Duration, DVO1, Maturity and Coupon (FRM Part 1)
1. Context
In this swatch, we look at a graphical analysis of how first-order yield based risk metrics (Duration and DVO1) vary with characteristics of the instrument (mainly maturity and coupon). The details of the reading in which this topic appears are given below:
| Area | Valuation and Risk Management |
| Reading | Interest Rate Risk |
| Reference | Tuckman, Bruce Ch4. One-Factor Risk Metrics and Hedges. In Fixed Income Securities, 3rd Edition, Hoboken, NJ: John Wiley & Sons, 2011 |
2. Symbols
We use the following symbols in the sections that follow:
| $y$ | Interest rate factor (yield, in current context) |
| $P(y)$ | Price rate function |
| $D$ | Duration |
| $DVO1$ | Dollar Value of 1 Basis Point |
| $c_i$ | Cash flow paid by fixed income security (bond) at date $t_i$ |
| $t_i$ | Payment dates |
2. DVO1: Definition and Calculation
If the price vs rate function of a fixed income security is denoted by $P(y)$, then let $\Delta P$ and $\Delta y$ denote the change in the price and rate (or factor, measured in basis points) respectively. Then $DVO1$, acronym that stands for “dollar value of an 01” is a price sensitivity defined as the change in value of a fixed income security for a one-basis point decline in rates (a basis point refers to 0.01%): $$\begin{matrix} DVO1 &=& -\frac{\Delta P}{10,000\times \Delta y}\\ &=&-\frac{\left(P(y+\Delta y/2)-P(y-\Delta y /2)\right)}{10,000\times \Delta y} \end{matrix}$$ Owing to the (-) sign included in the above definition, DVO1 is positive if price increases when rates decline and negative otherwise. This sign convention is adopted so that DVO1 is positive most of the time.
3. Duration: Definition and Calculation
Duration (or effective duration) measures the percentage change in value of a security for a unit change in rates, as given below: $$\begin{matrix} D &=& -\frac{1}{P} \, \frac{\Delta P}{\Delta y}\\ &=&-\frac{1}{P(y)} \, \frac{P(y+\Delta y/2)-P(y-\Delta y/2)}{\Delta y}\\ \end{matrix}$$
4. Duration: Variation with Instrument Characteristics and Yield
5. DVO1: Variation with Instrument Characteristics
