In Lesson 3, we will learn about Annual Percentage Rates (APR), Salvage Values, Bond Investments, Financial Costs, and Opportunity Costs of Capital. APR is another rate that is important to this class. Bond is a common investment tool these days. After this lesson, students will also be able to distinguish the financial cost of capital and the opportunity cost of capital. Similar to the previous lesson, the introduction in this class will be based on examples, textbook reading, and assigned reading materials.
At the successful completion of this lesson, students should be able to:
This lesson will take us one week to complete. Please refer to the Course Syllabus for specific time frames and due dates. Specific directions for the assignment below can be found within this lesson.
Reading | Read Chapter 3 of the textbook. |
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Assignment | Homework and Quiz 3. |
If you have any questions, please post them on our discussion forum (not email), located under the Modules tab in Canvas. I will check that discussion forum daily to respond. While you are there, feel free to post your own responses if you, too, are able to help out a classmate.
Annual Percentage Rate (APR) is usually used for loans, mortgages, and so on. APR represents an annualized expression of the cost of borrowing money.
When you take out a loan or mortgage on a property, in addition to the interest, you are required to pay some other transaction costs such as points*, loan origination fees, a home inspection fee, mortgage insurance premiums, … . Considering these costs, the amount of money that you will receive is actually somewhat less than what you requested. APR is the expression that reflects some of these costs, and under the Federal Truth in Lending Law, Regulation Z, the lender is required to provide this information to the borrower. Since APR includes mentioned transaction costs, it is higher than interest rates. You can think of APR as the rate of return on the loan taking process considering its costs.
* Loan points are a percentage of the loan value that is deducted as transaction cost.
APR can be a good tool for comparing different loans offered by lenders. But there are two issues that need to be considered before comparing APRs:
Annual Percentage Rate Video (1:34) [1]
Investopedia presents: Annual Percentage Rate
The annual percentage rate or APR is the cost per year of borrowing. By law, all financial institutions must show customers the APR of a loan or credit card. Which clearly indicates the real cost of the loan. APR is not the same as the interest rate on a loan. Loans charge and interest rate but usually also charge other fees such as closing costs, origination fees, or insurance costs which are typically wrapped into the loan. If two loans have the same interest rate but one has much higher fees than the other, simply shopping by interest rates won’t give an accurate comparison of the loan's true cost. That’s why there is an APR. By factoring in other fees APR gives a more accurate estimate of the cost per year of a loan. For this reason, the APR is generally higher than the interest rate.
For example, a mortgage company may offer a customer an interest rate of 4% on a mortgage loan of $100,000 but after closing costs and other fees, the loan may have an APR of 4.1%. Unfortunately, not all financial institutions include the same fees in their APR calculation, so APRs are not always a perfect comparison tool. When comparing load or credit card APR’s ask what fees are included so your comparison is accurate.
Calculate the APR for a 5-year, $25,000 loan with the interest rate of 6% (compounded annually), considering 1.5 points and loan originating fee of 250 dollars. Assume all the costs are deducted at the time of taking the loan (present time).
Note: 1.5 points equals a cost of 1.5% of the loan value.
First, the uniform series of annual payments needs to be calculated.
Regarding Table 1-12 and Equation 1-6
Then, we have to calculate the costs and deduct them from the loan:
So, borrower will receive $24,375 at the present time and pay $5,934.91 to the bank, each year, starting from end of the year 1:
Now, we have to calculate the rate of return for such a project.
Loan-cost= 24,375 | A=5,934.91 | A=5,934.91 | A=5,934.91 | A=5,934.91 | A=5,934.91 |
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0 | 1 | 2 | 3 | 4 | 5 |
Present value of loan – present value of the costs = present value of all annual payments
With the trial and error technique, explained in the Lesson 2 section “Break-Even and Rate of Return (ROR) Calculations II,” we can calculate i =6.94% as the APR for loan.
Please watch the following video, Calculating APR for a loan or mortgage (4:43).
Rate of return for an investment can be determined by the try and error method that is previously explained. Also, a convenient way to learn to calculate rate of return is to use Microsoft Excel or Google Sheets and apply Internal a Rate of Return (IRR) function to the cash flow.
Note: You have to enter the occurred amounts in the spreadsheet in the form of cash flow (you can enter the years in horizontal or vertical direction). It means inflow and outflow of cash should be entered with different signs (depending on the project). So, you can enter the loan with negative signs at the present time and annual payments in following years with positive signs.
More information about the IRR function in provided in following links.
IRR Function in Microsoft Excel [2]
Please watch the following video, Internal Rate of Return (1:58).
Figure 3-1 displays the APR calculations for Example 3-1.
Please read the materials and watch the videos in the following links:
Investopedia Dictionary: Bond [4] (1:46).
Investopedia Dictionary: Bond Yield [5] (1:56).
Coming Soon
Investopedia presents: Bond Yields
Two popular bond yield measures are the current yield and the yield to maturity. Current yield is the interest it pays annually divided by the bond’s current price. This calculation tells investors what they will earn from buying a bond and holding it for one year. Jane is thinking about buying a bond for $100 with a $10 annual coupon she divided $10 by $100 to find it’s current yield is 10%. Since bond prices constantly change due to market and economic conditions jane might not actually earn 10% her actual return will depend on how long she holds the bond and its price when she sells it. Jane might sell the bond in two years for $75. So, while she earned $20 for the 2 years she held it since she sold it for $25 less that she bought it for she actually lost $5. The current yield helps her approximate what she might earn which helps her decide whether or not to invest. Since she wants to buy a bond Jane also needs to consider yield to maturity: YTM which is how much she’ll earn if she holds the bond until it matures, that is, if she doesn’t sell before the maturity date of the bond. YTM is expressed as an annual rate and it accounts for what all of a bonds future coupon payments are worth today at their present value. Jane needs to know the bonds market price, par value, coupon interest rate and time to maturity to calculate YTM. She plugs these values into a computer program that assumes coupon payments are reinvested at the same rate as the bonds current yield of 10%. YTM is a complex calculation but it gives Jane a better idea of her future returns and lets her compare bonds with different maturities and coupons.
Please watch the following video, Investing Basics: Bonds (3:56).
A bond is a financial tool that can help the government and corporations raise money for their investments. A bond is a document that simply means “I owe you” or “IOU.” The Government and corporations issue the bond for a specified period of time (can be weeks to years). Buyers pay the bond at face value (the price that is written on the bond) and purchase the bond once it is issued. In the end of the specified period (known as maturity date), buyers receive the face value. In return, bond issuers agree to pay a fixed annual amount as interest, called bond’s coupon. Some bonds allow the interest rate to be adjusted with inflation rate. And some bonds can be converted to common stock or other securities after a period of time. A good thing about a bond is that buyers don’t necessarily need to wait until the maturity date; they can sell their bonds before the maturity dates in the market. The price of a bond (a bond that is not new) depends on the financial market and interest rates in the market and can be higher or lower than its face value. If the interest rate in the market drops, then the bond can be sold at a higher price than the face value, and vice versa.
The organization that issues the bond usually backs (supports) it with some selected asset as collateral in case of bankruptcy. And if the issuer organization doesn’t provide real tangible assets for supporting the bond, the bond is called a “junk bond.” In general, bonds with a higher level of risk pay higher interest rates.
Brokers and investors usually measure economic performance in terms of compound interest rate of return, which is referred as “yield to maturity” (YTM), as well as the “current yield." Most bonds, debentures, and notes pay interest on a semi-annual basis, but related interest rates are described nominally. This means that the evaluation of a bond must be made on a semi-annual basis and then expressed as a nominal value.
The U.S. Government offers different types of securities [7] including:
Please read the materials provided in the above links.
If you would like to know more about the history of bonds and the bond market, you can find some interesting documentaries on YouTube.com.
Calculate the rate of return for a new bond with a face value of $1000 dollars and a maturity date of 10 years that pays 30 dollars every six months.
C = $1000 | I=$30 | I=$30 | I=$30 | L = $1000 | |
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C: Cost
I: Interest Income (semi-annual)
L: Maturity Value
Present value of cost = present value of income
According to Table 1-12:
With the trial and error method, we can calculate that i = 3% per semi-annual period. So, the nominal rate of return equals 2*3 = 6% per year compounded semi-annually. In bond broker terminology, the term “yield to maturity” is used to describe this nominal rate of return and may be listed by acronym “YTM.”
The following figure shows how you can calculate rate of return using IRR function in Microsoft Excel. Please notice the figures and signs, especially the first and last years.
As explained before, buyers can sell their bonds in the market before their maturity dates.
Assume person A buys the new bond that is explained in Example 3-2. After two years (in the end of the year), person A decides to sell the old bond to person B for 800 dollars. Calculate the rate of return of investment for person B.
Person B investment can be shown as:
C = $800 | I=$30 | I=$30 | I=$30 | L = $1000 | |
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0 | 1 | 2 | ... | 16 |
We can write the equations for this investment as:
Present value of cost = present value of income
The trial and error technique or IRR function in Microsoft Excel gives that i = 4.82% per semi-annual period and a nominal rate of return 2*4.82 = 9.64%per year compounded semi-annually.
Note: the only thing different from previous the calculation is the time and investment cost.
Please watch the following video, Calculating return on a bond investment (7:53).
Assume interest rates in the financial market dropped, which causes the price of an old bond to increase. So, person A in Example 3-2 can sell the old bond after two years (in the end of the year) to person B for 1200 dollars. Calculate the rate of return of investment for person B.
Similar to Example 3-3, person B's investment can be shown as:
C = $1200 | I=$30 | I=$30 | I=$30 | L = $1000 | |
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0 | 1 | 2 | ... | 16 |
Present value of cost = present value of income
And rate of return per semi-annual period will be i = 1.58% and the nominal rate of return is: 2*1.58 = 3.16%per year compounded semi-annually.
Now assume this situation: Since the interest rate dropped in the financial market, the issuer organization can call the old bonds after 4 years (from now -- total maturity period of 6 years). This means that at that time, the issuer organization takes the bond and pays the face value. Please calculate the rate of return for person B’s investment if he buys the old bond at $1200.
Person B's investment can be shown as:
C = $1200 | I=$30 | I=$30 | I=$30 | L = $1000 | |
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0 | 1 | 2 | ... | 8 |
Note that the old bond will be called in 4 years from now after person B buys it.
Present value of cost = present value of income
The rate of return for person B’s investment will be i = 0.45% per semi-annual period and the nominal rate of return: 0.9% per year compounded semi-annually.
As briefly explained in the first lesson, the financial cost of capital for a project (for a privately owned company) can be the average cost of financing current projects (or under consideration projects). The opportunity cost of capital or minimum rate of return (denoted as “i*”) reflects other opportunities that exist for the investment of capital now and in the future. The opportunity cost of capital for an investment is higher and more important than the financial cost of capital. An investor will invest in a project only if the rate of return is higher than opportunity cost capital (minimum rate of return).
Rate of return is a decision method to accept or reject a project and it is not a reliable method to rank several projects in terms of investment. Also, the rate of return for a current project is not necessarily applicable to future projects. For example, if an investment project has the rate of return of 5%, but another investment with similar (or lower) risk (such as interest paid by the bank to the money in your account or interest from buying Treasury Bond) has the rate of return of 6%, then the minimum rate of return and opportunity cost of capital will be 6%, and the project is not acceptable for investment.
If a company doesn’t have budget constraints, then it would keep investing in a new project until the rate of return on the next project is less than the cost of raising money. See Figure 3-3, below.
But this assumption is not usually realistic, and in the real world, there is always a budget constraint. As Figure 3-4 shows, budget constraint causes the cost of the capital curve to move upward and also to the left. In this case, the financial cost of capital needs to be adjusted to a minimum acceptable rate of return (MARR). The minimum acceptable rate of return reflects the project’s rate of return that is given up for the project under consideration.
However, if the project that is under consideration is the only possible project or it is not comparable with other projects, or there is enough funding available for all other projects with a higher rate of returns, then the opportunity cost of capital can be equal to the financial cost of capital.
As explained in the first lesson, Net Present Value (NPV) is the cumulative present worth of positive and negative investment cash flow using a specified rate to handle the time value of money.
Or
Or
If the calculated NPV for a project is positive, then the project is satisfactory, and if NPV is negative then the project is not satisfactory.
The following video, NPV function in Excel, explains how NPV can be calculated using Microsoft Excel (8:04).
In the video NPV and IRR in Excel 2010 [13] (8:59) you can find another useful video for calculating NPV using Excel NPV function. In this video, cash flow is formatted in the vertical direction (there is absolutely no difference between vertical and horizontal formatting, using spreadsheet).
In the following video, IRR function in Excel, I'm explaining how to calculate the Rate of Return for a given cash flow using Microsoft Excel IRR function (4:19).
Please calculate the NPV for the following cash flow, considering minimum discount rate of 10% and 15%.
C=60,000 | C=50,000 | I=24,000 | I=24,000 | ... | I=24,000 |
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0 | 1 | 2 | 3 | ... | 10 |
C: Cost, I:Income
If using spreadsheet, following method can be more convenient:
Figure 3-5 illustrates the calculation of the NPV function in Microsoft Excel. Please note that in order to use the NPV function in Microsoft Excel, all costs have to be entered with negative signs.
Benefit Cost Ratio (B/C ratio) or Cost Benefit Ratio is another criteria for project investment and is defined as present value of net positive cash flow divided by net negative cash flow at i*.
For the project assessment:
Present Value Ratio (PVR) can also be used for economic assessment of project(s) and it can be determined as net present value divided by net negative cash flow at i*.
Calculate the B/C ratio and PVR for the cash flow in Example 3-6.
Figure 3-6 illustrates the calculation of the B/C function in Microsoft Excel. Please note that you need to use the absolute value in the denominator or multiply the answer by -1.
Figure 3-7 illustrates the calculation of the PVR function in Microsoft Excel. Please note that you need to use the absolute value in the denominator or multiply the answer by -1.
In Lesson 3, we have learned that annual percentage rates (APR) represent an annualized expression of the cost of borrowing money, and how to calculate an APR based on a leader's cash flow. The salvage value is also introduced, which presents a positive cash flow for the project. Bonds are a very popular tool for corporations and governments to raise debt capital and we have learned the cash flows of a bond. The old bond rate or return with or without call privileges is also introduced. We also learned the concepts and effects of financial cost and opportunity cost of capital and in the last part we figured out how to evaluate a project(s) using Net Present Value, Benefit Cost Ratio, and Present Value Ratio.
You have reached the end of Lesson 3! Double-check the to-do list on the Lesson 3 Overview page [14] to make sure you have completed all of the activities listed there before you begin Lesson 4.
Links
[1] https://www.investopedia.com/video/play/annual-percentage-rate-apr/
[2] https://support.office.com/en-us/article/IRR-function-64925eaa-9988-495b-b290-3ad0c163c1bc
[3] http://www.excelfunctions.net/Excel-Irr-Function.html
[4] http://www.investopedia.com/terms/b/bond.asp
[5] http://www.investopedia.com/terms/b/bond-yield.asp
[6] https://www.youtube.com/user/TDAmeritrade/about
[7] http://www.treasurydirect.gov/indiv/products/products.htm
[8] http://www.treasurydirect.gov/indiv/products/prod_tbills_glance.htm
[9] http://www.treasurydirect.gov/indiv/products/prod_tnotes_glance.htm
[10] http://www.treasurydirect.gov/indiv/products/prod_tbonds_glance.htm
[11] http://www.treasurydirect.gov/indiv/products/prod_tips_glance.htm
[12] http://www.treasurydirect.gov/indiv/products/prod_frns_glance.htm
[13] http://www.youtube.com/watch?v=qAhV3xG0i8s
[14] https://www.e-education.psu.edu/eme460/node/524