7

Assume a number of bonds with three constant variables, par value $par$, coupon value $C$ (paid annually), and interest rate $r$, and one changing variable, time to maturity $n$ First off, the relevant formulas: The price $P$ of each bond, as you've already written it, is $$ P=C*[\frac{1}{r}-\frac{1}{r}*\frac{1}{(1+r)^{n}}]+\frac{par}{(1+r)^{n}} $$ The ...


6

Maccauly Duration means nothing else than that after the given amount of years, you will have your capital investment back as nominal amount. If you have \$100 invested, and you have a duration of two years, after two years you will have gotten \$100 repaid, not directly dependent of interest rate or payment scheduling (indirectly they are of course!). I ...


6

The Macaulay duration is a measure of how sensitive a bond's price is to changes in interest rates. Duration is related to, but differs from, the slope of the plot of bond price against yield-to-maturity. The slope of the price-yield curve is $-\frac{D}{1+r}P,$ where $D$ is Macaulay duration, $P$ is bond price, and $r$ is yield. Here's how the definition ...


6

Yes, you are correct. Duration is additive, so your aggregate portfolio duration is the weighted average of your individual durations as you present in point 2. That holds assuming a close to flat yield curve and parallel (additive) shifts. If that's not the case, the situation gets a bit more complex. Unfortunately, right now I couldn't find any ...


5

It is useful in risk reports because it tells a trader the interest rate risk of each bond in his portfolio. A trader then only needs to multiply the duration by the expected yield change to calculate the price change. Scenario analysis is then easier. Hedging a bond portfolio with duration measures is common. But as these must rely on the assumption that ...


5

After struggling through the Pianca paper due to its poor proofing ($F$ is never defined but appears to be face value, and $n$ is implied to be the number of periods remaining but is instead maturity), I seem to have it worked out. Using the lambertW function in gsl, I have it replicated in R: # Estimate duration using various closed-form formulae # ...


5

There are different measures and interpretations of duration. One, as has been pointed out already, has a formula weighting coupons and final contractual cashflow. Other definitions of duration take a broader perspective and relate it to the interest rate sensitivity of the security and not to a particular formula. These go by names such as effective or ...


5

The answer is NO, with very few exceptions There might be bonds with negative coupon(s), and the Bloomberg search even finds some, but there are plenty of reasons why negative coupons are impractical. Instead of having negative coupons on the issue, there are bonds with low or 0 coupons, issued at a premium and having a negative yield. Here are some of ...


4

Like Aksakal already mentioned in his comment it might depend on the duration formula you use. (see e.g. the wikipedia page or here) It can also depend on the type of instrument as mentioned by Richard. This topic has also been already discussed on the Wilmott Forum (their proposed solution is a reverse floater) Theoretically bonds with embedded options (...


4

You don't say which duration, but it's generally okay to use effective duration: $$ duration (eff) = \frac{-1}{P(r)}*\frac{Price(r+b) - Price(r-b)}{2*b} $$ where $r$ = rate and $b$ = yield shock. Although, to address Brian's point, the mortgage contains an embedded call option that creates negative convexity, so the three re-pricings, $P(r)$, $P(r+b)$, $...


4

Yes, it is definitely possible to do so. With a long fixed-income portfolio, you'd typically be buying puts on treasury futures or writing calls on them (writing calls may not be feasible if you're an institutional investor due to regulatory reasons). In general, duration for long puts/short calls would be negative. However see caveats below: Typically, ...


4

If you're able to work with the results from the paper cited (Pianca, Maximum Duration of Below Par Bonds: A Closed-Form Formula), congratulations! You have the hard part done! Maximum durations for par and premium bonds are trivial. Here is a figure directly from the cited paper: Some points about the figure: the market interest rate used is $i=10\%$ $...


4

The intuition behind Macaulay Duration is the average time it takes to get all the cash flows from a bond. Think of it as computing the centre of gravity for a see-saw. You can find the image depicting the same here: This should immediately tell you that Macaulay Duration for Zero coupon bond is the maturity of the bond. In continuous discounting ...


4

Duration is not linear. It is the weighted average of the duration of the underlyings with the weightings being their values. To get a linear system multiply the durations by the associated pvs and match that quantity instead.


4

The change of the price $P(y)$ if the yield changes from $y$ to $y+\Delta y$ is $$ \frac{P(y+\Delta y) - P(y)}{P(y)} = - D \Delta y + \frac12 C \Delta y^2, $$ where $D$ is the duration and $C$ is convexity. For small $\Delta y$ the square is much smaller. Thus the duration component dominates.


4

Let's step back and look at the reason for making a DV01 calculation first before answering the question; The reason for making a DV01 calculation is to quantify what market movements has impact on the valuation of the trade. Since the 'flat' forecast curve won't be affected by market movements the answer is (using pre-2008 methodology): The floating ...


4

To directly answer the question: bond A= one day to maturity , price 100, yield 2%. Bond B: 10 years to maturity, price 100 yield 2%. This is perfectly possible. Bond B has greAter convexity but it also has substantially more risk.


4

This is an approximation (to first order) based on the idea that the option gives you access to the underlying, but with leverage. Let the duration of the underlying be $D_B$. The expression $\lambda=\Delta_c\frac{B}{C}$ is called the elasticity of the option (link), defined as "the percentage change in option value per percentage change in the underlying ...


3

If I understand correctly the question, you wish to completely hedge the interest rate risk (defined as a parallel shift in the yield curve). If that is the case, you should use modified duration, which is the price sensitivity, rather than the MacAulay duration. They are usually close in value, but not quite the same. Fortunately, you can easily transform ...


3

Macaulay duration is simply a weighted average. $MacD(A,B)=\frac{V(A) \cdot MacD(A)+ V(B) \cdot MacD(B)}{V(A)+V(B)}$


3

DV01 is the dollar variation in a bond's value per unit change in the yield. https://en.wikipedia.org/wiki/Bond_duration IR DV01 is the dollar value change for a 1bp upward parallel shift in interest rates. http://dataforthoughts.blogspot.it/2009/09/economics-of-negative-bond-cds-basis.html


3

You can do either. It depends on what you're trying to do and how you build your curve. If you're trying to match bond index duration, then shocking par curve is the way to go, because index providers, such as Barclays (now Bloomberg Barclays), Citi, and BofAML, all shock the par yield curve when reporting their option-adjusted duration statistics. However, ...


3

Modified duration is the right concept to use to estimate change in price in response to an infinitesimal change in yield. It works very well for a small change in yield (say a few basis points). However with a bigger yield change it gets less accurate, arguably with a 1% change in yield it is no longer satisfactory. What to do? Modified duration is the ...


3

Adding to the answer of Tim: If you consider a fixed-rate bond then IR-duration and spread-duration have the same effect on the bond. For a floating-rate bond, on the other side, you have IR-risk only until the next reset of the floating rate and thus very small IR-duration. The credit risk, however, is much higher than IR-risk and you can measure this ...


3

By definition, modified duration is $$ D_\text{mod} = \frac{1}{P} \frac{dP}{dy} $$ where $P$ is the dirty price of a bond. Clean price is the standard quoting convention for the vast majority of bond markets (though not all), but nearly all analytics, be it yield to maturity, DV01, or duration, are all computed using dirty price.


3

The duration of a bond is the percentage change in the value of the bond for a 1% change in yield. For example, a 10Y bond with a duration of 8Y will lose approximately 8% for every 1% increase in yield (equivalently 0.08% for every 1bp increase in yield). Eurodollar futures are derivatives, not cash instruments, so they do not have a duration (their value ...


2

You are correct: none of the durations are the slope of (the tangent to) the price/yield curve. Rather the slope is the "dollar duration" = modified duration * Price *-1. This will tend to betray rather large numbers; e.g., under continuous compounding the modified/Macaulay duration of a 100 par 10-year zero coupon bond is 10.0 years. The slope (of the ...


2

There are (at least) two factors here. One is the difference in convexity between the vanilla bond and the cheapest-to-deliver underlying the futures. The second is potential changes in which bond is cheapest to deliver. The former is simple enough to calculate, and you will need to dynamically hedge with futures to offset that risk For the latter, you will ...


2

Duration is also additive if you are dealing with key rate durations. In this case, Effective Duration is the weighted average of your key rate durations.


2

There are a couple limitations to bond portfolio immunization. Let's start by analyzing cash-flow matching which is a dedication strategy that is the alternative to immunization. Cash-flow matching can completely eliminate interest rate risk. The cash-flow match is setup such that the liabilities (outflows) are precisely offset by portfolio inflows on the ...


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