Question

e) How to annualize a bi-weekly return?

f) Bond Duration and types of Duration

g) Features of Macaulay’s Duration Measure

h) Term Structure of Interest Rate and Term Structure Theory

i) Convexity of a Bond

j) Convexity of a Callable Bond[hint: see RB textbook]

k) Immunization Strategy

l) Zero-coupon bond vs. Perpetuity and their duration measures

Answer 1

f) Bond duration measures how long it takes, in years, for an investor to be repaid the bond’s price by the bond’s total cash flows. Using duration, you can estimate how much a bond’s price is likely to rise or fall if interest rates change (the bond’s price sensitivity), and it can be thought of as a measurement of interest rate risk. Remember that interest rates and bond prices move in opposite directions: When interest rates rise, bond prices fall, and vice versa. As maturity increases, duration also increases and the bond’s price becomes more sensitive to interest rate changes.

There are four main types of duration calculations, each of which differ in the way they account for factors such as interest rate changes and the bond's embedded options or redemption features. The four types of durations are Macaulay duration, modified duration, effective duration and key-rate duration.

g) The Macaulay duration (named after Frederick Macaulay, an economist who developed the concept in 1938) is a measure of a bond's sensitivity to interest rate changes. Technically, duration is the weighed average number of years the investor must hold a bond until the present value of the bond’s cash flows equals the amount paid for the bond.

h) The term structure of interest rates is the relationship between interest rates or bond yields and different terms or maturities. When graphed, the term structure of interest rates is known as a yield curve, and it plays a central role in an economy. The term structure reflects expectations of market participants about future changes in interest rates and their assessment of monetary policy conditions.

The shape of the yield curve has two major theories, one of which has three variations.

• Market Segmentation Theory: Assumes that borrowers and lenders live in specific sections of the yield curve based on their need to match assets and liabilities. The theory goes further to assume that these participants do not leave their preferred maturity section. Thus, the yield curve shape is determined by supply and demand at different maturities.

  The Market Segmentation Theory could be used to explain any of the three yield curve shapes.

• Expectations Theories (3): There are three variations of the Expectations Theory, one being “pure” and the other two “biased”. All three variations share a common assumption that short term forward interest rates reflect market expectations of short term rates will be in the future.
  1. Pure Expectations Theory (“pure”): Only market expectations for future rates will consistently impact the yield curve shape. A positively shaped curve indicates that rates will increase in the future, a flat curve signals that rates are not expected to change, and an inverted yield curve points to interest rates falling in the future.
  2. Liquidity Preference Theory (“biased”): Assumes that investors prefer short term bonds to long term bonds because of the increased uncertainty associated with a longer time horizon. Therefore investors demand a liquidity premium for longer dated bonds. This theory has a natural bias toward a positively sloped yield curve.
  3. Preferred Habitat Theory (“biased”): Postulates that the shape of the yield curve reflects investor expectations of future interest rates, but rejects the notion of a liquidity preference because some investors prefer longer holding periods. The Preferred Habitat Theory relies heavily on the notion that investors will match assets and liabilities.

i) bond convexity is a measure of the non-linear relationship of bond prices to changes in interest rates, the second derivative of the price of the bond with respect to interest rates (duration is the first derivative). In general, the higher the duration, the more sensitive the bond price is to the change in interest rates. Bond convexity is one of the most basic and widely used forms of convexity in finance.