7
votes
Is duration of a bond a convex function?
The generic bond pricing function is
$$
PV = \sum_i^n c_iD(t_i)+D(t_n)
$$
Convexity of PV01
Let's identify its duration with the negative of its first derivate, and let's set $D(t_i)=e^{-rt_i}$
$$
D\...
- 6,175
5
votes
How can we have negative probabilities in finance? Can we have negative payments in bonds? If not, how else can we have negative probabilities?
The answer is NO, with very few exceptions
There might be bonds with negative coupon(s), and the Bloomberg search even finds some, but there are plenty of reasons why negative coupons are impractical....
- 2,914
5
votes
Investment Grade Bond vs Junk Bond, whose duration is larger?
There are different measures and interpretations of duration. One, as has been pointed out already, has a formula weighting coupons and final contractual cashflow. Other definitions of duration take a ...
- 266
5
votes
Accepted
Duration vs. Convexity Contradiction
The change of the price $P(y)$ if the yield changes from $y$ to $y+\Delta y$ is
$$
\frac{P(y+\Delta y) - P(y)}{P(y)} = - D \Delta y + \frac12 C \Delta y^2,
$$
where $D$ is the duration and $C$ is ...
- 13.4k
5
votes
Accepted
What curve are you shifting when you calculate DV01 for a swap?
Let's step back and look at the reason for making a DV01 calculation first before answering the question;
The reason for making a DV01 calculation is to quantify what market movements has impact on ...
- 86
5
votes
Can two bonds have same yield and price but different convexity?
To directly answer the question: bond A= one day to maturity , price 100, yield 2%. Bond B: 10 years to maturity, price 100 yield 2%. This is perfectly possible. Bond B has greAter convexity but ...
- 15.2k
5
votes
Clean vs. Dirty Price and its impact on duration
By definition, modified duration is
$$ D_\text{mod} = \frac{1}{P} \frac{dP}{dy} $$
where $P$ is the dirty price of a bond.
Clean price is the standard quoting convention for the vast majority of bond ...
- 11.1k
5
votes
Accepted
Origin of the $-\frac{1}{P}$ in Macaulay Duration?
We want the duration $D$ to satisfy $$\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 ...
- 14.9k
5
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. ...
- 5,523
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 ...
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( ...
- 6,104
5
votes
Is duration of a bond a convex function?
Recall that duration is defined as the average time to receive the cashflow, with the weights being the present values of the cashflows. So when interest rates rise very high, the long dated ...
- 15.2k
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 ...
- 6,833
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 ...
- 13.4k
4
votes
Accepted
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 $\...
- 9,212
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 ...
- 5,718
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 ...
- 153
4
votes
Accepted
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 ...
- 161
4
votes
Accepted
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 ...
- 6,175
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 ...
- 9,212
3
votes
Accepted
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 ...
- 722
3
votes
Macaulay Duration: Duration for 2 bonds
Macaulay duration is simply a weighted average.
$MacD(A,B)=\frac{V(A) \cdot MacD(A)+ V(B) \cdot MacD(B)}{V(A)+V(B)}$
- 768
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 ...
- 187
3
votes
Accepted
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, ...
- 520
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,...
- 11.1k
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). ...
- 9,212
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 ...
- 5,592
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 ...
- 31
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 ...
- 8,217
3
votes
ATM interest rate swap dv01 vs off-market swap dv01
You need Gamma to answer this question really. Gamma tells you how much your delta moves for a change in rates. Taking a 5y \$ receiver swap with a DV01 of \$4333.60 on 10MM notional we get a Gamma ...
- 1,230
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