tax costs for our budget. We are also planning no change in Household Utilities. These costs can fluctuate from year to year as well. However, you can take measures to reduce costs, such as using less electricity, turning off heat when no one is in the house, keeping track of your wireless minutes so you do not go over the maximum allowed in your plan, and so on. As a result, there is no change in planned spending for Household Utilities because we will assume that any rate increases will be offset with a decrease in usage. The third item that is planned not to change is Insurance. Insurance policies for cars and homes can change, but as is true for taxes, the changes are predictable. Therefore, we are assuming no changes in our insurance policy.

The first big change that is noticeable in the worksheet is the Food and Entertainment categories in rows 5 and 6 (see definitions in Table 2.1 “Spend Category Definitions”). The Percent Change column indicates that there is an 11.1% decrease in Entertainment spending and an 11.1% increase in Food spending. This is logical because if you plan to eat in restaurants less frequently, you will be eating at home more frequently. Although this makes sense in theory, it may be hard to do in practice. Dinners and parties with friends may be tough to turn down. However, the entire process of maintaining a budget is based on discipline, and it certainly takes a significant amount of discipline to plan targets for yourself and stick to them.

A few other points to note are the changes in the Gasoline and Vacation categories. If you commute to school or work, the price of gas can have a significant impact on your budget. It is important to be realistic if gas prices are increasing, and you should reflect these increases in your budget. To compensate for the increased spending for gas, the spending plan for vacations has been reduced by 25%. Budgeting often requires a certain degree of creativity. Although the Vacation budget has been reduced, there is still money you can set aside to make plans for spring break or winter break.

Finally, the budget shows a decrease in Miscellaneous spending of 19.8%. This was defined as a group containing several expenses, such as textbooks, school supplies, software updates, and so on (see Table 2.1 “Spend Category Definitions”). You may be able to reduce your spending in this category if you can use items such as online textbooks. This reduction in spending can free up funds for Clothes, a spend category that has increased by 20%. We will continue to develop the Personal Budget workbook further in Section 2.3 “Functions for Personal Finance”.

The Fundamentals of Loans and Leases

One of the functions we will add to the Personal Budget workbook is the PMT function. This function calculates the payments required for a loan or a lease. However, before demonstrating this function, it is important to cover a few fundamental concepts on loans and leases.

A loanA contractual agreement in which money is borrowed from a lender and paid back over a specific period of time. is a contractual agreement in which money is borrowed from a lender and paid back over a specific period of time. The amount of money that is borrowed from the lender is called the principalThe amount of money borrowed from a lender. of the loan. The borrower is usually required to pay the principal of the loan plus interest. When you borrow money to buy a house, the loan is referred to as a mortgageA loan used to purchase a home or property.. This is because the house being purchased also serves as collateral to ensure payment. In other words, the bank can take possession of your house if you fail to make loan payments. As shown in Table 2.5 “Key Terms for Loans and Leases”, there are several key terms related to loans and leases.

Figure 2.29 “Example of an Amortization Table” shows an example of an amortization tableA schedule of payments broken down by interest and principal for a loan. By law, a lender is required to provide an amortization table to a borrower. for a loan. A lender is required by law to provide borrowers with an amortization table when a loan contract is offered. The table in the figure shows how the payments of a loan would work if you borrowed $100,000 from a lender and agreed to pay it back over 10 years at an interest rate of 5%. You will notice that each time you make a payment, you are paying the bank an interest fee plus some of the loan principal. Each year the amount of interest paid to the bank decreases and the amount of money used to pay off the principal increases. This is because the bank is charging you interest on the amount of principal that has not been paid. As you pay off the principal, the interest rate is applied to a lower number, which reduces your interest charges. Finally, the figure shows that the sum of the values in the Interest Payment column is $29,505. This is how much it costs you to borrow this money over 10 years. Indeed, borrowing money is not free. It is important to note that to simplify this example, the payments were calculated on an annual basis. However, most loan payments are made on a monthly basis.

A leaseA contract in which the lessee uses an asset such as a car or a piece of equipment and agrees to make regular payments to the owner or the lessor. The lessee is often required to return the leased asset to the lessor at the conclusion of the lease contract. is a contract in which you, the lessee, use an asset such as a car or a piece of equipment and you agree to make regular payments to the owner or the lessor. When you lease a car, the manufacturer or a leasing company retains ownership of the vehicle and you agree to make regular payments for a specific period of time. The amount of money you pay depends on the price of the car, the terms of the lease contract, and the car’s expected residual value at the end of the lease. The calculation of lease payments is similar to the calculation of loan payments. However, when you lease a car, you pay only the value of the car that is used. For example, suppose you are leasing a car that is priced at $25,000. The lease contract is for 4 years at an interest rate of 5%. The residual value of the car is $10,000. This means the car will lose $15,000 of its value over 4 years. Another way to state this is that the car will depreciate $15,000. A lease will be structured so that you pay this $15,000 in depreciation. However, the interest charges will be based on the purchase price of $25,000. We will look at a demonstration of leasing a car as well as buying a home in the next section.

The PMT (Payment) Function for Loans

If you own a home, your mortgage payments are a major component of your household budget. If you are planning to buy a home, having a clear understanding of your monthly payments is critical for maintaining strong financial health. In Excel, mortgage payments are conveniently calculated through the PMT (payment) function. This function is more complex than the statistical functions covered in Section 2.2 “Statistical Functions”. With statistical functions, you are required to add only a range of cells or selected cells within the parentheses of the function. With the PMT function, you must accurately define a series of arguments in order for the function to produce a reliable output. Table 2.6 “Arguments for the PMT Function” lists the arguments for the PMT function. It is helpful to review the key loan and lease terms in Table 2.5 “Key Terms for Loans and Leases” before reviewing the PMT function arguments.

We will use the PMT function in the Personal Budget workbook to calculate the monthly mortgage payments for a house. These calculations will be made in the Mortgage Payments worksheet and then displayed in the Budget Summary worksheet through a cell reference link. So far we have demonstrated several methods for adding functions to a worksheet. The following steps explain a new method using the Insert Function command for adding the PMT function:

1. Click the Mortgage Payments worksheet tab.
2. Click cell B5.
3. Click the Formulas tab on the Ribbon.

4. Click the Insert Function button (see Figure 2.30 “Mortgage Payments Worksheet”). This opens the Insert Function dialog box, which can be used for searching all functions in Excel.

   Figure 2.30 Mortgage Payments Worksheet

   

5. In the “Search for a function:” input box at the top of the Insert Function dialog box, type mortgage payments (see Figure 2.31 “Insert Function Dialog Box”). Note that the current description in the “Search for a function:” input box will already be highlighted. You can begin typing and the description will be replaced with your entry.
6. Click the Go button in the upper right side of the Insert Function dialog box. This adds all the Excel functions that match your description in the “Select a function:” box in the lower half of the Insert Function dialog box (see Figure 2.31 “Insert Function Dialog Box”).
7. Click the PMT option in the “Select a function:” box in the lower half of the Insert Function dialog box.

8. Click the OK button at the lower right side of the Insert Function dialog box. This will open the Function Arguments dialog box.

   Figure 2.31 Insert Function Dialog Box

   

   Mouseless Commands

   Insert Function

   • Hold the SHIFT key while pressing the F3 key.
9. Click the Collapse Dialog button next to the Rate argument in the Function Arguments dialog box. This will be the first argument defined for the function.
10. Click cell B3 on the worksheet. This is the rate being charged on the loan.
11. Type a forward slash (/) for division.
12. Type the number 12. Since our goal is to calculate the monthly payments for the loan, we need to divide the rate, which is stated in annual terms, by 12. This converts the annual rate to a monthly rate.
13. Press the ENTER key on your keyboard. This returns the Function Arguments dialog box to its expanded form. You will also see that the Rate argument is now defined.
14. Click the Collapse Dialog button next to the Nper argument in the Function Arguments dialog box. This is the second argument we define in the function.
15. Click cell B4 on the worksheet. This is the term or the amount of time we have to repay the loan.
16. Type an asterisk (*) for multiplication.
17. Type the number 12. Since our goal is to calculate the monthly payments for the loan, we need to multiply the terms of the loan by 12. This converts the terms of the loan from years to months.
18. Press the ENTER key on your keyboard. This returns the Function Arguments dialog box to its expanded form. You will also see that the Nper argument is now defined.
19. Click the Collapse Dialog button next to the Pv argument in the Function Arguments dialog box. This is the third argument we will define in the function.
20. Type a minus sign (−). When defining the Pv argument of the PMT function, any cell location or value must be preceded with a minus sign.
21. Click cell B2 on the worksheet. This is the principal of the loan.
22. Press the ENTER key on your keyboard. You will now see the Rate, Nper, and Pv arguments defined for the function.
23. Click the OK button at the bottom of the Function Arguments dialog box. The function will now be placed into the worksheet. Since we are not paying any lump sums of money at the end of the loan, there is no need to define the Fv argument. Also, we will assume that the monthly mortgage payments will be made at the end of each month. Therefore, there is no need to define the Type argument.

Figure 2.32 “Function Arguments Dialog Box for the PMT Function” shows the completed Function Arguments dialog box for the PMT function. Notice that the dialog box shows the values for the Rate and Nper arguments. The Rate is divided by 12 to convert the annual interest rate to a monthly interest rate. The Nper argument is multiplied by 12 to convert the terms of the loan from years to months. Finally, the dialog box provides you with a definition for each argument. The definition appears when you click in the input box for the argument.

Figure 2.33 “Mortgage Payments Worksheet with the PMT Function” shows the final appearance of the Mortgage Payments worksheet after the PMT function is added. The result of the function in cell B5 will be displayed in the Budget Summary worksheet.

The PMT (Payment) Function for Leases

In addition to calculating the mortgage payments for a home, the PMT function will be used in the Personal Budget workbook to calculate the lease payments for a car. The details for the lease payments are found in the Car Lease Payments worksheet. Similar to the statistical functions, we can type the PMT function directly into a cell. However, you must know the definitions for each argument of the function and understand how these arguments need to be defined based on your objective. The terms for loans and leases are in Table 2.5 “Key Terms for Loans and Leases”, and the definitions for the arguments of the PMT function are in Table 2.6 “Arguments for the PMT Function”. The following steps explain how the PMT function is added to the Personal Budget workbook to calculate the lease payments for a car:

1. Click cell B6 in the Car Lease Payments worksheet.
2. Type an equal sign (=).
3. Type the letters PMT.
4. Type an open parenthesis (().Excel then provides a tip box showing the arguments of the function.
5. Click cell B4. This is the interest rate being charged for the lease.
6. Type the forward slash (/) for division.
7. Type the number 12. Since our goal is to calculate the monthly lease payments, we divide the interest rate by 12 to convert the annual rate to a monthly rate.
8. Type a comma. When you type a function containing arguments, you must separate each argument with a comma. This signals to Excel that one argument has been defined and you are ready to define the next argument in the function.
9. Click cell B5. This is the term or the length of time for the lease contract. Since the term is already expressed in months, we can just reference cell B5 and move to the next argument.
10. Type a comma. This advances the function to the Pv argument.
11. Type a minus sign (−). Remember that cell locations or values used to define the Pv argument must be preceded with a minus sign.
12. Click cell B2 on the worksheet, which is the price of the car.
13. Type a comma. This advances the function to the [Fv] argument.
14. Click cell B3 on the worksheet. This is the residual value of the car. Note that cell location and values used to define the [Fv] argument are NOT preceded by a minus sign.
15. Type a comma. This advances the function to the [Type] argument.
16. Type the number 1. We will assume that the lease payments will be due at the beginning of each month.
17. Type a closing parenthesis ()).
18. Press the ENTER key.

Figure 2.34 “PMT Function Constructed to Calculate Lease Payments” shows how the PMT function should appear before pressing the ENTER key. Notice the commas that separate each argument of the function. Also, the tip box will show the current argument being defined in bold font.

Figure 2.35 “Results of the PMT Function in the Car Lease Payments Worksheet” shows the result of the PMT function. The monthly payments for this lease are $206.56. This monthly payment will be displayed in the Budget Summary worksheet.

Linking Worksheets (Creating a Summary Worksheet)

So far we have used cell references in formulas and functions, which allow Excel to produce new outputs when the values in the cell references are changed. Cell references can also be used to display values or the outputs of formulas and functions in cell locations on other worksheets. This is how data will be displayed on the Budget Summary worksheet in the Personal Budget workbook. Outputs from the formulas and functions that were entered into the Budget Detail, Mortgage Payments, and Car Lease Payments worksheets will be displayed on the Budget Summary worksheet through the use of cell references. The following steps explain how this is accomplished:

1. Click cell C3 in the Budget Summary worksheet.
2. Type an equal sign (=).
3. Click the Budget Detail worksheet tab.
4. Click cell D12 on the Budget Detail worksheet.
5. Press the ENTER key on your keyboard. The output of the SUM function in cell D12 on the Budget Detail worksheet will be displayed in cell C3 on the Budget Summary worksheet.

Figure 2.36 “Cell Reference Showing the Total Expenses in the Budget Summary Worksheet” shows how the cell reference appears in the Budget Summary worksheet. Notice that the cell reference D12 is preceded by the Budget Detail worksheet name enclosed in apostrophes followed by an exclamation point (‘Budget Detail’!) This indicates that the value displayed in the cell is referencing a cell location in the Budget Detail worksheet.

As shown in Figure 2.36 “Cell Reference Showing the Total Expenses in the Budget Summary Worksheet”, the Budget Summary worksheet is designed to show the expense budget for the mortgage payments and the auto lease payments. However, you will recall that we used the PMT function to calculate the monthly payments. In the Budget Summary worksheet, we need to show the total annual payments. As a result, we will create a formula that references cell locations in the Mortgage Payments and Car Lease Payments worksheets. The following steps explain how this is accomplished:

1. Click cell C4 in the Budget Summary worksheet.
2. Type an equal sign (=).
3. Click the Mortgage Payments worksheet tab.
4. Click cell B5 in the Mortgage Payments worksheet.
5. Type an asterisk (*) for multiplication.
6. Type the number 12. This multiplies the monthly payments by 12 to calculate the total payments required for the year.
7. Press the ENTER key on your keyboard. The value of multiplying the monthly mortgage payments by 12 is now displayed on the Budget Summary worksheet.
8. Click cell C5 on the Budget Summary worksheet.
9. Type an equal sign (=).
10. Click the Car Lease Payments worksheet tab.
11. Click cell B6 in the Car Lease Payments worksheet.
12. Type an asterisk (*) for multiplication.
13. Type the number 12. This multiplies the monthly lease payments by 12 to calculate the total payments required for the year.
14. Press the ENTER key on your keyboard. The value of multiplying the monthly lease payments by 12 is now displayed on the Budget Summary worksheet.

Figure 2.37 “Formulas Referencing Cells in Mortgage Payments and Car Lease Payments Worksheets” shows the results of creating formulas that reference cell locations in the Mortgage Payments and Car Lease Payments worksheets.

We can now add other formulas and functions to the Budget Summary worksheet that can calculate the difference between the total spend dollars vs. the total net income in cell D2. The following steps explain how this is accomplished:

1. Click cell D6 in the Budget Summary worksheet.
2. Type an equal sign (=).
3. Type the function name SUM followed by an open parenthesis (().
4. Highlight the range C3:C5.
5. Type a closing parenthesis ()) and press the ENTER key on your keyboard. The total for all annual expenses now appears on the worksheet.
6. Click cell D7 on the Budget Summary worksheet.
7. Type an equal sign (=).
8. Click cell D2.
9. Type a minus sign (−) and then click cell D6.
10. Press the ENTER key on your keyboard. This formula produces an output of $1,942, indicating our income is greater than our total expenses.

Figure 2.38 “Formulas Added to Show Income Is Greater Than Expenses” shows the results of the formulas that were added to the Budget Summary worksheet. The output for the formula in cell D7 shows that the net income exceeds total planned expenses by $1,942. Overall, having your income exceed your total expenses is a good thing because it allows you to save money for future spending needs or unexpected events.

We can now add a few formulas that calculate both the spending rate and the savings rate as a percentage of net income. These formulas require the use of absolute references, which we covered earlier in this chapter. The following steps explain how to add these formulas:

1. Click cell E6 in the Budget Summary worksheet.
2. Type an equal sign (=).
3. Click cell D6.
4. Type a forward slash (/) for division and then click D2.
5. Press the F4 key on your keyboard. This adds an absolute reference to cell D2.
6. Press the ENTER key. The result of the formula shows that total expenses consume 94.1% of our net income.
7. Click cell E6.
8. Place the mouse pointer over the Auto Fill Handle.
9. When the mouse pointer turns to a black plus sign, left click and drag down to cell E7. This copies and pastes the formula into cell E7.

Figure 2.39 “Calculating the Savings Rate” shows the output of the formulas calculating the spending rate and savings rate as a percentage of net income. The absolute reference shown for cell D2 prevents the cell from changing when the formula is copied from cell E6 and pasted into cell E7. The results of the formula show that our current budget allows for a savings rate of 5.9%. This is a fairly good savings rate. In the next section we will discuss how these savings can grow over time by exploring the time value of money concepts.

Time Value of Money Concepts

In reviewing the Budget Summary worksheet in Figure 2.39 “Calculating the Savings Rate”, you will notice that the range B9:D14 contains data that can be used to assess a savings plan. We can project how much money can be saved over a specific period of time given set contributions and a rate of return. This calculation is accomplished through the future value, or FV, function. We will use the FV function in cell D10 of the Budget Summary worksheet to calculate our savings plan projection. However, before we use the FV function, it is important to review a few basic concepts regarding the time value of money, as shown in Table 2.7 “Key Terms for Time Value of Money Concepts”.

Table 2.7 “Key Terms for Time Value of Money Concepts” provides definitions for several terms used when addressing the time value of money concepts. The time value of moneyThe opportunity to increase the value of money over time through investments that provide a constant or average positive rate of return. is the opportunity to grow your money over time given a constant or average rate of return. For example, consider the data shown in Figure 2.40 “Time Value of Money Example for a One-Time Investment”. This data assumes that a person makes a one-time investment of $100 in a bond mutual fund that returns 5% interest per year. Notice that the interest paid in Column E increases every year. This is because the interest is reinvested in the mutual fund, which increases the total value of the investment. For example, the interest earned in year 1 is based on a $100 investment. Therefore, the interest paid is $5.00, or 5% of $100. However, in year 2, when the $5.00 interest payment is reinvested, the total investment increases to $105. Therefore, in year 2 the interest paid increases to $5.25, or 5% of $105. The value of the investment at the end of 5 years is $127.63. This is the value that can be calculated using the FV function.

Figure 2.41 “Time Value of Money Example for an Annuity Investment” shows another example demonstrating the time value of money concept. Instead of making a one-time investment, we will assume that a person invests $100 at the beginning of every year in the same bond mutual fund. This is referred to as an annuityAn investment made in regular payments over a period of time. because the person is making reoccurring investments over a specific period of time. Notice that the value of this investment after 5 years is $580.19. Also, the total interest earned on this investment is $80.19 as opposed to the $27.63 earned on the one-time investment in Figure 2.40 “Time Value of Money Example for a One-Time Investment”.

The FV (Future Value) Function

Establishing a personal savings plan is one of the most important financial exercises you can do. For example, a savings plan is critical for establishing financial security for your retirement years. Many people mistakenly believe that saving for retirement is something you do when you get older. However, the greatest financial gains for your retirement can be achieved if you start saving in the earliest years of your career. Now that you have an understanding of the time value of money, you can see that the more years you can earn interest on your investments and reinvest those earnings, the more money you will have when you retire. Savings plans are also important for other key life events, such as going to college or buying a home. The FV function is a convenient tool that can help you establish savings goals and project the value of your investments over time. Similar to the PMT function, the FV function requires you to accurately define specific arguments in order to produce a reliable result. Table 2.8 “Arguments for the FV Function” provides definitions for each of the arguments in the FV function. It is helpful to review the time value of money terms in Table 2.7 “Key Terms for Time Value of Money Concepts” before using the FV function.

With respect to the Personal Budget workbook, we will use the FV function to project the value of the savings plan in 10 years. We will type the function directly into the Personal Budget worksheet for this demonstration. However, you can use any of the methods demonstrated in this chapter for future use. The following steps explain how this function is added to the worksheet:

1. Click cell D10 in the Budget Summary worksheet.
2. Type an equal sign (=).
3. Type the letters FV followed by an open parenthesis (().
4. Click cell D13. This is the expected rate of return for the investments.
5. Type a comma.
6. Click cell D12. This is the amount of time the investments are expected to grow.
7. Type a comma.
8. Type a minus sign (−). All values or cell locations used to define the Pmt argument must be preceded by a minus sign.
9. Click cell D7. This is the change in cash that was calculated by subtracting the total expenses from the net income. We are expecting to save this amount of money for the 10-year period this investment is being measured.
10. Type a comma.
11. Type a minus sign (−). All values and cell locations used to define the Pv argument must be preceded by a minus sign.
12. Click cell D14. Since the savings plan has a current balance, we use this to define the Pv argument of the function. This is equivalent to starting with a lump-sum investment.
13. Type a closing parenthesis ()). There is no need to define the last argument of the function because we will assume that the savings in cash achieved in our budget will be invested at the end of each year of the savings plan.
14. Press the ENTER key. Check that cell D11 is activated.
15. Type an equal sign (=).
16. Click cell D10.
17. Type a minus sign (−) and then click cell D9. This subtracts the savings plan from the current savings plan projection.
18. Press the ENTER key.

Figure 2.42 “Results of the Savings Plan Projections” shows the results of the FV function. Notice that the current savings plan projection is $25,606. This is $606 higher than the target of $25,000 entered into cell D9, which shows that the current budget is working to achieve the goals of this savings plan. In other words, given the current net income, we are saving enough money to achieve our savings plan goals.

There are two important factors to notice with regard to this plan. The first factor is that our spending plan allows us to save enough money so that it can be invested to achieve our target of $25,000. The second factor is that the expected rate of return is 3.5%. This is a relatively low expected rate of return and could be achieved by investing in relatively low-risk investments such as bonds as opposed to stocks. This rate can be considered good because we can achieve our savings goals without having to make high-risk investments that could result in a significant loss of our savings.

Goal Seek (What-If Scenarios)

We used several formulas and functions to complete the Personal Budget workbook shown in Figure 2.42 “Results of the Savings Plan Projections”. All the formulas and functions entered contain cell references that allow for a variety of what-if scenarios. Goal Seek is a tool that can be used in the process of conducting these what-if scenarios. Goal Seek maximizes the benefits of Excel’s cell-referencing capabilities by changing inputs to precise values to achieve specific outputs produced by formulas or functions. We will begin by changing one of the inputs in the Personal Budget workbook through the following steps:

1. Click the Budget Detail worksheet tab.
2. Click cell D9.
3. Type the number 2000. Instead of planning a decrease in our vacation spending, we will see what happens to our budget if we spend the same amount as last year, which was $2,000.
4. Press the ENTER key.

Figure 2.43 “Budget Detail Worksheet “ and Figure 2.44 “Budget Detail Worksheet “ show the Budget Detail worksheet before and after the change in the annual vacation budget. By comparing these two figures you can see that by changing just one input, many of the outputs produced by the formulas and functions in the worksheet changed. The following is a list of the changes that occurred in the worksheet:

• The formula output in cell F12 now shows that we are planning a 1.1% increase in our total spending as opposed to a −1.7% decrease.
• The formula output in cell F9 changes from −25% to 0%.
• The SUM function in cell D12 changes from $17,950 to $18,450.
• The SUM function in cell C12 changes from $1,496 to $1,538.
• The AVERAGE function in cell D14 changes from $1,994 to $2,050.

In addition to the changes in the Budget Detail worksheet, outputs of formulas and functions on the Budget Summary worksheet also change when the Annual Spend for the Vacation category was increased. To see the changes, compare Figure 2.42 “Results of the Savings Plan Projections” to Figure 2.45 “Budget Summary Worksheet “. There were a total of fourteen changes in the outputs of formulas and functions on the Budget Summary worksheet. In total, there were twenty-one outputs that changed in the Personal Budget workbook as a result of changing just one input.

One of the most notable changes on the Budget Summary worksheet is the Savings Projection in cell D10. By spending an additional $500 a year on vacation plans, the projected savings value in 10 years decreases by $5,865. However, what if the rate of return were to increase? An increase in the rate of return could recover the decrease in the future value of our savings plan. We can use a tool such as Goal Seek to determine exactly how much the rate of return would have to increase to achieve our savings plan target of $25,000. The following steps explain how to use Goal Seek to accomplish this goal:

1. Click the Budget Summary worksheet tab.
2. Click the Data tab of the Ribbon.
3. Click the What-If Analysis button in the Data Tools group of commands.

4. Click Goal Seek from the list options (see Figure 2.46 “Selecting Goal Seek from the What-If Analysis Options”). This opens the Goal Seek dialog box.

   Mouseless Commands

   Goal Seek

   • Press the Alt key on your keyboard and then the letters A, W, and G one at a time.

   Figure 2.46 Selecting Goal Seek from the What-If Analysis Options

   

5. Click the Collapse Dialog button next to the “Set cell:” input box on the Goal Seek dialog box.
6. Click cell D10 on the Budget Summary worksheet.
7. Press the ENTER key on your keyboard.
8. Place the mouse pointer over the “To value” input box in the Goal Seek dialog box and click.
9. Type the number 25000 in the “To value” input box in the Goal Seek dialog box.
10. Click the Collapse Dialog button next to the “By changing cell” input box in the Goal Seek dialog box.
11. Click cell D13 on the Budget Summary worksheet.
12. Press the ENTER key on your keyboard.
13. Click the OK button on the Goal Seek dialog box.
14. Click the OK button on the Goal Seek Status dialog box (see Figure 2.48 “Solution Calculated by Goal Seek”). The status box is telling you that Excel found a value for cell D13 that produces an output of $25,000 for the FV function in cell D10.
15. Figure 2.47 “Final Settings for the Goal Seek Dialog Box” shows the final settings for the Goal Seek dialog box before clicking the OK button.

Figure 2.48 “Solution Calculated by Goal Seek” shows the solution Goal Seek calculated for the rate of return. Notice that in order to achieve the target savings plan of $25,000, the rate of return must increase to 7.8%. Initially, it appears that we can spend the additional $500 a year on vacations and still achieve our savings goal of $25,000. However, achieving a 7.8% annual rate of return will require us to make riskier investments with our savings. Thus, there is a greater possibility that we could lose a substantial amount of our savings. This is the downside of decreasing your overall savings rate. If you save less money, it forces you to take higher risks with the money you have in order to achieve higher rates of return. Unfortunately, many people end up on the losing end of these risks, which severely compromises their ability to reach their savings goals.