Present Value Concepts

Calculating the Present Value of Notes Payable

Long-term notes payable are normally repayable in a series of periodic payments. These payments can consist of either (1) fixed principal payments plus interest, or (2) blended principal and interest payments. The accounting treatment of notes is similar to that for bonds. The present value of a note is a function of the same three variables: (1) the payment amounts, (2) the length of time until the amounts are paid, and (3) the market interest rate. Examples of long-term notes payable include unsecured notes, mortgages, which are secured notes on real property (e.g. house), and loans (e.g., student or car).

Note Payable

Let's assume Heathcote Company obtains a 5-year note payable, with an 8% interest rate, to purchase a piece of equipment costing \$25,000. If we first assume that repayment is to be in fixed principal payments plus interest, paid annually, then the payment amount would be \$5,000 (\$25,000 ÷ 5 years) plus 8% interest on the outstanding balance. Illustration B-14 details the total cost of this loan over the five-year period:

 Illustration B-14  Cost of equipment loan-fixed principal payments plus interest

1 Interest is calculated on the outstanding principal balance at the beginning of the year. In Year 1, it is \$25,000 x 8% = \$2,000. In Year 2, it is \$20,000 (\$25,000 - \$5,000) x 8% = \$1,600, and so on.

2 These factors are taken from Table B-1, for i = 8% and the appropriate n, or number of periods.
If we next assume that repayment is to be in blended principal and interest payments, we can use present value concepts to calculate the amount of the annual payment.

If we divide the total loan amount of \$25,000 by the present value factor for an annuity from Table B-2 for i = 8% and n = 5, then we can determine that the annual payment is \$6,261 (\$25,000 ÷ 3.99271). Illustration B-15 details the total cost of this loan over the five-year period:

 Illustration B-15  Cost of equipment loan-blended principal payments and interest

1 These factors are taken from Table B-1, for i = 8% and the appropriate n, or number of periods.
2 The total doesn't work out exactly because of rounding. If full digits were used, the total would be \$25,000.
Both options result in a present value cost of \$25,000, but the blended principal and interest payment option results in a higher total cash outflow, \$31,305 compared to \$31,000 for the fixed principal payments plus interest option. Note the difference also in the annual cash flows required under each option. For example, the fixed principal and interest payment option results in a higher cash outflow in the first year and lower cash outflow in the last year.

These examples illustrate the effect of the timing and amount of cash flows on the present value. Basically, payments made further away from the present are worth less in present value terms, and payments made closer to the present are worth more.

Businesses sometimes offer financing with stated interest rates below market interest rates to stimulate sales. Zero interest bearing notes do not specify an interest rate to be applied, but have an increased face value. Examples of notes with stated interest rates below market are seen in advertisements offering "no payment for 2 years". The present value of these notes is determined by discounting the repayment stream at the market interest rate.

As an example, assume that a furniture retailer is offering "No payment for 2 years, or \$150 off on items with sticker price of \$2,000." The implicit interest cost over the 2 years is \$150, and the present value of the asset is then \$1,850 (\$2,000 - \$150). From Table B-1, the effective interest rate can be determined:

 PV ÷ FV = discount factor \$1,850 ÷ \$2,000 = 0.925

Looking at the n = 2 row, this would represent an interest rate of approximately 4%. Most notes have in interest-cost, whether explicitly stated or not.

The Canadian federal government, in an effort to encourage post secondary education, offers loans to eligible students with no interest accruing while maintaining student status. When schooling is complete, the entire loan must be repaid within 10 years. Repayment of the loan can be at a fixed rate of prime + 5%, or a floating rate of prime + 2 1/2%. Let's assume that you attend school for 4 years and borrow \$2,500 at the end of each year. Upon graduation, you have a total debt of \$10,000. If you had not been eligible for the student loan, and had borrowed the money as needed through a conventional lender who agreed to let the interest accrue at 9% over the 4 years, with no payments required, how much would you owe upon graduation?

Compound interest would accrue as shown in Illustration B-16.

 Illustration B-16  Cost of student loan

The loan would total \$11,433 at the end of Year 4. This amount can be determined using present value concepts as follows:

 Illustration B-17  Cost of student loan

1 n=1, i=9%; calculation done for each year separately, based on the balance at the beginning of each period.

The same answer is reached by calculating the PV of an annuity of \$25,00 over 4 years at 9%, and then calculating the FV of this figure, as follows:

 * From Table B-2; the present value of an annuity of \$2,500 per year for 4 years at 9% is \$2,500 X 3.23972. = \$8,099.30. * From Table B-1, the future value, (4 years at 9%), of \$8,099.30 is \$8,099.30 / 0.70843 = \$11,432.75.

This amount, if repaid over the next 10 years, at 9%, would require annual payments of \$1,781.49 (\$11,433 ÷ 6.41766 from Table B-2, i = 9%, n = 10) per year. Alternatively, the student loan of \$10,000, if repaid over the next 10 years at 9%, would require annual payments of \$1,558.20 = \$10,000 ÷ 6.41766 from Table B-2, i = 9%, n = 10) per year. This represents a nominal savings of \$2,232.90 [10 x (\$1,781.49 - \$1,558.20)]. This example illustrates the effect of market interest rates on present value calculations.