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# Example 21 - Calculate PV of a Principal + Interest Loan

This example applies to our online demo Time Value of Money Calculator. The C-Value! program for Windows works in a similar way and has a few more features. Note, our online demo TVM calculator is limited to calculations using interest rates between 2.0% and 8.9%

## Calculating the value of an existing fixed principal + interest only loan requires the use of a special series.

For greater detail about how values are entered into the TVM calculator, please see Example 1 - conventional mortgage or loan.

Background: The issue is, how do you calculate the present value of a loan which has terms that call for the payback to be for a fixed principal amount? That is, using your personally selected discount rate, how do you determine the present value of the loan? The difficulty is, a fixed principal loan does not have level payment amounts, because the principal paid amount is fixed while the interest varies as the principal is paid down. Therefore, the periodic payment varies (principal + interest = payment amount) with each payment. Normally, if you wanted to find the discounted value of such a loan, you would have to enter each anticipated payment amount into a PV calculator. However, if you are using our Time Value of Money Calculator, you won't have to enter each payment. The calculator will calculate the cash flow at the loan's prevailing interest rate and then discount the cash flow so that you can find its value using your desired discount rate.

This example walks you through this calculation. Assume you are employed by an investment firm which purchases debt obligations. You need to value the debt held by a mortgage bank. The facts of the loan are as follows: The loan is a five-year, \$800,000 obligation with an origination date of June 1, 2012. Semiannual principal plus interest payments start on December 1, 2012 and continue through June 1, 2017. The nominal annual interest rate is 8.25%.

This example illustrates the use of the "Existing Fixed Special Series" to evaluate the value of the loan using a user selected discount rate of 6.5% for a loan collecting 8.25% interest.

To make this evaluation, follow these steps:

1. Click the [New] button to clear any previous entries.
2. Set "Rounding" to "Ignore" by clicking on the "Rounding" button.
1. For this example, we are assuming a normal compute method and a 365 day year.
1. Click on {Setup} menu. Select {Set calculation options...}
2. Click "Compute/Amortization Methods" tab. Set "Normal"
3. Click "Days Per Year". Set "365"
4. Click [OK] to close.

We will setup two cash flow calculations to help you better understand this calculation. Strictly speaking, the first calculation isn't necessary to value the loan. We include the calculation here as a check on the second discounted calculation and to provide a better illustration.

Step 1 - compute the fixed principal + interest loan using its original terms.

1. Set "Compounding Frequency" to "Semiannually"
2. Enter "8.25%" for "Nominal Annual Rate"
3. Create a "Loan" event in row one of the cash flow input area.
1. Set the "Date" to June 1, 2012 (mm/dd/yyyy)
2. Set the "Amount" to 800,000.00
3. Set the "# Periods" to 1.
1. Move to the second row of the cash flow input area. Select "Payment" for the "Event" type.
1. Set the "Date" to December 1, 2012
2. Set the "# Periods" to 10. The end date will display as 06/01/2017
3. Set "Frequency" "Semiannually" if necessary
1. Set the "Fixed Principal + Interest Special Series". Click in the second row, under "Special Series" to open the "Special Series" window.
1. Select the "Principal + Interest" tab.
2. Click on "Activate Fixed Principal Plus Interest series...". (A check will appear in the checkbox.)
3. Set "Principal payment amount" to "Unknown" (Type a "U".)
4. Click [OK] to close the window.
5. The words "Fixed Principal (+Int)" will appear in the amount column of the second row.

The calculator will now look like this.

1. Calculate the unknown amount. The result will be \$80,000 for the principal part of the regular payment. The amount of the regular payment will change each period due to the declining principal balance which causes the amount of interest due each period to decline. Thus, a smaller and smaller interest amount is being added to a constant principal amount of 80,000.
2. Click on the Amortization tab. Print a copy of the schedule which will be referenced in step 2. Note that the two interest only payments are \$33,000 each.
1. Click the "Schedule" tab.
2. Click the "Print Preview" button. Note the final total for all payments — \$981,500 as well as the consistent \$80,000 month to month principal payments and the varying periodic payment amount.

Step 2 - Compute the value of the debt assuming your selected discount rate.

1. Click {File}{New}.
1. Set compounding to "Semiannually".
2. Enter 6.5% for the "Nominal Annual Rate". This is your discount rate. (Note: if you express your discount rate at a different compounding frequency or assuming simple interest, then use our "Equivalent Interest Rate Calculator" to convert your discount rate to mach the loan's compounding frequency.)
3. Create a "Loan" event in row one of the cash flow input area.
1. Set the "Date" to September 30th, 2012 (mm/dd/yyyy) — some time has passed since the loan's origination date. We enter here the date we want to evaluate the value of the loan.
2. Set the "Amount" to "Unknown".
3. Set the "# Periods" to 1.
1. Move to the second row of the cash flow input area. Select "Payment" for the "Event" type.
1. Set the "Date" to December 1, 2012
2. You can leave "Amount" set to \$0.00 for now.
3. Set the "# Periods" to 10
4. Set "Frequency" to "Semiannually" if necessary. The end date will display as 06/01/2017
1. Set the "Existing Fixed Special Series". Click in the second row, under "Special Series" to open the "Special Series" window.
1. Select the "Existing Fixed" tab.
2. Click on "Activate Existing Fixed series for the currently highlighted event...". (A check will appear in the checkbox.)
3. Set "Note balance" to "800,000"
4. Interest "Rate for note" is "8.25%"
5. "Principal Payment" is "\$80,000"
6. Click [OK] to close the window.
7. The words "Existing Fixed" will appear in the amount column of the second row.
1. Calculate the unknown value. The result or fair value of the loan is \$850,780.90 at your selected discount rate of 6.5%. In other words, you can purchase this loan for \$850,780.90 on Sept. 30th, and you'll earn a rate of return of 6.5% on your investment.
1. Click the "Print Preview" button. Note the final total for all payments — \$981,500 — the same as the original loan (of course this would not be the case if you are calculating the PV of the loan after a payment has been made).

You may be wondering why the calculated value is greater than the balance of the loan. This actually makes perfect sense. The loan has a cash flow based on an 8.25% nominal annual rate. The discount rate is only 6.5%. Therefore, since the cash flow from the loan is greater than your investment requirements, you should be willing to pay more than the balance of the outstanding debt. Had your discount rate been say 10.0%, then the value of the loan (to you) would be less then the balance of the debt.

NOTE: You could also value a fixed principal + interest loan by entering each cash flow amount. But since the cash flow amounts are changing it can become somewhat tedious to enter say forty-eight payments or even ten payments. You should be able to see, after studying this example, how the "Existing Fixed Special Series" saves you this tedium.

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