Treasury bills, commercial paper, and bankers acceptances in the U.S. are quoted on a discount rate (DR) basis. The price of the security is a discount from the face value.

(1.5)

Here, PV and FV are the two cash flows on the security; PV is the current price and FV is the amount paid at maturity. The term in brackets is the amount of the discount – it is the future (or face) value times the annual discount rate times the fraction of the year. Interest is not “added on” to the principal; instead it is included in the face value.

The pricing equation for discount rate instruments expressed more compactly is:

(1.6)

Suppose that the money manager buys the 180-day CP at a discount rate of 3.80%. The face value is $1,000,000. Following market practice, the “amount” of a transaction is the face value (the FV) for instruments quoted on a discount rate basis. In contrast, the “amount” is the original principal (the PV at issuance) for money market securities quoted on an add-on rate basis. The purchase price for the CP is $981,000.

What is the realized rate of return on the CP, assuming the mutual fund holds it to maturity (and there is no default by the issuer)? We can substitute the two cash flows into equation 1.4 to get the result as a 360-day AOR so that it is comparable to the bank CD.

Notice that the discount rate of 3.80% on the CP is a misleading growth rate for the investment – the realized rate of return is higher at 3.874%.

The rather bizarre nature of a money market discount rate is revealed by rearranging the pricing equation 1.6 to isolate the DR term.

(1.7)

Note that the DR, unlike an AOR, is not an APR because the second term in parenthesis is not the periodic interest rate. It is the interest earned (FV – PV), divided by FV, and not by PV. This is not the way we think about an interest rate – the growth rate of an investment should be measured by the increase in value (TV – PV) given where we start (PV), not where we end (TV). The key point is that discount rates on T-bills, commercial paper, and bankers acceptances in the U.S. systematically understate the investor's rate of return, as well as the borrower's cost of funds.

The relationship between a discount rate and an add-on rate can be derived algebraically by equating the pricing equations 1.3 and 1.6 and assuming that the two cash flows (PV and FV) are equivalent.

(1.8)

The derivation is in the Technical Appendix. Notice that the AOR will always be greater than the DR for the same cash flows, the more so the greater the number of days in the time period and the higher the level of interest rates. Equation 1.8 is a general conversion formula between discount rates and addon rates when quoted for the same assumed number of days in the year.

We can now convert the CP discount rate of 3.80% to an add-on rate assuming a 360-day year.

This is the same result as given earlier – there the AOR equivalent is obtained from the two cash flows; here it is obtained using the conversion formula. If the risks on the CD and the CP are deemed to be equivalent, the money manager likes the CD. Doing the bond math, the manager expects a higher return on the CD because 3.90% is greater than 3.874%, not because 3.90% is greater than 3.80%. The key point is that add-on rates and discount rates cannot be directly compared – they first must be converted to a common basis. If the CD is perceived to entail somewhat more credit or liquidity risk, the investor's compensation for bearing that relative risk is only 2.6 basis points, not 10 basis points.

Despite their limitations as measures of rates of return (and costs of borrowed funds), discount rates are used in the U.S. when T-bills, commercial paper, and bankers acceptances are traded. Assume the money market mutual fund manager has chosen to buy the $1,000,000,180-day CP quoted at 3.80%, paying $981,000 at issuance. Now suppose that the manager seeks to sell the CP five months later when only 30 days remain until maturity, and at that time the securities dealer quotes a bid rate of 3.35% and an ask rate of 3.33% on 1-month CP. Those quotes will be on a discount rate basis. The dealer at that time would pay the mutual fund $997,208 for the security.

How did the CP trade turn out for the investor? The 150-day holding period rate of return realized by the mutual fund can be calculated as a 360-day AOR based on the two cash flows:

This rate of return, 3.965%, is an APR for a periodicity of 2.4. That is, it is the periodic rate for the 150-day time period (the second term in parenthesis) annualized by multiplying by 360 divided by 150.

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