Query - Duration

[Avishek]

Hi David, Please take a look at the below Question. I have a feeling that I was correct. Duration has no Unit while the Yield should be decreasing with increasing Duration. Only Macaulay Duration has unit. Am I correct on this? Correct me please if am wrong.

Question: Which of the following statements regarding duration is FALSE?

A) Duration is a measure of percentage change in price for a given change in yield.

B) Duration is unitless.

C) Duration of a portfolio of bonds is equal to the market value weighted average of the duration of individual bonds in the portfolio.

D) Other things equal, bonds with longer durations tend to have higher yields.



Your answer: D was incorrect. The correct answer was B) Duration is unitless.

Duration is a measure of percentage change in price for a given change in yield. Hence it is not unitless like beta. Since duration is a measure of risk for bonds, bonds with higher duration tend to have a higher yield (other things being equal). Portfolio duration can be computed as simply market-value weighted durations of individual bonds in the portfolio.

 

[Avishek]

I guess am wrong. Duration comes in years!

 

[David Harper CFA FRM]

Avi,

No! What is source of question? (I hope/trust it's not ours...)

First, it's a bad question; (D) is imprecise and (D) is an understandable response! Jyothi is right to imply that not every sample question is gospel. The correct answer to this question is: (B) and (D). (D) absolutely CONTRADICTS the ASSIGNED READING, TUCKMAN SAYS the opposite...

First (B): don't say "duration comes in years." Macaulay duration is weighted avg. term to maturity; so, we can say MACAULAY DURATION is denominated in years.

But MODIFIED DURATION is: the approximate (i.e., linear approximation-remember?) percentage change in price for a 100 basis point change in yield." The temporal (years) definition is a fine, but inferior sort of slang (colloquialism)

Generally, a first derivative (duration) must always be in units (i.e., not unitless): we start with an equation that is unit-based. That is, Price (bond) = whatever. The price is expressed in dollars, so duration must be units too (dollars/yield). If it helps by way of example, consider that distance is always expressed in units (e.g., feet). The 1st derivative of distance (i.e,. velocity) must therefore also be given in units, too (feet/second); and the 2nd derivative also must be in units (feet/second ^2).

(D) is badly worded. They likely mean: a bond with longer duration tends to have a longer maturity. And longer maturity bonds (assuming a normal yield curve) will tend to have higher yields (e.g., a 10 year treasury, normally, yields more than a 2 year treasury).

But your FIRST INSTINCT was correct!

From Tuckman:

Inspection of equation (6.5) reveals that increasing yield lowers DV01. This fact has already been introduced: Chapter 5 discussed how bonds with fixed coupons display positive convexity, another way of saying that the derivative and DV01 fall as yield increases. As it turns out, increasing yield also lowers duration. The intuition behind this fact is that increasing yield lowers the present value of all payments but lowers the present value of the longer payments most. This implies that the value of the longer payments falls relative to the value of the whole bond. But since the duration of these longer payments is greatest, lowering their corresponding weights in the duration equation must lower the duration of the whole bond. Conversely, decreasing yield increases DV01 and duration.

David

 

[Avishek]

No David, This is certainly not from Bionic. It is from the Schwesers Pro Software. I went by elimination and found only D to be the last best option. Regarding option (B) I had doubts but clarified now.

Thanks again, Avi

 

[Atin]

Hi David

I have a question regarding the Duration explained in the screencast.
You explained
Duration = (V2 - V1)/2*(V0)*(Y2-Y1)

However the Tuckman text defined it as

Duration = (V2 - V1)/(V0)*(Y2-Y1)

Am I missing something here?

Thanks
Atin

 

[David Harper CFA FRM]

Hi Atin,

They are the same because mine is not (Y2-Y1) but rather (2)(yield shock). For example, if shock a 4% yield by 20 bps, then

Tuckman: 4.2% - 3.8%
Mine (Fabozzi): (2)(0.2%)

The reason i did this was for the sake of the convexity metric. I think Tuckman's convexity is hard to follow, where Fabozzi's convexity is easier and the yield shock (e.g., 20 bps) is squared. But that is just the rationale for doing that. The bottom line is they are the same.

David

 

[Atin]

Got it. Thanks