Listing Category: Fixed Income

Yield Measures (Fixed Income)

There are a number of types of yield measures for bonds; you need to know how they are calculated, the assumptions that underlie them, and their strengths and weaknesses.

Risks in Fixed Income Investing

There are quite a few risks inherent in investing in fixed income securities / portfolios.  Amongst them are these that I’ll discuss in this article: Go to Blog Post…

Par Curve, Spot Curve, and Forward Curve

When you hear someone talk about “the yield curve”, they usually mean the par yield curve (and, more specifically, the par yield curve for risk-free bonds (e.g., the U.S. Treasury par curve)), but there are occasions when they might mean the spot yield curve or the forward yield curve. We’ll go through a description of each curve, how they’re related (and how they differ), and, finally, what they look like. In all cases, remember that we’re talking about yields on bonds, so the values quoted for yields will be bond equivalent yields (BEYs) (as opposed to, say, effective annual yields (EAYs)).

Macaulay Duration, Modified Duration, and Effective Duration

In this article I’ll cover three quantities that go by the name of “duration”…


As we’ve seen in the article on duration, the duration of a bond (whether Macaulay duration, modified duration, or effective duration) is not constant; amongst the factors that cause (all types of) duration to change is the bond’s yield to maturity (YTM).

Calculating Forward Rates (from Spot Rates)

A forward interest rate is a discount rate that takes a single payment at one point in the future and discounts it to another (nearer) time in the future; they have their own special notation. For example, if we’re measuring time in years, the discount rate that would take a payment 6 years from now and discount it back to four years from now would be written as: 2f4: the 2-year forward rate (from year 4 to year 6) starting 4 years from now. The discount rate that would take a payment 5 years from now and discount it back to 2 years from now would be written as: 3f2: the 3-year forward rate starting 2 years from now. Note that these are annual rates; you need to discount for the appropriate number of years.

Bond Equivalent Yield (BEY)

Because a large number of coupon-paying bonds make their coupon payments semiannually (e.g., US Treasury Notes and Bonds, and many corporate bonds), and each coupon payment is ½ of the annual coupon (i.e., each coupon is calculated as 0.5 × annual coupon rate × par), the default yield convention for bonds is that the annual yield is quoted as the annual coupon rate; i.e., the annual yield is twice the effective semiannual yield. This convention for annual bond yields is called bond equivalent yield (BEY). Symbolically: