MONEY MARKET CALCULATION (ADD-ON RATES)

There are two distinct ways that money market rates are quoted: as an add-on rate and as a discount rate. Add-on rates generally are used on commercial bank loans and deposits, including certificates of deposit, repos, and fed funds transactions. Importantly, LIBOR is quoted on an add-on rate basis. Discount rates in the U.S. are used with T-bills, commercial paper, and bankers acceptances. However, there are no hard-and-fast rules regarding rate quotation in domestic or international markets. For example, when commercial paper is issued in the Euromarkets, rates typically are on an add-on basis, not a discount rate basis. The Federal Reserve lends money to commercial banks at its official “discount rate.” That interest rate, however, actually is quoted as an add-on rate, not as a discount rate. Money market rates can be confusing – when in doubt, verify!

First, let's consider rate quotation on a bank certificate of deposit. Add-on rates are logical and follow simple interest calculations. The interest is added on to the principal amount to get the redemption payment at maturity. Let AOR stand for add-on rate, PV the present value (the initial principal amount), PV the future value (the redemption payment including interest), Days the number of days until maturity, and Year the number of days in the year. The relationship between these variables is:

(1.2)

The term in brackets is the interest earned on the bank CD – it is just the initial principal times the annual add-on rate times the fraction of the year.

The expression in 1.2 can be written more succinctly as:

(1.3)

Now we can calculate accurately the future value, or the redemption amount including interest, on the $1,000,000 bank CD paying 3.90% for six months. But first we have to deal with the fraction of the year. Most money market instruments in the U.S. use an “actual/360” day-count convention. That means Days, the numerator, is the actual number of days between the settlement date when the CD is purchased and the date it matures. The denominator usually is 360 days in the U.S. but in many other countries a more realistic 365-day year is used. Assuming that Days is 180 and Year is 360, the future value of the CD is $1,019,500, and not $l,019,313as incorrectly calculated using the standard time-value-of-money formulation.

Once the bank CD is issued, the FV is a known, fixed amount. Suppose that two months go by and the investor – for example, a money market mutual fund – decides to sell. A securities dealer at that time quotes a bid rate of 3.72% and an ask (or offer) rate of 3.70% on 4-month CDs corresponding to the credit risk of the issuing bank. Note that securities in the money market trade on a rate basis. The bid rate is higher than the ask rate so that the security will be bought by the dealer at a lower price than it is sold. In the bond market, securities usually trade on a price basis.

The sale price of the CD after the two months have gone by is found by substituting FV = $1,019,500, AOR = 0.0372, and Days = 120 into equation 1.3.

Note that the dealer buys the CD from the mutual fund at its quoted bid rate. We assume here that there are actually 120 days between the settlement date for the transaction and the maturity date. In most markets, there is a one-day difference between the trade date and the settlement date (i.e., next- day settlement, or “T + 1”).

The general pricing equation for add-on rate instruments shown in 1.3 can be rearranged algebraically to isolate the AOR term.

(1.4)

This indicates that a money market add-on rate is an annual percentage rate (APR) in that it is the number of time periods in the year, the first term in parentheses, times the interest rate per period, the second term. FV-PV is the interest earned; that divided by amount invested PV is the rate of return on the transaction for that time period. To annualize the periodic rate of return, we simply multiply by the number of periods in the year (Year/Days). I call this the periodicity of the interest rate. If Year is assumed to be 360 days and Days is 90, the periodicity is 4; if Days is 180, the periodicity is 2. Knowing the periodicity is critical to understanding an interest rate.

APRs are widely used in both money markets and bond markets. For example, the typical fixed-income bond makes semiannual coupon payments. If the payment is $3 per $100 in par value on May 15 and November 15 of each year, the coupon rate is stated to be 6%. Using an APR in the money market does require a subtle yet important assumption, however. It is assumed implicitly that the transaction can be replicated at the same rate per period. The 6-month bank CD in the example can have its AOR written like this:

The periodicity on this CD is 2 and its rate per (6-month) time period is 1.95%. The annualized rate of 3.90% assumes replication of the 6-month transaction on the very same terms.

Equation 1.4 can be used to obtain the ex-post rate of return realized by the money market mutual fund that purchased the CD and then sold it two months later to the dealer. Substitute in PV = $1,000,000, FV = $1,007,013, and Days = 60.

The 2-month holding-period rate of return turns out to be 4.21%. Notice that in this series of calculations, the meanings of PV and FV change. In one case PV is the original principal on the CD, in another it is the market value at a later date. In one case FV is the redemption amount at maturity, in another it is the sale price prior to maturity. Nevertheless, PV is always the first cash flow and PV is the second.

The mutual fund buys a 6-month CD at 3.90%, sells it as a 4-month CD at 3.72%, and realizes a 2-month holding-period rate of return of 4.21%. This statement, although accurate, contains rates that are annualized for different periodicities. Here 3.90% has a periodicity of 2; 3.72% has a periodicity of 3; and 4.21% has a periodicity of 6. Comparing interest rates that have varying periodicities can be a problem but one that can be remedied with a conversion formula. But first we need to deal with another problem – money market discount rates.