In all previous lessons, we assumed that the money required for the investment is available in cash at no cost. However, it’s very common that an investment project is funded by a combination of borrowed money and equity capital. This way of funding a project is called “leverage” and “gearing.” The idea here is to try to increase (leverage) the profitability of the project by borrowing money. There are three main differences between funding an investment project by cash or borrowed money:
- Interest on borrowed money is an additional operating expense tax deduction that must be accounted for each evaluation period that mortgage payments are made.
- Loan principal payments are additional non-tax deductible capital costs that must be accounted as after-tax outflows of money each evaluation period that mortgage payments are made.
Compounding
In order to compare different alternatives in an economic evaluation, they should have the same base (equivalent base). Compound interest is a method that can help applying the time value of money. For example, assume you have 100 dollars now and you put it in a bank for interest rate of 3% per year. After one year, the bank will pay you . Then, you will put the 103 dollars in the bank again for another year. One year later, you will have . If you repeat this action over and over, you will have:
Which can be written as:
In general:
The value of money after nth period of time can be calculated as:
(Equation 1-1)Which F is the future value of money, P is the money that you have at the present time, and i is the compound interest rate.
Example 1-1:
Assume you put 20,000 dollars (principal) in a bank for the interest rate of 4%. How much money will the bank give you after 10 years?
So the bank will pay you 29604.8 after 10 years.
Discounting
In economic evaluations, “discounted” is equivalent to “present value” or “present worth” of money. As you know, the value of money is dependent on time; you prefer to have 100 dollars now rather than five years from now, because with 100 dollars you can buy more things now than five years from now, and the value of 100 dollars in the future is equivalent to a lower present value. That's why when you take loan from the bank, the summation of all your installments will be higher than the loan that you take. In an investment project, flow of money can occur in different time intervals. In order to evaluate the project, time value of money should be taken into consideration, and values should have the same base. Otherwise, different alternatives can’t be compared.
Assume you temporarily worked in a project, and in the end (which is present time), you are offered to be paid 2000 dollars now or 2600 dollars 3 years from now. Which payment method will you chose?
In order to decide, you need to know how much is the value of 2600 dollars now, to be able to compare that with 2000 dollars. To calculate the present value of a money occurred in the future, you need to discount that to the present time and to do so, you need discount rate. Discount rate, i, is the rate that money is discounted over the time, the rate that time adds/drops value to the money per time period. It is the interest rate that brings future values into the present when considering the time value of money. Discount rate represents the rate of return on similar investments with the same level of risk.
So, if the discount rate is i=10% per year, it means the value of money that you have now is 10% higher next year. So, if you have P dollars money now, next year you will have and if you have F dollars money next year, your money is equivalent to dollars at present time.
Going back to the example, considering the discount rate of 10%:
We can calculate the present value of $2600 occurred 3 years from now by discounting it year by year back to the present time:
Value of 2600 dollars in the 2nd years from now
Value of 2600 dollars in the 1st years from now
Value of 2600 dollars at the present timeSo, it seems at the discount rate of i=10%, present value of 2600 dollars in 3 years equals 1953.42 dollars, and you are better off, if you accept the 2000 dollars now.
With the following fundamental equation, present value of a single sum of money in any time in the future can be calculated. It means a single sum of money in the future can be converted to an equivalent present single sum of money, knowing the interest rate and the time. This is called discounting.
Equation 1-2P: Present single sum of money.
F: A future single sum of money at some designated future date.
n: The number of periods in the project evaluation life (can be year, quarter or month).
i: The discount rate (interest rate).Example 1-2:
Assuming the discount rate of 10 %, present value of 100 dollars which will be received in 5 years from now can be calculated as:
You can see how time and discount rate can affect the value of money in the future. 62.1 dollars is the equivalent present sum that has the same value of 100 dollars in five years under the discount rate of 10%
Note:
The concept of compounding and discounting are similar. Discounting brings a future sum of money to the present time using discount rate and compounding brings a present sum of money to future time.
Figure 1-1: Compounding and DiscountingCredit: Farid TayariInvestment capital costs must be adjusted for borrowed money inflows of money each evaluation period that loans are made.
To explore the effect of borrowed money on the project, we need to study four methods of loan amortization. Suppose an investor takes a $1000 loan with fixed annual interest rate of 8% to be repaid over four years.
Project Evaluation with Borrowed Money
PRESENTER: In this video, I'm going to explain how we can analyze a project considering borrowed money. In previous videos, we assumed that money required for the investment in a project is available in cash and the investor provides the entire money required for the investment without any cost. But in reality, it's very common that the project is funded by a combination of borrowed money-- loans-- and equity capital, or the money that the investor puts in the project.
So funding a project using borrowed money is called leverage or gearing. And the reason behind that is the higher portion of fund in the project coming from borrowed money, from loans, it is going to give the project higher profitability, and it is going to enhance the economics of the project.
So when an investor funds a project with borrowed money, two things have to be considered. First, the interest portion of borrowed money, similar to operating costs, can be expensed and deducted from revenue as tax deductions. And this is going to give leverage to the project and enhance the economics of the project because this is actually going to reduce the tax paid by the investor. The other difference is that the loan principal payments are non tax-deductible, and they can be deducted from the after-tax cash flow as similar to capital cost. But because they can be distributed over years, they can also contribute in enhancing the project's profitability and the economics of the project. I will work on an example in the following videos, and I will explain them later in this lesson.
So there are four main types of loans that I will explain in the following videos. The first one is a balloon payment loan; the second one is interest only loan; the third one is constant amortization loan; and the fourth one, which is the most common one, is the constant payment loan.
1. Balloon Payment Loan
In this method, the loan will be repaid in full (future value) at the end of the period. The payment at the end is called a balloon payment.
| Loan = $1000 with 8% interest | Balloon Payment =1361 | |||
| 0 | 1 | 2 | 3 | 4 |
So, in this case, the balloon payment equals $1361 at the end of year 4, with loan principal of $1000 and interest of $361.
2. Interest Only Loan
In this method, loan interest is paid at each period and the principal is paid in full at the end:
| Loan = $1000 with 8% interest | Interest = $80 | Interest = $80 | Interest = $80 | Principal= $1000 Interest = $80 |
| 0 | 1 | 2 | 3 | 4 |
Balloon Payment Loan and Interest Only Loan
So there are four main types of loan, balloon payment loan, interest only loan, constant amortization loan, constant payment loan. So the last one is the most common one, and I will explain it in the following videos. In this video, I'm going to explain the first two types.
Balloon payment loan. In this type of loan, the borrower receives the loan, takes the loan at the present time at year zero, and has to repay the loan in the end of the agreed period. The borrower has to pay the principal and interest for the loan in the end of the period.
So there is no installment. There is no monthly, or annual, or per period payments. The borrower takes the loan at the present time and repays the entire loan with interest in the end of the period.
So for example, if the borrower is going to take a loan of $1,000 at present time with 8%, let's say for four years, then the borrower doesn't need to pay anything at year one to three, but the borrower has to pay the entire loan with interest, the principal and interest, in the end of year four. So in order to calculate the money that the borrower has to pay to the lender, we have to multiply $1,000 of loan by the factor F over P, 8% of loan interest and after four years.
So this factor equals 1 plus 8 percent power 4, and the result is going to be $1,361. So from this $1,361, $1,000 is the principal and $361 is the interest of this loan. That has to be paid in the end of the period, which was year 4. So the borrower pays the principle of $1,000 plus $361 of interest to the lender in the end of the period.
The second type of loan is called interest only loan. In this type of loan, borrower takes the loan at present time, let's say $1,000. Then borrower returns this $1,000 in the end of their agreed period, but borrower has to pay equal amounts of annual interest to the lender.
So let's assume a borrower takes the loan of $1,000 with 8% interest at present time. Then, borrower has to pay $1,000, must apply 8%, which comes to $80 per year from year one to year four. Let's say four years is the time interval that is agreed between lender and borrower. And so borrower has to pay $80 per year from year one to year four to the lender. And also, the borrower has to pay the principal of $1,000 at the end of year four.
3. Constant Amortization Loan
In this method, an equal portion of the principal is paid at each period plus interest based on the remaining balance in the beginning of each period.
Payment at year 1:
Principal:
Interest:
Payment at year 2:
Principal:
Interest:
Payment at year 3:
Principal:
Interest:
Payment at year 4:
Principal:
Interest:
| Loan = $1000 with 8% interest | Principal= $250 Interest = $80 | Principal= $250 Interest = $60 | Principal= $250 Interest = $40 | Principal= $250 Interest = $20 |
| 0 | 1 | 2 | 3 | 4 |
Constant Amortization Loan
PRESENTER: So there are four types of loan-- balloon payment, loan interest on the loan, constant amortization loan, and constant payment loan. In the previous video, I explain the first two types-- balloon payment loan and interest on the loan.
In this video, I'm going to explain the constant amortization loan and in the next video, I'm going to explain the most common type of loan, which is constant payment loan.
In a constant amortization loan, borrower receives the loan, takes the loan let's say at the present time, and has to pay equal portion of principal per period, plus the interest that the interest is calculated based on the remaining balance.
So I'll explain constant amortization loan in an example. Let's assume an investor takes the loan of $1,000 that has an interest rate of 8%, and the loan has to be repaid over four years. And we are going to consider the constant amortization loan.
So in constant amortization loan, the principal paid in each period is constant, and it's not changing. So the first step is to calculate the principal.
The principal is calculated as the loan divided by the number of period that has to be repaid. So the loan was $1,000, and the loan has to be repaid over four years. So the principal is going to be 1,000 divided by four years, which is going to give $250 per year. So the principal is going to be constant from year one to year four and is $250.
The next step is going to be calculating the interest and payments for each period. The interest is the balance multiply the interest rate of the loan.
So the balance at year one is $1,000, multiply the interest rate for the loan, which is going to give us $80. So the payment equals the principal plus interest. So the borrower has to pay $330 at year one to the lender.
And then we calculate the balance after this payment is paid, so $1,000 minus balance equals the balance of previous year, which was $1,000 minus the principal paid. From $330, $250 was the principal that we calculated here. So the remaining is $750, which is the balance, which is going to be applied for the calculation of year two.
So for year two, the interest equals balance multiply the interest rate. The balance is the balance that we calculated at year one after borrower paid $330, so the balance is $750. Multiply the interest rate is going to be $60, and it is the interest that the borrower has to pay. So in total, borrower has to pay $250 of principal plus $60 of interest, which comes to $310 at year two for this loan.
So then we are going to calculate the balance after the loan payment is paid. The balance is going to be $750, which is the balance of the previous year minus the principal, which is going to be $500.
And year three, the principal was constant $250. In order to calculate interest, we need to multiply the balance of the previous year by the interest rate, which was 8%. So $500 multiplied by 8% gives $40 of interest.
The payment that has to be paid by the borrower to the lender is $250. The principal plus the interest-- 40, which comes to $290 for year three.
And then we need to calculate the balance. Balance is the balance of previous-- the balance of previous year after the payment is paid, which was $500 minus the principal. And the principal is constant, and it is $250 per year. And the remaining is $250. So the balance at year three, after the payment is paid, the loan payment is paid is $250.
For year four, again, the principal is constant-- $250. The interest is the balance at the previous year is 250 multiplied the interest, 8%, which gives $20. And the payments at year four is the principal plus interest-- $250 plus $20 gives $270.
And here, the balance should equal zero. The balance is the balance of previous year-- $250 minus the principal that is paid at year four. So if we calculated everything correctly, the balance at the end of year four has to be zero.
So this is the summary of this loan. Borrower has received the loan of $1,000 at 8%, and borrower has to pay the constant principal of $250 per year plus the interest. That is calculated based on the remaining balance. And as you can see, the interest is $80, $60, $40, and $20 from year one to year four, but the principal is constant from year one to year four.
4. Constant Payment Loan
This method is similar to what we learned in previous lessons, and equal annual payments, A, can be calculated based on Table 1-12 as:
Year 1:
Payment =
Interest
Principal
Balance
Year 2:
Payment
Interest
Principal
Balance
Year 3:
Payment
Interest
Principal
Balance
Year 4:
Payment
Interest
Principal
Balance
| Year | 1 | 2 | 3 | 4 |
|---|---|---|---|---|
| Payment | 301.92 | 301.92 | 301.92 | 301.92 |
| Interest | 80 | 62.25 | 43.07 | 22.36 |
| Principal | 221.92 | 239.67 | 258.85 | 279.56 |
| Balance | 778.08 | 538.41 | 279.56 | 0 |
| Loan = $1000 with 8% interest | Payment= $301.92 | Payment= $301.92 | Payment= $301.92 | Payment= $301.92 |
| 0 | 1 | 2 | 3 | 4 |
These methods consider a fixed annual interest rate of 8%. But there are types of loans that have variable interest rates, also called Adjustable Rate Mortgage (ARM), and interest rate changes periodically.
Constant Payment Loan, Interest and Principle
PRESENTER: So there are four types of loan-- balloon payment loan, interest only loan, constant amortization loan, and constant payment load, which is the most common one. I explained the first three types in previous videos, and in this video I'm going to explain the constant payment loan.
So in the constant payment loan, the payments that are paid per period are constant. So let's work an example and see how we can do the calculations for the constant payment loan. Let's assume an investor takes a loan of $1,000, and the interest rate is 8%, and that the loan has to be repaid over four years. And we consider constant payment loan.
So the first step is to calculate equal annual payments, A. And we can use factor A over P, or capital recovery factor, for calculating these payments. And this is the equation. i is the interest rate, and n is the number of years, period, that the loan has to be repaid. And P is the loan. So $1,000 might multiply this equation, which gives us about $302 per year for this loan.
So the payments are going to be equal, $302 per year. Then we need to calculate the interest and principal portion of these payments of $302 for each year. The interest is the balance multiplied the interest rate, which was $1,000 multiply 8%, which gives us $80 of interest for year one.
For calculating the balance, we have to deduct the interest from the payment. And we are going to have $302 minus 80, which comes to almost $222. After this payment is paid, the balance is going to be the balance of previous year, which was $1,000, minus the principal portion of the payment, which is $222. And the remaining is the remaining balance, which is 778.
For year two, payment is constant, is what we calculated. The interest is the remaining balance, which was 778, multiply the interest rate, which is going to be $62.25. For the principal, we deduct the interest from the payment, and the remaining is going to be the principal at year two. So the remaining balance is the balance of the previous year, 778, minus the principal portion of the payment, which was 339.68. And this is the balance at year two when we pay this $302.
For year three, the payment is constant, similar to year one and year two. The interest is the remaining balance, which is 538.41, the remaining balance that we have here, multiply the interest rate, and it is going to give us $43.07. And this is the interest portion of this payment.
The remaining is the principal portion. So we deduct the interest from the payment, and the remaining is going to be the principal portion of this payment. In order if we calculate the remaining balance, we deduct the principal from the balance of the previous year, which was here. So the balance of previous year minus the principle that we paid this year gives us the remaining balance.
For year four, payment is constant. The interest is the remaining balance multiply interest rate of 8%, which is going to be 22.36. And the principal is going to be the payment minus the interest. The remaining is 279.56. And if we calculate everything correctly, this number this principle for the last year, should be exactly same as the remaining, which means that we're going to have zero balance at year 4 after we pay the payments.
So we can summarize these loan calculations as-- so a loan is taken at 8%-- a loan of $1,000 is taken at with 8% at present time, and the payments of almost $302 is paid to repay the loan from year one to year four.