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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\...
Kermittfrog's user avatar
  • 6,792
6 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 ...
Helin's user avatar
  • 11.8k
6 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. ...
Jan Stuller's user avatar
  • 6,178
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 ...
horseless's user avatar
  • 266
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....
vanguard2k's user avatar
  • 2,925
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 ...
Fred's user avatar
  • 86
5 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 ...
Richi Wa's user avatar
  • 13.7k
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 ...
dm63's user avatar
  • 17.2k
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 ...
Kevin's user avatar
  • 16k
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 ...
Pythonista anonymous's user avatar
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( ...
Magic is in the chain's user avatar
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 ...
dm63's user avatar
  • 17.2k
5 votes
Accepted

Is the "$\textit{theoretical}$" $DV01$ of a bond an accurate estimate?

Question 1: Not that often, usually finite difference is use to compute it (bump and reprice). See for example this complete step by step explanation of Bloomberg's DV01 computation in SWPM. ...
AKdemy's user avatar
  • 9,024
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 $\...
Alex C's user avatar
  • 9,382
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 ...
Chris Taylor's user avatar
  • 5,931
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 ...
teoeme139's user avatar
  • 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 ...
wgajate's user avatar
  • 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 ...
Kermittfrog's user avatar
  • 6,792
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 ...
Stephen Ge's user avatar
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,...
Helin's user avatar
  • 11.8k
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). ...
Alex C's user avatar
  • 9,382
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 ...
Antoine Conze's user avatar
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 ...
Mario Marra's user avatar
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 ...
Attack68's user avatar
  • 10.8k
3 votes
Accepted

Macaulay Duration - Liability matching

Can someone provide a detailed example to prove the following statement: "When the investment horizon is equal to the Macaulay duration of the bond, coupon reinvestment risk offsets price risk.&...
Pontus Hultkrantz's user avatar
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 ...
oronimbus's user avatar
  • 1,896
3 votes

Simplified formula for duration of interest rate swap

That's an interesting approximation, I have not yet seen this one. The PV of a fix-for-float IRS, in a single-curve-world, is: $$ PV=\sum_i\Delta^{float}_iF_iD(t_i)-q\sum_j\Delta^{fix}_jD(t_j)=1-D(t_n)...
Kermittfrog's user avatar
  • 6,792
2 votes

What's the intuition behind DTS(duration times spread) in fixed income?

The DTS paper (Ben Lor, Dynkin et al) describes how DTS (duration times spread) can be used as both an exposure to a common factor, and a measure of specific risk. Assume that relative spread changes ...
Chris Taylor's user avatar
  • 5,931
2 votes

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

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 ...
Lior Yochpaz's user avatar
2 votes

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

Duration is technically independent of credit risk. ANY bond's duration is just a matter of coupon, price, discount rate. However, many issued high yield bond ARE typically shorter, because of a. ...
Larasing's user avatar
  • 193

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