5 votes
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Duration vs. Convexity Contradiction

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 ...
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5 votes

Investment Grade Bond vs Junk Bond, whose duration is larger?

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 ...
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  • 266
5 votes

Why is the duration of a bond important?

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 ...
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  • 2,059
5 votes

What is the intuition behind the fact that Modified duration = Macaulay Duration / (1+r)?

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 ...
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  • 301
5 votes

Can the duration of a bond be greater than Time to Maturity

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 ...
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  • 3,337
5 votes

How can we have negative probabilities in finance? Can we have negative payments in bonds? If not, how else can we have negative probabilities?

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....
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  • 2,874
5 votes
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What curve are you shifting when you calculate DV01 for a swap?

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 ...
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  • 86
5 votes

Clean vs. Dirty Price and its impact on duration

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 ...
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5 votes
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Origin of the $-\frac{1}{P}$ in Macaulay Duration?

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 ...
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5 votes

How can a deep discount bond with a longer time to maturity have a LOWER duration than an otherwise identical bond with a shorter time to maturity?

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 ...
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5 votes

How can a deep discount bond with a longer time to maturity have a LOWER duration than an otherwise identical bond with a shorter time to maturity?

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( ...
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4 votes
Accepted

A question on immunization and Macaulay duration

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 ...
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4 votes

Why is 'duration' not the same as 'spread duration' for risky bonds

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 ...
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4 votes

Can two bonds have same yield and price but different convexity?

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 ...
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4 votes
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How to derive and interpret the duration of a call option?

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 $\...
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  • 9,077
4 votes

US 10yr future and ED future

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 ...
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  • 5,603
4 votes

Can you calculate modified duration for swaps?

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 ...
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  • 153
4 votes
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Duration of a floating rate bond with spread

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 ...
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  • 161
4 votes
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Is the risk the same for two different tenor bonds with the same DV01?

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 ...
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  • 5,333
3 votes

Duration of a floating rate bond

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 ...
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  • 9,077
3 votes
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Duration of perpetual bond

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 ...
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  • 712
3 votes

Macaulay Duration: Duration for 2 bonds

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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  • 758
3 votes
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Bond Portfolio Immunization - Duration Matching

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, ...
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  • 520
3 votes

Interpretation of Macaulay Duration

There are many ways to understand the Macaulay Duration, one of them is from "the interest rate risk" point of view. For a fixed coupon bond, there are two risks that is caused by the change ...
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3 votes

To compute key rate duration, shall I use par curve or zero curve?

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,...
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3 votes

duration and modified duration

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). ...
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  • 9,077
3 votes

High convexity vs low convexity bond definition

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 ...
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3 votes

How should we calculate the duration of a convertible bond?

Unfortunately, convertible bonds are quite complex so you don't have simple formulas or approaches as with vanilla bonds. However, this does not mean you are powerless. You can follow different ...
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3 votes

US 10yr future and ED future

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 ...
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  • 7,997
3 votes

Modified Duration and how it explains bond price sensitivity to changes in the yield to maturity

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. ...
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