金融代写|金融数学代写Financial Mathematics代考|Duration

Duration is the premier measure of the interest rate risk of a fixed-income security. Duration, $D$, has several interpretations. The most common interpretation is that it is the weighted average of the longevities of a fixedincome asset or portfolio. The weights of each cash flow’s time-to-maturity are defined as the portion of the bond’s value attributable to the cash flow as expressed in Equation 2.15:
$$D=\sum_{i=1}^n w_i \times t_i$$
where $w_i=C_{t_i} e^{-y t_i} / \sum_{i=1}^n C_{t_i} e^{-y t_i}$ and $t_i$ is the time when cash flow is to be received, and $t_n=T$.
Note that Equation $2.15$ uses a bond’s yield to maturity for simplicity as the discount rate in calculating all of the weights. In theory, each cash flow should be discounted by a zero coupon rate corresponding to its longevity. Using zero coupon rates raises three problems: (1) Zero coupon rates are somewhat difficult to estimate, (2) estimates of zero coupon rates can vary based on the method used, and (3) if each cash flow is discounted with estimated zero coupon rates, the sum of the discounted cash flows may not exactly equal the bond’s price and the weights may not sum to one. Accordingly, Equation $2.15$ provides the convenient and popular yieldbased method of estimating the duration of a bond with fixed promised cash flows.

A second important interpretation of duration $(D)$ is that it is a measure of price risk from interest rate shifts. Specifically, duration may be defined and interpreted as the elasticity (percentage change) of a fixed-income value $(B)$ with respect to the underlying yields $(y)$. If the duration is estimated using the term structure of spot rates rather than yields then duration may be defined and interpreted as the elasticity of a fixed-income value $(B)$ with respect to a parallel shift in the term structure of interest rates.

金融代写|金融数学代写Financial Mathematics代考|Convexity

The use of duration in the approximation of bond returns can be viewed as a first-order Taylor expansion. In order to model the potential effects of large interest rate shifts, a second-order Taylor expansion may be used. Equations $2.20$ and $2.21$ provide the second-order risk measure (convexity) and the second-order approximation formula.
$$\begin{gathered} C=\frac{1}{B} \frac{\partial^2 B}{\partial y^2} \ \Delta B \approx-B D \Delta y+B \frac{1}{2} C(\Delta y)^2 \end{gathered}$$
Convexity approximates the curvature of a bond’s price-yield relationship. The formula for convexity is based on squaring the times-to-receipt of a bond’s cash flows. Roughly speaking, the convexity of a $T$ year zero coupon bond is approximately $T^2$. For two portfolios with equal durations, the portfolio with higher dispersion in the times-to-receipt of cash flows will tend to have greater convexity.

Duration is limited in its ability to approximate accurately in the case of: (1) large interest rate shifts and (2) non-parallel interest rate shifts. It should be noted that convexity not only provides a second-order approximation of the effects of large parallel interest rate shifts, it also approximates a first-order effect of slope changes in the term structure. Therefore, convexity and higher-order duration terms can be used in the case of managing the interest rate risk of fixed-income portfolios when greater accuracy is sought.

金融代写|金融数学代写金融数学代考|持续时间

$$D=\sum_{i=1}^n w_i \times t_i$$

金融代写|金融数学代写金融数学代考|凸性

$$\begin{gathered} C=\frac{1}{B} \frac{\partial^2 B}{\partial y^2} \ \Delta B \approx-B D \Delta y+B \frac{1}{2} C(\Delta y)^2 \end{gathered}$$

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