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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.


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 ...


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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 (...


5

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 ...


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

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 ...


5

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 ...


5

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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It's a very good question. This is also mentioned in "Bond Math: the theory behind the formulas" - but the author doesn't get into a lot of details, he just mentions it as some kind of a mathematical oddity, if I remember correctly. If there is a rigorous proof with a closed-form formula, the maths is beyond me. What I can do is help you think ...


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Probably easier to see with the $Dur, which can be expressed as follows (assuming principal=1): ${\rm Dur}=\frac{c}{y^2}\left(1-{\frac { yT+y+1}{ \left( 1+y \right)^{T+1} }}\right)+\frac{T} {\left( 1+y \right) ^{T+1}}={\rm Cpn \,Contrib+Princp\, Contrib}$ The numerator in the principal component is linear (T), so as T grows, the denominator will start to ...


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.


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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 ...


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.


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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.


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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 ...


4

If you know how to calculate them for bonds, you know how to calculate them for swaps. Assuming you refer to fixed-income swaps where a party receives a fixed rate and pays a floating rate or vice versa, the duration of a swap is the duration of the long position and the duration of your short position, which in this case will be a negative duration. Let's ...


4

To a first order of approximation, $dV=\frac{\partial V}{\partial r}dr$, and assuming normally distributed rate shifts, $dr\sim N(0,\sigma_r^2)$, then your risk is -- again to a first oder of approximation -- $\sigma_V^2=DV01^2\sigma_r^2$. Hence, your two risks may be the same if the curve shifts are parallel along the curve. What is more likely, though, is ...


3

You were on a right track. In the first approach you've shown Modified Duration of perpetuity is $ModDur=\frac{1}{r}$. In your second approach keep in mind that $ModDur=\frac{MacDur}{(1+y_k/k)}$ so for annual compounding your second approach should converge to $MacDur=ModDur \cdot (1+r) = \frac{1+r}{r}$, which should be the case. $$S_m=\sum_{k=1}^mkx^k=x+2x^...


3

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


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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

Yes. the duration of a floating rate bond is the time t until the next coupon payment, as your equation shows. The payments that come after are not known yet and will be determined based on interest rates then prevailing, so they carry no duration risk. In general floating rate bonds are what people buy when they want the smallest duration possible. Long ...


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 ...


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Do not forget the effect of passing time (the theta) on your portfolio. If two portfolios have the same value and duration, then the portfolio made up of the difference has locally zero sensitivity to yields and is delta hedged. Since the sum of (modified) theta (the derivative to time minus the position funding, in essence the carry) and local yield ...


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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Is formula (1) correct? Yes, follows from first definition - floater with deterministic spread is composed (sum) of two components: (1) pure floater and (2) deterministic coupon strip via contractual spread payment. Is formula (2) correct? Yes, by taking the derivative of an exponential function. what other case where duration of floating rate bond not ...


3

Bond price in terms of yield (denoted "$y$") is just the Present Value (PV) of the Bond coupons (denoted "$C$") and the final Notional (denoted $N$), discounted at the yield. Suppose the bond matures in 10 years time, then the present value can be written as (yields expressed as annualized for simplicity): $$PV=\sum_{i=1}^{10} \frac{C}{(1+...


2

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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