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6.2 Interest

PLEASE NOTE: This book is currently in draft form; material is not final.

Learning Objectives

  1. Define the concepts of principal and interest.
  2. Describe how an interest rate is derived from a PV and FV.
  3. Describe how a market interest rate is determined.

Continuing our discussion of John and Mary: suppose John declines the original deal, and Mary counterproposes, “Could you lend me just $8, then? I’ll still pay you the full $15 next week.” We now know that $8 is greater than the present value of $15 to Mary. If John were to now accept the proposal, we would further know that the PV of $15 to John is somewhere between $8 and $10, since he accepted $8 but rejected $10. If Mary continued to make proposals to John, she would eventually find the point where John is indifferent between giving the agreed sum now and receiving $15 later. This point of indifference must be the present value to John of $15 in a week. Of course, John could do the same to Mary to find her PV. As long as the PV of $15 in a week for Mary is less than John’s PV, they should be able to come to an agreement. Of course, if they’re exactly equal, then they could agree, but both would be indifferent to doing the trade.

“Okay, I Guess….”

In everyday conversation, when we speak of being indifferent, that usually means something negative. When we say “indifferent” in economics and finance, we mean it literally: just as happy to do the trade as not. If we got one penny more, we’d always do the trade, and one penny less and we’ll say, “No thank you!”

Let’s assume Mary’s PV is exactly $9. We can now quantify the opportunity cost for Mary as ($15−$9) = $6. Mary is willing to pay $6 future dollars to borrow $9 for a week. In a loan, we call the original borrowed amount ($9) principalIn a loan, the amount originally borrowed. and the difference between the amount borrowed and the amount repaid (the extra $6) interestIn a loan, the difference between the amount borrowed and the amount repaid.. If we look at the ratio of the interest to the principal, we get an interest rateThe ratio of the interest to the principal for a loan. Typically expressed as an APR.. In this case, the interest rate for the week is ($6 / $9) = 66.67%.

Mary will, as a borrower, try to get the lowest interest rate she can find; perhaps Martha down the hall will be willing to loan her $9 and only ask for $14 in a week. If so, she will only pay $5 interest, for a weekly interest rate of ($5 / $9) = 55.56%. And John, as the lender, wants to receive the highest interest rate he can for his loan. If there are other potential borrowers, he’ll pick the one willing to accept the highest rate (assuming, of course, that his PV of the principal plus interest is greater than the amount borrowed). As we add more potential lenders and borrowers to the mix, the invisible hand takes over and we should find an equilibrium interest rate, or market interest rate, emerge. Note that interest rates are implied by the market, based on the participants’ relative valuations of cash at different time periods. We will often, as a shorthand, speak of the interest rate as a given, but it is good to remember that ultimately the interest rate is the derived number.

If the market is large enough, then if Mary wanted to borrow twice as much ($18), she should be able to still apply the same interest rate to her principal to find the interest owed. So if Martha is willing to lend Mary $18 at 55.56%, Mary will owe her: principal + interest = $18 + ($18 × 55.56%) = $18 + $10 = $28, or twice Martha’s original agreed FV of $14. Note that if we know the interest rate for the period, r, then the relationship between the PV and the FV will be:

Equation 6.1 Interest Rate Matches Time Period

principal + interest = FV
PV + (PV × r) = FV PV × (1 + r) = FV

Interest, at a fundamental level, represents the costs (especially opportunity costs) of an investment; if the market is competitive, then a competitive interest rate can be obtained from another choice of investment for the resources. Included in these costs are the perceived risks of an investment. For example, Mary has always been a reliable borrower in the past, but if John believes that there is a chance Mary might not pay him back, then he might demand a higher FV for the same PV, effectively raising the interest rate.

Key Takeaways

  • principal + interest = FV
  • Interest is derived from the difference between PV and FV and is compensation for the costs (including opportunity costs and the cost of perceived risk) of an investment.
  • The market interest rate is derived from the relative values of PV and FV by the participants.

Exercises

  1. If a bank account begins the year with a $50 balance, but ends the year with $55, what is the implied annual interest rate?
  2. If interest rates are currently 5% per year, than what is the interest earned on a $200 investment in one year? What is the total FV?
  3. If at the end of the year an investment is worth $480, and the implied interest rate is 20% per year, what was the PV of the investment at the beginning of the year?