The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. The relation between coupon and convexity Ask Question. Asked 1 year, 11 months ago. Active 1 year, 5 months ago. Viewed times.
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And can anyone proof why in formula? Oreo A. Oreo 4 4 silver badges 15 15 bronze badges. Oreo Nov 26 '17 at OP also does not want to go into details, he wants "just the proff of those three statements". Oreo Nov 27 '17 at Oreo After careful inspection, I tend to agree. For instance, since zero-coupon bonds only pay the face value at maturity, the duration of a zero is equal to its maturity.
It also follows that any bond of a certain duration will have an interest rate sensitivity equal to a zero-coupon bond with a maturity equal to the bond's duration. Duration is also often interpreted as the percentage change in a bond's price for a small change in its yield to maturity YTM. It should not be surprising that there is a relationship between the change in bond price and the change in duration when the yield changes, since both the bond and duration depend on the present values of the bond's cash flows.
Before , it was well known that the maturity of a bond affected its interest rate risk, but it was also known that bonds with the same maturity could differ widely in price changes with changes to yield. On the other hand, zero-coupon bonds always exhibited the same interest rate risk. Therefore, Frederick Macaulay reasoned that a better measure of interest rate risk is to consider a coupon bond as a series of zero-coupon bonds, where each payment is a zero-coupon bond weighted by the present value of the payment divided by the bond price.
Hence, duration is the effective maturity of a bond, which is why it is measured in years. Not only can the Macaulay duration measure the effective maturity of a bond, it can also be used to calculate the average maturity of a portfolio of fixed-income securities. The Macaulay duration is calculated by 1 st calculating the weighted average of the present value PV of each cash flow at time t by the following formula:.
For a interest rate that is continuously compounded, the weighted average is equal to the following:.
Dispersion and convexity
The duration can be calculated as follows:. Because the bond price is equal to the total present value of all bond payments, the bond price will change inversely to changes in yield, which can be calculated approximately by the following equation:. So if interest rates increased by 0. Compare this calculation with the bond price as given by the sum of the present value of its payments:.
As you can see, bond prices as calculated using Macaulay duration is very close to the price calculated with the present values of the cash flows when the interest rate change is small. In fact, when rounded, the values are equal. The duration adjustment is a close approximation for small changes in interest rates. However, duration changes as well, which is measured by the bond's convexity discussed later. Because duration also changes, larger changes in interest rates will yield larger discrepancies between the actual bond price and the price calculated using duration.
Duration can also be approximated by the following formula:. Duration is measured in years, so it does not directly measure the change in bond prices with respect to changes in yield. Nonetheless, interest rate risk can easily be compared by comparing the durations of different bonds or portfolios. Modified duration, on the other hand, does measure the sensitivity of changes in bond price with changes in yield. Like Macaulay duration, modified duration is valid only when the change in yield is small and the yield change will not alter the cash flow of the bond, such as may occur, for instance, if the price change for a callable bond increases the likelihood that it will be called.
Of course, interest rates usually only change in small steps, so duration is an effective tool to measure interest rate risk. Note that modified duration is always slightly less than duration, since the modified duration is the duration divided by 1 plus the yield per payment period. Convexity adds a term to the modified duration, making it more precise, by accounting for the change in duration as the yield changes—hence, convexity is the 2 nd derivative of the price-yield curve at the current price-yield point.
Note that the price-yield curve is convex, and that the modified duration is the slope of the tangent line to a particular market yield, and that the discrepancy between the price-yield curve and the modified duration increases with greater changes in the interest rate.
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It can easily be seen that modified duration changes as the yield changes because it is obvious that the slope of the line changes with different yields. The gap between the modified duration and the convex price-yield curve is the convexity adjustment, which — as can be easily seen — is greater on the upside than on the downside. Although duration itself can never be negative, convexity can make it negative, since there are some securities, such as some mortgage-backed securities that exhibit negative convexity , meaning that the bond changes in price in the same direction as the yield changes.
Because duration depends on the weighted averages of the present value of the bond's cash flows, a simple calculation for duration is not valid if the change in yield could result in a change of cash flow. Valuation models must be used in calculating new prices for changes in yield when the cash flow is modified by options. The effective duration aka option-adjusted duration is the change in bond prices per change in yield when the change in yield can cause different cash flows. Deferred callable bonds Deferred callable bonds are similar to callable bonds, except that these have an initial period of call protection, which is a time period in which the bonds are not callable.
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Convertible bonds 3 Convertible bonds provide the bondholder the option to exchange the bond for aspecified number of shares of the issuing firm. Conversion ratio: the number of shares that each bond can be exchanged for 2. Market conversion value: current value of shares for which the bond can be exchanged 3. Conversion premium: excess of bond value over its conversion value Holders of convertible bonds benefit from price appreciation of the stock, and so the convertible bonds offer lower coupon rates.
Floating rate bonds Floating rate bonds provide interest payments that are based on the current market rates. There is therefore less interest rate risk: as interest rates rise, the increase in interest offsets the higher discounting rate.
Wealth Management
Inverse floaters Inverse floaters are similar to floating rate bonds, but the coupon falls when interest rates rise. Catastrophe bonds Catastrophe bonds are discussed elsewhere in the syllabus. They provide a higher coupon rate because if there is a catastrophe, the bondholder will lose a portion or all of the investment. Indexed bonds Indexed bonds make payments that are based on a general price index, or the price of a specific commodity. Preferred stocks Preferred stocks are technically equity, but are often grouped with bonds. This is because preferred stocks promise to pay a specified stream of dividends just like bonds promise to pay a stream of coupon payments.
An increase in the yield produces a smaller price change than the same size decrease in yield 3. As the term of the bond increases, the price becomes more sensitive to yield changes steeper yield curve 4.
The sensitivity of bond prices to yield increases at a decreasing rate as the maturity increases convexity 5. Lower coupon bonds are more sensitive to changes in yields 6. The sensitivity of bond prices to yield are inversely related to the yield. Higher yield bonds are less sensitive to yield changes flatter yield curve. The duration of a zero-coupon bond equals its time to maturity 2. Duration decreases as the coupon rate increases 3. Duration generally increases as the maturity increases 4.
Positive And Negative Convexity
Duration increases as the yield to maturity reduces 5. The Homer-Liebowitz taxonomy of active bond portfolio strategies: substitution swap : exchange one bond for a nearly identical bond based on perceived mispricing intermarket spread swap : profiting from temporary imbalance in the historical spread of yields between two sectors. Intermarket spread swap - You buy the corps and sell the government bonds?