Question

Describe the differences between Macaulay, modified and effective duration.

Explain the difference between positive and negative convexity.

Answer 1

Effective duration is considered to be a measure for bond duration with embedded options. It takes into account fluctuations in movements of price of a bond relative to changes in yield to maturity (YTM) of the same. Put simply, it considers possible fluctuations which are associated with the expected cash flows from a bond. It is computed as enumerated below:

Where,

· V_–Δy is the value of the bond when there is a fall in yield by y%

· V_+Δy is the value of the bond when there is a rise in yield by y%

· V₀ is the present value of the bond cash flows

· Δy is the changes in yield.

The metric of modified duration is considered to be a more precise indicator of sensitivity of prices as against Macaulay duration. It is mainly used for bonds, but is also applied to other various securities which are viewed as a function of yield. It provides an estimate of the percentage change in the value of a bond associated with a change in rate of interest. It is recognized as an extension of the Macaulay duration. It is computed as enumerated below:

Macaulay duration is considered as the weighted average of time until the cash flows from a fixed interest bearing security are received. It is computed with respect to units in time such as years. It mainly provides an estimate of the time required for repayment of the price of a bond by its total cash flows.

If the duration of a bond is being increased with the increase in yield, the bond is considered to have negative convexity. Put simply, bond prices would decline at a higher rate with higher yields as against fall in yields. Thus, if any bond has negative convexity, it would lead to increase in its duration with the fall price in price. The opposite is considered to be true as there is rise in rate of interest.

If the duration of a bond is being increased with the decrease in yield, the bond is considered to possess positive convexity. Put simply, bond prices would rise at a higher rate with fall in yields as against rise in yields. It leads to higher increases in prices of bonds. When the bond has a positive convexity, it will usually undergo higher price rises when yields fall as compared to that with price decreases as there is a rise in yield.