Time Value of Money Concepts

Interest Rate (i) Used in TVOM Calculations: the Issue

Of the five variables in a TVOM calculation, the one that can be the most subjective, and therefore the one with the greatest potential for the introduction of error, is the interest rate (i).

In some cases the interest rate is defined as part of the problem; e.g., find the payment amount on a 9%, 30-year $500,000 mortgage. However, in other cases the interest rate is more a matter of professional judgment; e.g., determine the current price of a $10,000 payment due in 10 years.

Consider the following situation:

A fixed-income security with a face value of $1,000 pays $25 semi-annually and matures in 5 years. What price would you pay for this security today?

This security pays a nominal rate of interest of 5% calculated as (25 x 2)/1,000. But if we want to know what it is worth today, is 5% really an appropriate value for the interest rate (i)? What if the current yield on 30-day T-bills is 3%? And the yield on 5-year Treasury notes is 6.5%? And comparable corporate notes are paying 8%?

Which rate is most appropriate for valuing the security?

Interest rates (i) of 3%, 5%, 6.5% and 8% will produce prices of $1,092, $1,000, $937, and $878 respectively, a spread of $214 that equates to more than 20% of the face value. (If you are unsure how these prices (PVs) were computed, see example problem #35.)

But perhaps none of these rates is appropriate. Perhaps some form of composite rate or combination of rates should be used. For example, what if we used the a 5-year note rate to price the cash flows in years 4 and 5, a 2-year note rate to price the the cash flows in years 2 and 3 and T-Bill yields to price the cash flows in year 1?

Unfortunately there is no hard and fast rule that will tell us what interest rate to use. That choice is a matter of judgment.

The material presented here is intended to assist with making an informed decision with regard to the choice of interest rate.


Focus: Interest Rates and TVOM

It is important to keep in mind that the purpose of this web site is to instruct students in, and provide a reference on, time value of money (TVOM) calculations.

If we are to stay true to this purpose then it would not be wise to stray too far into topics that are related but fundamentally distinct from TVOM. For example, we don't want to delve too far into economic theory or go too deep into the practical aspects of the mortgage or bond industry.

Instead we want to concentrate on the mechanics of the TVOM calculations with our primary focus being on the mathematics.

Having said that, it is important to recognize that a certain amount of background information is helpful and probably even necessary to properly apply TVOM theory.

And this is what I am attempting to do here with interest rates; provide an overview of key issues that are useful in understanding how TVOM calculations are performed.

To this end, the following topics related to interest rates are presented:

  1. Interest Rate Definitions
  2. Benchmark Interest Rates
  3. Factors Affecting Interest Rates (Risks)
  4. Yield Curve
  5. Summary and Rule of Thumb


1. Interest Rate Definitions

  • Interest.

    Interest is the fee or cost paid or charged for borrowing money (capital); it is the amount a borrower pays a lender for the use money.

    Interest is compensation to the lender and expense to the borrower.


  • Principal.

    Principal is the amount of money (capital) which is owed by the borrower to the lender and upon which interest is charged.


  • Rate of Interest.

    The rate of interest or "interest rate" is the percentage of principal charged or paid as fee over an interval of time, typically a year.

    For example, if $50 were charged to borrow $1,000 principal for one year, the interest rate is 5%.

    Interest rate is sometimes used interchangeably with the term yield (see below).


  • Yield.

    Yield is a measure of return on a security. Generally "return" refers to income return (interest and dividends) as opposed to capital gain or loss.

    There are two basic variations of yield:

  • Cost yield is equal to the annual payout (interest or dividends) divided by the cost of the security.

  • Current yield is equal to the annual payout (interest or dividends) divided by the current price of the security.

  • For example, if you pay $1,000 for a bond at issue that pays $50 annually then the cost yield is 5% (50/1,000). However, if interest rates rise to the point where the price of the bond falls to $950, then the current yield is 5.26% (50/950).

    As mentioned previously, yield can be synonymous with interest rate and there are many times when we will use a yield as our value for i in a TVOM calculation.


  • Yield to Maturity(YTM).

    The yield to maturity is the return a bond holder earns under the assumptions that 1) the bond is held to maturity; 2) all coupon payments (interest) and return of principal are timely; and 3) all coupon payments are reinvested at the stated YTM.

    In other words, "YTM" is the i in the equation below that sets the sum of the present value of all interest payments plus the present value of the return of principal at maturity equal to the bond's current price...

    Note that a significant assumption for the YTM is that all interest payments are reinvested at the stated YTM.

    Also note that the YTM is essentially the internal rate of return (IRR) of the bond's cash flows.


  • Annual Percentage Rate (APR).

    The Annual Percentage Rate is the annual interest rate after inclusion of fees and other costs.

    The concept of an APR is an attempt by regulators to create a standard means of expressing the true cost of borrowing. However, because lenders have some discretion as to what costs are included in the calculation (e.g., points are required but application fees are not), APRs reported by different lenders may not be directly comparable.

    The Federal Truth in Lending law requires mortgage lenders to disclose the APR in their advertisements.


  • Nominal rate of interest.

    The nominal interest rate is the rate applied to principal each compounding period.

    Regardless of the compounding frequency, the nominal rate is almost always given as an annual rate.

    For example, when we refer to a nominal rate of "8% quarterly" we actually mean 2% applied to principal each quarter. The nominal or stated rate of 8% is applied proportionately across all four quarters.

    The nominal rate does not directly take into account compounding; it is simply the rate applied to principal each compounding period. (See effective interest rate below.)

    The nominal rate is also referred to as the stated or applied interest rate.

    In TVOM problems the value for i is typically a nominal rate.


  • Effective rate of interest (ieff).

    The effective interest rate is the nominal interest rate (i) taking into account the effect of compounding as defined here:

    So if we have a nominal rate of "12% monthly," we calculate an annual effective rate of interest of 12.68% as shown below:

    Just to reiterate, in the equation above 12% is the nominal rate of interest and 12.68% is the effective rate of interest.

    The effective rate is higher than the nominal rate because of compounding. In the example above we are applying 12% on a monthly basis but the additional interest on interest results in an annual effective return that is larger than 12%.

    The effective rate is almost always given as an annual rate and is sometimes referred to as the effective annual rate.

    The effective rate provides a means of comparing interest rates of different compounding frequencies.

    It also serves as a means of equivalence in that it is possible through the annual effective rate to find a monthly rate of interest that is equivalent to a known quarterly rate of interest.


  • Important Point. The concepts of nominal and effective interest rates - and the distinction between the two - are critical to understanding TVOM theory.

    The definitions presented above may not be sufficient for readers to fully grasp all of the subtleties associated with nominal and effective interest rates. If you really want to gauge your comprehension of this critical issue, you can review the following example problems:

  • QID 4. You have an option to purchase a $5,000 note that is due in 3 years for $4,100. Alternatively you can invest the $4,100 in a CD that pays 3% every 6 months over the same period. Which offers the higher return? Which is the preferred investment? View solution here.

    Issue: Calculate the nominal interest rate (i) and effective interest rate (ieff) and compare alternative investments.

  • QID 9. Find the present value of $1,000 due at the end of 10 years if interest is calculated at a) a nominal annual rate of 6% compounded monthly and b) an effective quarterly rate of 1.5%. View solution here.

    Issue: Compare the present value (PV) of a single sum under different compounding frequencies; distinguish between nominal and effective interest rates.

  • QID 15. What is the effective annual rate of interest being earned by an investor who receives $5 interest each month on a $500 note? How much would the monthly payment be if the investor were earning 10% compounded quarterly? View solution here.

    Issue: Calculate the effective rate of interest (ieff) given a fixed monthly interest amount and principal balance; calculate the amount of the interest payment under quarterly compounding.

  • QID 16. An investor loaned $5,000 10 years ago. The investment earned an effective annual rate of interest of 9% for the past 6 years and is now worth $9,950. a) Assuming monthly compounding, what is the annual nominal interest rate earned for the first 4 years; b) what is the annual effective rate of interest for the first 4 years? View solution here.

    Issue: Calculate the PV of an investment and the nominal (i) and effective (ieff) rates of interest; distinguish between nominal and effective rates of interest.

  • QID 21. One institution offers an annual rate of 5% compounded monthly while another offers a rate of 5.1% compounded quarterly. Which offers the higher return? View solution here.

    Issue: Calculate the effective interest rate (ieff) under different compounding frequencies and nominal interest rates.

  • QID 26. How much interest will be earned on $1,000 at 5.5% compounded daily over 180 days? What is the effective annual rate of interest earned? (Assume a 360 day year). View solution here.

    Issue: Calculate the dollar amount of interest earned and the effective annual rate of interest (ieff) .


  • 2. Benchmark Interest Rates

    Listed below are various rates that can be considered when choosing an appropriate value for i in a TVOM calculation.

    Historical data on a variety of interest rates is available through the Federal Reserve's Data Download Program.


  • Inflation Rate.

    Inflation is the rate at which the general level of prices are rising in the economy.

    Inflation is commonly expressed as the change in the level of prices as measured against an aggregate of base prices.

    There are various means of making this measurement but the most common are the Consumer Price Index (CPI) and the broader GDP Deflator.


  • Discount Rate.

    The discount rate is the rate charged to banks on short-term borrowings directly from the US Federal Reserve Bank (the Fed). The central banks of other nations also charge interest on borrowings by their member banks.

    The discount rate is set directly by the Federal Reserve Bank and generally applies to overnight borrowings.

    Note because of imprecise terminology the term "discount rate" may also refer to the value used for i in a TVOM calculation. This is particularly true in the case of present value (PV) calculations where some future value is being "discounted" back to its current value.


  • Federal Funds Rate.

    The federal funds rate is the rate charged among banks for overnight loans. Thus, it is the rate charged when banks borrow from each other, not from the Fed.

    Unlike the discount rate, the fed funds rate is not set directly by the Fed but is a target rate (as illustrated by the variations in the chart below) that the Fed may influence through its open market operations.


  • LIBOR.

    LIBOR is the London Interbank Offered Rate at which banks lend money to other banks in the London money market.

    The rate is published daily and is an average of inter-bank rates paid on Eurodollar CDs (US dollar denominated CDs issued by major international banks).

    The LIBOR rate is generally regarded as the proxy for the global rate at which large international banks can borrow from each other. It is increasingly popular as the reference rate for international financial transactions. In addition, derivative securities such as futures, currencies, swaps, and forwards are increasingly priced by reference to a LIBOR rate.


  • T-Bill Rate.

    T-Bills are US government securities issued by the Department of the Treasury with maturities of 30, 91, and 182 days. They are issued at a discount meaning they do not pay interest during their term but are redeemed at maturity for their face value.

    The amount of interest earned on a T-Bill is equal to the face value less the discounted purchase price. The interest rate or yield is calculated as:

    T-Bill rates are set through weekly auctions and such rates are viewed internationally as proxies for the riskless rate for their given maturities.


  • Repo Rate.

    The Repo rate is the rate implied or explicit in a repurchase agreement (repo).

    In a repo one party sells securities to another party for cash with the obligation to repurchase the securities for a greater price in the future. The excess of the repurchase price over the original price is deemed to be interest, the amount of which is determined by the repo rate.


  • Call Money Rate.

    The Call Money Rate is the rate banks charge to brokers on loans secured by stock and securities collateral.

    Brokers typically add 100 basis points to this rate as the margin rate they charge on loans to their clients. The call money rate may also be referred to as the "broker rate".


  • Prime Rate.

    Traditionally the US prime rate is the rate charged by banks to their most creditworthy customers.

    It is normally 300 basis points above the federal funds rate and typically changes only when the Fed changes the fed funds rate or when the Fed injects or withdraws reserves from the banking system through its "open market" operations.

    The prime rate is typically used as a benchmark for setting rates on other borrowings such as credit cards and adjustable rate mortgages (ARMs).

    As illustrated below, prime rates will vary (sometimes considerably) by nation.


  • 3. Factors Affecting Interest Rates (Risks)

    In identifying the impact that prevailing interest rates have on the value of i used in our TVOM calculations, we are generally more concerned with the differences among interest rates than with the general level of interest rates.

    There are three primary factors that account for differences among interest rates:

  • inflation
  • maturity
  • credit risk
  • While there are other factors that contribute to differences among interest rates, for purposes of this discussion we will focus on the three mentioned above.

    Each of these factors is considered a risk for which a premium must be paid to compensate the investor for assuming that risk. These premiums and the effect on interest rates are illustrated in the chart below:

    Starting at the bottom of the chart, the real rate of interest is determined by economic activity. It represents the true cost of borrowed funds and is based on the rate of return a business earns on its capital as well as consumers' preference to consume rather than save. Generally speaking, real rates are low in recession and high during periods of economic growth.

    1. Inflation is the rate of change in prices. Investors demand an inflation premium to compensate them for the loss of purchasing power they incur when they loan funds.

    The real rate of interest plus the inflation premium is referred to as the nominal rate of interest.

    Again we encounter some imprecise terminology. The "nominal" rate mentioned here refers to an economic concept and should not be confused with the nominal interest rate (i.e., the stated or applied interest rate) used in TVOM calculations that we defined earlier.

    The 90-day T-Bill rate is often used as a proxy for the nominal rate of interest.

    If the difference between the nominal and real rates of interest is equal to the rate of inflation, then the chart above reflects an expected rate of inflation of 3.2%.

    Note the real rate of interest cannot be observed directly. Typically it is "backed into" by subtracting the expected rate of inflation from the T-Bill rate.

    2. The term premium reflects the risk introduced by the investment's time horizon, its time to maturity.

    In general, the greater the time to maturity, the higher the premium.

    Investors demand a term premium because the longer a security is held, the greater the price change for a given change in interest rates and therefore the greater the market risk. Thus, investors usually demand higher yields at longer maturities.

    This is why a "normal" yield curve is upward sloping. (See Yield Curve below.)

    3. The risk premium reflects the likelihood of default.

    Less creditworthy borrowers have a higher risk so a higher yield is required to compensate investors.

    U.S. Treasuries have virtually no credit risk so their yields will always be lower than comparable private sector securities.

    Spreads between Treasuries and corporate securities tend to widen when interest rates are high as investors feel other issues are vulnerable and seek a safe place for their money.

    When choosing an appropriate interest rate (i), we should be careful to consider the context in which the TVOM calculation is being performed and the possible influence that each of the three risk premiums (credit, term, and inflation) may introduce.

    Example. We need a interest rate to value a corporate payout that will be made in 5 years. In this case we might use the current yield on similar corporate obligations because this rate should reflect all three risk premiums, just like the payout we are trying to value.

    But what if we can't find a comparable corporate obligation?

    The only quotes we have are for AAA rated corporations while the payout we are valuing will be made by a BBB company.

    In order to take this difference into account, we might observe the risk premium differential between AAA and BBB corporate obligations and use it to adjust the AAA rate upward to reflect the additional risk assumed.

    Note that not all three risks are present in every security. There is no credit premium, at least not in theory, associated with Treasuries and there is no term premium inherent in T-Bill rates.


    4. Yield Curve

    The effect of the "term premium" discussed previously can be observed in the yield curve.

    The yield curve is a graphical representation of the relationship between interest rates and maturity. In other words, it shows the particular yield associated with a given maturity, say 5% for a 30 year bond.

    While yield curves are available for all types of issues, it is the yield curve for US Treasuries, published daily in the Wall Street Journal, that most people associate with the term "yield curve."

    The graphic below shows the US Treasury yield curve before and after the Federal Reserve cut both the discount rate and the fed funds rate in September of 2007:

    Note that between July 2006 and July 2007 the yield curve was relatively flat implying that there was in effect no term premium associated with Treasury issues during this period. In other words, 3-month T-Bills were paying the same 5% as 30-year bonds.

    A flat yield curve is traditionally considered an anomaly and typically does not persist for long as can be seen in the October curve.

    The October curve also illustrates that the Fed's influence is primarily on short term rates and may not necessarily extend to long term rates which are set by market forces.

    Since the yield curve provides a means to measure the term premium it is not uncommon for bonds and other fixed income contracts to be "priced off the yield curve" by setting their yield equal to the yield at the same maturity point on the yield curve and then adjusting for credit quality and other factors.


    5. Summary and Rule of Thumb

    In concluding I will try, as best I can, to put this complicated subject into perspective and offer a rule of thumb that, hopefully, can serve as an aid in selecting the appropriate interest rate (i) for use in your TVOM calculation.

    The first question to ask in choosing an interest rate is: What is the objective? In other words, what are you trying to accomplish? The objective will determine the need for precision and, thus, the degree of effort expended in selecting a value for i.

    If the objective is a quick and dirty comparison of two alternative investments on a relative basis, then it can be argued that the value chosen for i is less critical than if the objective is to obtain a precise valuation. Any inaccuracies resulting from using a reasonable but less than perfect i will tend to cancel out as the same error will be present in both valuations.

    On the other hand, if the objective is to value a security or liability as precisely as possible, then the need for accuracy in choosing a value for i takes on much greater significance.

    Still, the amount of effort that goes into this process can vary considerably. As a minimum, the credit, term, and inflation factors and their associated premiums discussed previously should be considered. The degree of effort into how these premiums are determined and applied can vary considerably.

    In some cases, it may be sufficient just to add a credit premium to the yield on a Treasury security of appropriate maturity. In other cases, complex economic models may be required to forecast future interest rates and economic activity.

    If all this seems confusing, a simple but less than perfect rule of thumb is to set the interest rate equal to the "required yield."

    The required yield is the rate the investor requires, or can reasonably expect, from the investment. This rate can be determined by observing the yields on comparable investments in the market ("comparable" loosely meaning those of the same maturity and credit quality) and, with consideration for current and anticipated economic activity, choosing the one that is most reasonable.

    Granted this rule will rarely result in a precise valuation but, when faced with a lack of detailed information, it does at least provide something to work with and can be refined as the need dictates.