New Video: Derivatives of Bond Prices w.r.t. Parallel Shift in Yield Curve

The series of payments that arises from a coupon bond is an example of a cashflow that can be assigned a “duration“.

A coupon bond can be purchased as an investment that pays “coupons”, which are fixed amounts of money, at certain equally-spaced times during the term of the investment, plus a redemption value at the end of the term. This series of payments is an example of a discrete cashflow that can be assigned a certain rate of return, or yield, with respect to the initial price of the bond.

The duration of the payments of a coupon bond is a quantity that measures, in a precise and useful way, a type of weighted average of the payment times. It turns out that the higher the duration of a coupon bond, the more that coupon bond’s resale price will be sensitive to changing market conditions. That is, a bond with high duration has a high duration risk, which can be either good or bad for an investor, depending on whether market yield rates go down or up, respectively. When market yield rates go down, bond prices go up, which is good for an investor who wants to resell the bond. When market yield rates go up, bond prices go down, which is bad for an investor who wants to resell the bond.

There are two types of duration that are related to each other: Macaulay Duration and Modified Duration. The Macaulay Duration is directly defined to be a weighted average of the times of payment. To be technical, the Macaulay Duration is a sum $D_{mac}(i)=\displaystyle\sum_{t=1}^{n}w_{t}\cdot t$ , where the times of payment are $t=1,2,3,\ldots,n$ and the weights are $w_{1},w_{2},w_{3},\ldots, w_{n}.$ These weights are the sizes of the present values of each of the payments at the yield rate, relative to the price of the bond. If the periodic yield rate is $i$ and the periodic (present value) discount factor is $v=(1+i)^{-1}$ , then $w_{t}=\frac{K_{t}v^{t}}{\sum_{m=1}^{n}K_{m}v^{m}},$ where $K_{t}$ is the payment at time $t.$ Note that $\displaystyle\sum_{t=1}^{n}w_{t}=1.$

What is the Modified Duration, then? If we let $i$ be an arbitrary yield rate and let $P(i)$ be the price of the given bond (and its payments) at that arbitrary yield rate, then the Modified Duration, as a function of $i$ , is the relative rate of decay of the price $D_{mod}(i)=-\frac{P'(i)}{P(i)}.$

If this is evaluated at the yield rate at which the bond is purchased, this gives the Modified Duration for that particular bond. Is this also a weighted average of times of payments? It is when it is ultimately derived that the Modified Duration equals the Macaulay Duration divided by $1+i,$ (that is, $D_{mod}(i)=\frac{D_{mac}(i)}{1+i}$ ).

Both types of durations can be used to estimate changes in the price of a bond for a given change in the yield. The Macaulay Duration can be used in the first-order Macaulay approximation of a bond price and the Modified Duration can be used in the first-order modified approximation of a bond price (there are second-order approximations as well).

The videos I made in the following playlist delve into the details of all these ideas in problem-solving contexts.

Duration of Bonds (Macaulay and Modified) Problem-Solving Video Playlist

As of today, the 18th of January in 2019, the last video I added to this playlist delves into how to differentiate a bond price with respect to a change in the yield rate that is the same for bonds of different terms (a “parallel shift“), and how to graphically and financially interpret what this means. That is, I describe how to differentiate the price with respect to $\Delta i,$ though this quantity is called $\alpha$ in the video.

Financial Math for Actuarial Exam 2 (FM), Video #167. Exercise *7.1.11 (modified) from “The Mathematics of Investment and Credit”, 7th Edition, by Samuel A. Broverman.

Also considered in this video is what happens when the price of the bond is determined by the so-called term structure of spot rates of zero-coupon bonds. Once again, I describe how to differentiate the price with respect to $\Delta i=\alpha$ (a parallel shift), and how to graphically and financially interpret what this means.

Finally, I relate these calculations to both the Macaulay Duration and the Modified Duration of the bond. This last part is extra-technical, but is helpful for reviewing these concepts and confirming the correct answers.

In all of this, Wolfram Mathematica is of great help for calculations and graphing.