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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 ...

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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 ...

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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 ...

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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 (...

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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 ...

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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 ...

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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 ...

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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 ...

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We want the duration $D$ to satisfy $$\mathrm{d}P=-PD\mathrm{d}y,$$ i.e. it tells us the proportional change in the bond price if the interest rate (yield) changes. The minus is due to the inverse relationship between bond yield and bond price. Thus, $$D=-\frac{1}{P}\frac{\mathrm{d} P}{\mathrm{d} y}.$$ Duration can be seen as a linear approximation to the ...

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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, $P(r)$, $P(r+b)$, $... 4 Duration is not linear. It is the weighted average of the duration of the underlyings with the weightings being their values. To get a linear system multiply the durations by the associated pvs and match that quantity instead. 4 The change of the price$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 ... 4 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 ... 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 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 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 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 If your notional is 100mm, and you buy a 10Y treasury note worth 10mm (10% of 100mm) then you own 100 contracts (since each contract specification is officially a nominal of \$100,000), and the DV01 is approximately \$8000/bp. If you sell 10mm of a EDH0 then you have sold 10 contracts (since each contract specification is officially a \$1mm nominal) and the ...

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You are right that if we exactly want to know the price of a bond after a change in the yield curve, we have to calculate it - and we can. What we can say about duration: it is a linear approximation of the price change if yield change, this works rather fine with plain vanilla bonds but things get more difficult e.g. with callable bonds. keeping the eye ...

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The simple but accurate answer should be that Macaulay Duration is the weighted average maturity of cash flows (in years). That is how it is defined in almost every text book and looked at by most market practitioners. That is why its quoted in years and it gives an indication of when, on a weighted basis, cash flows are paid out (mature). For example, in ...

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In Bonds with a negative yield the duration should be longer than the maturity. Duration is the length of time for the return of the fund. As long as the coupons are positive, the investor returns the fund before the final redemption, in a negative interest rate situation the negative interest rate eats from the fund and the duration should be longer ...

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I'm not a bond trader and haven't looked at this in years, so my quantities may not be defined exactly as per convention, but it is generally correct. To answer your question, you should restate the present value of the bond using exponentials. This new formulation is exactly equivalent to what you wrote but much more tractable algebraically (note that my ...

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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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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 ...

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