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Money and time are directly and inescapably related. The longer money is left on deposit, the more it earns; similarly, the longer it takes to repay a loan, the more it costs. Although this concept—that the benefit or cost of money increases over time—is easily explained, it is not always understood. This chapter explains how the time value of money works and provides formulas for calculating interest in various ways.
In the calculation of interest cost, time is the most critical element, even more so than the rate. These two factors—time and rate—define the true cost of money. When an organization borrows money through working capital loans, equipment financing, or any other vehicle, there is a tendency to focus on the interest rate only. Though the rate is important, there is more to consider, including the monthly payment required and the length of time it is going to take to retire the loan. At 7.5%, for example, a 10-year repayment is going to cost twice as much in interest as a loan for the same amount with a four-year repayment.
* Example: You borrow $20,000 from your local lender. You have a choice: repayment in four years at $483.58 per month or repayment in eight years at $277.68 per month. Your first reaction is that the lower payment is desirable. However, when you add up your total of payments for each of these loans, you discover the truth: The total for the four-year term is $23,211.84 (48 months $483.58), and the total for the eight-year term is $26,657.28 (96 months $277.68). The difference in total interest is $3,445.44. The interest cost for the longer-term loan is twice as much as for the shorter-term loan.
Selecting a repayment period is a matter of balance between the affordability of the monthly payment and the overall cost of interest. This decision is the essence of the time value of money. So in calculating the cost of repayment for this $20,000 loan, you need to evaluate the interest rate and monthly payment; however, you also need to compare the total cost of interest based on different loan repayment terms.
In addition to the monthly payment and overall interest cost, the method of interest calculation affects the total of payments as well. You need to employ different interest compounding methods, not to mention calculating the cost of borrowing money, for various reporting and budgeting purposes.
Time Value of Money: The Concept
A combination of elements defines the true cost or benefit of money. The cost is incurred when you borrow and the benefit results from savings. There are four elements:
1. Amount borrowed
2. Repayment term
3. Interest rate
4. Compounding method
1. Amount Borrowed
The most easily understood element of all is the amount borrowed. Most people understand that the more money they borrow, the higher the repayment is going to be. This simplicity is obscured, however, by the varying payment levels for different lengths of repayment.
* Example: At 7.5%, a $20,000 loan requires monthly payments of $483.58 over four years. However, you can borrow $30,000 and pay only $416.52 per month or less in monthly payments. The drawback, however, is that repayment of the $30,000 will take eight years, and the total interest is $9,985.92. (The $9,985.92 is almost equal to the additional amount borrowed: $30,000 $20,000). The smaller loan with faster repayment costs $3,211.84, or interest equal to about one-third of the longer-term loan with smaller payments.
Which loan is better suited to your needs? For most business owners and managers, the commitment to debt service that is twice the length of the original $20,000 loan has to be a primary consideration. The amount borrowed is $10,000 more, but you are committed to repayments for twice as many years.
Developing a rationale to justify the lengthier borrowing schedule is possible.
* Example: If you originally wanted only $20,000, why not borrow $30,000 and invest the difference? The payments are about the same amount, but the $10,000 is enough to repay all of the interest on the higher loan.
This argument overlooks two important facts, however. First, although the higher loan amount creates enough cash to pay the interest, you also have to repay the additional $10,000 borrowed, and that translates to twice the length of repayment. Second, will you really save the difference? As many business managers have realized, setting up a reserve and leaving it in place is difficult. Over time, management is going to be tempted to use the fund for other necessities, and ultimately the end result is the same: The longer-term loan is going to be more expensive and require a lengthier repayment commitment.
2. Repayment Term
Picking a repayment term should never be based on the monthly payment alone; it should include an analysis of cash flow requirements and limitations (see Chapter 4), as well as the affordability of borrowing. You may want to borrow money for any number of reasons, but all should be analyzed with a series of key questions:
Can I afford the repayments?
How does a loan affect my cash flow?
Have I identified how the loan will increase profits? (Profitability can be affected by expanded markets, greater efficiencies, or improved products or services.)
The repayment term might seem like a no-brainer: You want to get a loan repaid as quickly as you can afford, at the lowest interest cost, and with the least impact on cash flow. However, the question also has to depend on affordability and cash flow, not merely on the concept that "more is good" when it comes to adding debt. This common belief can not only be destructive to your ability to fund repayments while maintaining cash flow, but it can also ignore how much negative impact debt might have on future expansion and profits.
3. Interest Rate
The interest rate you are required to pay to borrow money (or that you are paid to save or invest) makes a tremendous difference over time. Some loans can be negotiated for a lower interest rate in exchange for more rapid repayment, saving money over the full term. For example, the difference between 7.0% and 7.5% is about $5.19 per month over 10 years. For a $20,000 loan, that comes out to a difference of $622.80. For a $200,000 loan, the difference is about $6,228 for that 0.5% difference in the rate. So negotiating a rate downward by a half percentage point makes a difference, and the larger the loan is, the more the dollar value of the savings.
The interest rate can also be either fixed or adjustable. Although these terms are most often associated with residential mortgage loans, they can also be applied to business loans of many types and have varying terms. An interest-only loan can be renegotiable after a few years. However, the rate you will be expected to pay is likely to change based on the interest market at the time. In this respect, the interest rate—unless fixed for the full term of the loan—is the great variable in the evaluation.
4. Compounding Method
The previous cases have all been based on monthly compounding of interest. In other words, the nominal rate (the annual rate stated by the lender) is divided by 12 (months), and the resulting monthly interest is calculated against the current loan balance. This method results in an annual rate higher than the nominal rate. As you might expect, the higher your interest rate, the more expensive monthly compounding is going to be.
Banks may charge monthly compounding rates for the money they loan, while paying you only quarterly compounded interest for funds you leave on deposit. Though this is not equitable, the banks also know that you need the loan at least as much as they want to grant it. Most managers pay little attention to the compounding method because it does not make much difference in the actual rate. For example, 7.5% compounded monthly comes out to an annual rate of 7.76% (compounding is explained later in this chapter). In comparison, quarterly compounding produces an annual rate of 7.71%, or only 0.005% less. The difference over the loan's repayment term adds up.
Simple Interest. To calculate interest, whether on a loan or a savings account, the basic formula—simple interest—is easy. Just multiply the stated interest rate by the principal amount (the amount borrowed).
Simple interest
P x R = I
where: P = principal R = interest rate I = interest
On a spreadsheet, enter the following:
A1 P B1 R C1 = SUM(A1*B1)
* Example: The amount you are thinking about borrowing for a short-term working capital loan is $5,000. The rate you are quoted is 8.0%. Simple interest is calculated as:
$5,000 X 8.0% = $400
The spreadsheet values are:
A1 5,000 B1 0 .08
BASIC MATH REVIEW
When multiplying by a percentage, convert the stated rate to decimal form. Shift the decimal two places to the left or divide by 100; either method produces the same result.
Percentage Conversion to Decimal: Decimal Shift
r.0% = 00r.0 = 0.0r
Percentage Conversion to Decimal: Divide by 100
r / 100 = D
where: r = percentage rate D = decimal equivalent
* Example: At 8.0%:
8.0% / 100 = 0.08
The recalculated decimal equivalent is used as the multiplier in the simple interest calculation. To make this calculation on a spreadsheet program, enter the following values:
A1 R B1 = SUM(A1/100)
Based on the preceding example, A1 is the value 8.00, and this results in a shift to C1 of 0.08.
Simple interest may be used for calculations in some loans, especially those due in one year or less. However, it is rarely used for most business loans. This calculation works as a sensible starting point for more complex interest calculations and for making comparisons between the stated, or nominal, rate and the annual compound rate.
Daily Compound Interest. Most interest is compounded more than once per year. The most common rates are monthly and quarterly. The periodic rate of interest (the rate paid per partial-year period based on compounding method) is the rate per cycle of compounding. For example, monthly compounding is equal to one-twelfth of the stated annual rate, which is each month's periodic rate. Quarterly compounding has a periodic rate of one-quarter (of the year). So there are 12 periods for monthly compounding and four for quarterly compounding. To find the periodic rate, divide the stated rate by the number of periods in the year.
Periodic Rate
R / p = i
where: R = nominal interest rate p = number of periods i = periodic interest rate
On a spreadsheet program, enter the following values:
A1 R B 1p C1 = SUM(A1/B1)
* Example: Your stated interest rate is 7.5%. Compounding takes place monthly, meaning there are 12 periods in the year. The periodic rate in this case is:
7.5% / 12 = 0.625%, or 0.00625 decimal
Spreadsheet values are:
A1 7.5 B1 12
Recall the conversion formula. To convert 7.5% to decimal form, shift the decimal two places to the left or divide by 100:
7.5 / 100 = 0.075 decimal
Next, the decimal equivalent is divided by the number of periods. For monthly compounding, divide by 12:
0.075 / 12 = 0.00625%
You need to know the periodic rate to calculate interest for each period and to figure out the compound annual rate. The method requiring the greatest amount of calculation, daily compounding, has a periodic rate of either 360 or 365. Using 365 is called the full-year method, and using 360 is known as the banker's year method.
To calculate daily compounding (using the 365-day method), first divide the full year's interest rate by 365. This produces the daily periodic rate.
Daily Periodic Rate (365 Days)
R / 365 = i
where: R = stated annual interest rate i = periodic interest rate (365 days)
On a spreadsheet program, enter:
A1 R (in decimal form) B1 = SUM(A1/B1)
* Example: Your stated interest rate is 7.5% (or a decimal equivalent of 0.075). The method used for calculating interest is daily, based on the 365-days-per-year rate. The daily period rate is:
0.075 / 365 = 0.0002055
Once you compute the daily rate, each day's interest is computed with a series of steps:
1. Add 1 to the daily rate. This is the first day's multiplier for a debt: 0.0002055 + 1 = 1.0002055
2. Multiply the sum in the previous step by the amount of the debt. For example, if the amount borrowed is $8,000, the first day's debt (principal plus interest) interest is: 1.0002055 X $8,000.00 = $8,001.64
3. To calculate subsequent days of the accumulated debt, multiply the preceding answer by the initial daily rate in step 1: 1.0002055 X $8,001.64 = 8,003.28
To calculate the effective interest for several days, you can use a shortcut method. Multiply the daily rate by the number of additional days and then by the initial sum.
* Example: If you want to calculate the interest as of the fifth day, multiply the daily rate by itself four times (for days two through five) and then by the principal amount:
1.0002055 X 0.0002055 X 1.0002055 X 1.0002055 X 1.0002055 X $8,000.00 = $8,008.22
A shorthand version of this formula is:
1.0002055^{5} X $8,000.00 = $8,008.22
This can be verified by checking the steps for each of the five days:
Day Rate Total
$8,000.00 1 1.0002055 8,001.64 2 1.0002055 8,003.29 3 1.0002055 8,004.93 4 1.0002055 8,006.58 5 1.0002055 8,008.22
The formula for calculating daily compounding is:
Daily Compounding
[1 + (R / i)^{n}] X P = C where: R = stated annual interest rate i = periodic interest rate (365 days) ^{n} = number of periods to be compounded P = principal ITLITL = compounded value
This series of calculations can also be placed on a worksheet and calculated using the formula feature. For spreadsheet programs, the following formulas are needed based on the placement of information in named cells:
Daily Compounding
A1 ANNUAL INTEREST RATE DIVIDED BY 365 = DAILY RATE, PLUS 1 = SUM(I/365) 1 B1 PRINCIPAL AMOUNT C1 ACCUMULATED AMOUNT = SUM(A1*B1) A2 = A1 B2 = C1 COPY C1 PASTE TO C2 COPY A2, B2, AND C2 PASTE TO ROW 3, COLUMNS 1, 2, AND 3 REPEAT PASTE FOR EACH ROW
This process is carried forward to as many days as you need. A fast shortcut for finding the effective daily rate for a large number of days is to multiply the daily rate (A3) by itself for as many days as needed (remembering that the initial sum is the first day).
* Example: For the rate applicable on the 20th day, multiply the rate 19 more times by itself. You can do this on any calculator by entering the amount, then the multiplication ( ) button, and then the equals ( ) button 19 times. In the case of the 7.5% annual (compounded daily), the 20th day's rate is:
1.0002055^{20} = 1.0041180
Next, multiply this by $8,000.00:
1.0041180 X $8,000.00 = $8,032.94
The outcome for 20 days based on the spreadsheet formula is summarized in Table 1-1.
The formula for calculating the daily debt (principal plus interest) is also called the accumulated value of 1. A visual representation of the concept of compounding interest on a single deposit is shown in Figure 1-1.
(Continues...)
Excerpted from The Manager's Pocket Calculator by Michael C. Thomsett Copyright © 2010 by Michael C. Thomsett. Excerpted by permission of AMACOM. All rights reserved. No part of this excerpt may be reproduced or reprinted without permission in writing from the publisher.
Excerpts are provided by Dial-A-Book Inc. solely for the personal use of visitors to this web site.
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