83. Duration concept, application, concept of convexity, and how convexity affects macalay’s duration
83. Duration concept, application, concept of convexity, and how convexity affects macalay’s duration
Duration: A measure of the sensitivity of the price (the value of principal) of a fixed-income investment to a change in interest rates. Duration is expressed as a number of years. Rising interest rates mean falling bond prices, while declining interest rates mean rising bond prices. There are two types of duration.
McCauley Duration shows average term (in years) of payments to serve and repay the debt, weighted upon their discounted value. Macaulay duration is the name given to the weighted average time until cash flows are received, and is measured in years. Macaulay duration is a time measure with units in years, and really makes sense only for an instrument with fixed cash flows. For a standard bond the Macaulay duration will be between 0 and the maturity of the bond. It is equal to the maturity if and only if the bond is a zero-coupon bond.
When yields are continuously-compounded Macaulay duration and modified duration will be numerically equal. When yields are periodically-compounded Macaulay and modified duration will differ slightly, and in this case there is a simple relation between the two. Modified duration is used more than Macaulay duration. Macaulay duration is the weighted average maturity of cash flows. Consider some set of fixed cash flows. The present value of these cash flows is:
Macaulay
duration is defined as:
where:
i - indexes the cash flows,
PVi - is the present value of the i-th cash payment from an asset,
ti-is the time in years until the i-th payment will be received,
V-is the present value of all cash payments from the asset.
The
Macaulay duration will equal the final maturity if and only if there
is only a single payment at maturity. In symbols, if cash flows are,
in order,
,
then:
.
Modified Duration shows sensitivity of bond’s dirty price (the price which includes the interest which haven’t been paid yet) to changes in yield. In other words it shows how much a bond’s dirty price will change, if yield (interest rates) changes to a certain extent. Modified duration is the name given to the price sensitivity and is the percentage change in price for a unit change in yield. Modified duration, on the other hand, is a derivative (rate of change) or price sensitivity and measures the percentage rate of change of price with respect to yield. The concept of modified duration can be applied to interest-rate sensitive instruments with non-fixed cash flows, and can thus be applied to a wider range of instruments than can Macaulay duration. The formula is:
where YTM-yield to maturity, n – frequency of coupon payment within a year, P – dirty price, ∆P – change of dirty price, ∆YTM – change in yield to maturity.
Convexity.Duration is linear a measure of how the price of a bond changes in response to interest rate changes. As interest rates change, the price does not change linearly, but rather is a convex function of interest rates. Convexity is a measure of the curvature of how the price of a bond changes as the interest rate changes. Specifically, duration can be formulated as the first derivative of the price function of the bond with respect to the interest rate in question, and the convexity as the second derivative. For any given bond, a graph of the relationship between price and yield is convex. This means that the graph forms a curve rather than a straight-line (linear). The degree to which the graph is curved shows how much a bond's yield changes in response to a change in price. If we graph a tangent at a particular price of the bond (touching a point on the curved price-yield curve), the linear tangent is the bond's duration, which is shown in red on the graph below. The exact point where the two lines touch represents Macaulay duration.
