Duration of Bond = .75 x maturity?

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jlearman

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I’m working on queston 6 of the 2010 morning exam, and I’m confused about the answer. The answer requires you to figure out that the duration of a fixed payment, 3.5%, 4 year bond = 4. However, from the swaps section, we’re taught that the duration of the fixed leg is .75 x maturiy. So, shouldn’t the duration of a 4 year fixed bond be .75 x 4 = 3?
What am I missing here?
 
jlearman wrote:
I’m working on queston 6 of the 2010 morning exam, and I’m confused about the answer. The answer requires you to figure out that the duration of a fixed payment, 3.5%, 4 year bond = 4. However, from the swaps section, we’re taught that the duration of the fixed leg is .75 x maturiy. So, shouldn’t the duration of a 4 year fixed bond be .75 x 4 = 3?
What am I missing here?
Click to expand...
for a zero-coupon bond is a very specific case where maturity = duration
 
this is an assumption because its a fixed rate bond (not a floating one which is always getting reset)
 
The Righteous Hacksaw wrote:
this is an assumption because its a fixed rate bond (not a floating one which is always getting reset)
Click to expand...
Watch out, their is no assumption here, this question is a 1 time paiment of 80M$ in 4 years.
the duration of this is exactly 4.
 
When calculating the duration of a swap you have to consider both sides of the arrangement.
You are correct that the fixed payer side of the swap in this example is 0.75 x 4 = 3.
However, you will be receiving floating payments as well. In a perfect world if the payments are completely floating and adjust each minte, duration would be zero. However, in real world payments are typically quarterly or semi-annual (meaning the floating payer side is exposed to some duration).
Therefore, the formula for the floating side duration is 1/2 the length of the payment period. For example, if the floating payment is made semi-annually, the duration of that side would be (1/2) x 0.5 = 0.25.

That is subtracted from from the fixed side to get the net duration of the swap.
3.00 - 0.25 = 2.75
 
Wow, you guys responded quickly. Disregard my answer if it’s not applicable. I haven’t looked at the example yet.
 
.75 is not an assumption? Where does it come from them on the exam? I didnt see anything on the item set to conclude .75 with?
 
The Righteous Hacksaw wrote:
.75 is not an assumption? Where does it come from them on the exam? I didnt see anything on the item set to conclude .75 with?
Click to expand...
0.75 is an assumption about the fixed side of a SWAP. ( this is an assumption )
the question he is talking about is a zero-coupon bond of 4yr. ( this is not a assumption. )
 
Thanks for the responses… all makes sense. To sum it up:
Duration of a Fixed rate bond with COUPON Payments = .75 x maturity
Duration of ZERO COUPON bond = maturity
Duration of Swap = Fixed leg - Floating = (.75 x maturity) - (.5 x reset frequency) *** because fixed leg has coupon payments
 
summerside182]</p> <p>
The Righteous Hacksaw[/I said:
wrote:
.75 is not an assumption? Where does it come from them on the exam? I didnt see anything on the item set to conclude .75 with?
Click to expand...
0.75 is an assumption about the fixed side of a SWAP. ( this is an assumption )
the question he is talking about is a zero-coupon bond of 4yr which you have to know that zero coupon bond duration = maturity . ( this is not a assumption. )
Click to expand...
 
jlearman wrote:
Thanks for the responses… all makes sense. To sum it up:
Duration of a Fixed rate bond with COUPON Payments = .75 x maturity
Duration of ZERO COUPON bond = maturity
Duration of Swap = Fixed leg - Floating = (.75 x maturity) - (.5 x reset frequency) *** because fixed leg has coupon payments
Click to expand...
yes, so much confusion on this thread. lol
 
summerside182 wrote:
for a zero-coupon bond is a very specific case where maturity = duration
Click to expand...
Where did this defined? I mean where can you find it in the book?
 
Correct.
0.75 for the fixed side is not a hard rule. It’s an assumption that would be provided in the example. It will depend on what the vignette says.
 
Ericz, it comes down to the definition or duration (likely level 1 or 2). Duration for zeros will be the maturity.
 
ericz wrote:
summerside182 wrote:
for a zero-coupon bond is a very specific case where maturity = duration
Click to expand...
Where did this defined? I mean where can you find it in the book?
Click to expand...
try it on your calculator. this is just what it is. it is not explained anywhere in the L3 books.
FV = 100
pmt = 0
pv = CPT
I = 5%
n = 5
vary i from 4.75 to 5.25 and calculate the % variation. it will be approximatly 5%*0.25 = 1.25%
this is not exactly 1.25 because of convexity.
 
jlearman wrote:Macaulay Duration of ZERO COUPON bond = maturity
Click to expand...
Fixed that for you.
Modified duration of a zero is slightly less than maturity (you divide by (1 + YTM)).
 
S2000magician wrote:
jlearman wrote:Macaulay Duration of ZERO COUPON bond = maturity
Click to expand...
Fixed that for you.
Modified duration of a zero is slightly less than maturity (you divide by (1 + YTM)).
Click to expand...
yes sorry bout that, but still use duration = maturity at the exam :)
 
macaulay and modified duration will be equal when the yield is continuously compounded. interesting little tidbit based on the differentiation properties of the exponential function. completely not relevant to the exam tomorrow.
 
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