# Interest Rates, Market Pricing, and Compounding

7 Pages Posted: 21 Oct 2008

See all articles by Robert S. Harris

## Robert S. Harris

University of Virginia - Darden School of Business

## Robert M. Conroy

University of Virginia - Darden School of Business

### Abstract

Using examples from financial markets, this note examines links among market prices, stated interest rates, and compounding assumptions. The note emphasizes how interest rates are expressions of market prices, and pays particular attention to the role of compounding assumptions. Market prices are converted into stated interest rates for different compounding assumptions. Guidance is offered on how to make intelligent comparisons across markets that use different compounding conventions. The note is useful as background for the study of a wide array of pricing issues in fixed-income and options markets. See also "Interest Rate Derivatives" (UVA-F-1431).

Excerpt

UVA-F-1517

Interest Rates, Market Pricing, and Compounding

Bond-market pricing is typically stated in terms of interest rates. For instance, in U.S. bond markets, you might hear “the bond is priced to yield 10%.” This practice converts a bond's traded dollar price into an interest-rate equivalent, which allows ready comparison across different bonds and between bonds and other types of investments. Importantly, the relevant interest rates for pricing are market rates and are driven by current market conditions. These interest rates measure prospective returns to investors and, on the flip side, the costs of obtaining funds to bond issuers. Using interest-rate terminology to capture pricing applies to a host of financial markets.

This note explores compounding, a particular facet of the language and calculation of interest rates. As it turns out, to fully understand what a stated interest rate tells us about market pricing, we need to be specific about compounding assumptions. In bond markets (and almost all financial markets), interest rates are always stated on an annual basis. So, on the one hand, when you hear “priced to yield 10%,” this is understood to mean 10% per year. On the other hand, the compounding period (i.e., the time that elapses before we earn interest on interest) is not always the same. Conventions for compounding can differ from market to market and sometimes are not stated explicitly. For instance, the convention for U.S. bonds is semiannual compounding, while European bond markets use annual compounding.

An Example and the Mechanics of Inferring Interest Rates

To illustrate the interaction between the stated interest rate and the compounding period, let's consider a two-year zero-coupon bond that pays off \$ 10,000 two years from today and sells today for a price of \$ 9,245.56. To re-express this dollar price as an annual interest rate, we can use familiar present-value calculations. We express the current dollar price as the present value of the cash payoff in two years, and then solve for the annual interest rate that makes the math work out. This annual rate is often referred to as the bond's yield to maturity. But we can express this annual rate in many different ways, depending on what compounding period we choose. To formalize the relationships, let c represent the compounding basis. This could be annual, semiannual, quarterly, monthly, weekly, daily, or an even shorter period; n is the number of compounding periods per year. T is the number of years, and i is the stated annual rate of interest. Using this notation, we can express the bond's price in terms of an annual interest rate as follows:

. . .

Keywords: bond pricing, interest rates, compounding, effective interest rates, market prices and interest rates, valuation

Suggested Citation

Harris, Robert S. and Conroy, Robert M., Interest Rates, Market Pricing, and Compounding. Darden Case No. UVA-F-1517, Available at SSRN: https://ssrn.com/abstract=1279953 or http://dx.doi.org/10.2139/ssrn.1279953

44
Abstract Views
749
PlumX Metrics