Convexity-free

[hellohi]

Dear @David Harper CFA FRM

in the original reading of "Mortgages and Mortgages-Backed Securities" Tuckman wrote this paragraph related to "Constant maturity mortgage (CMM)" as a product other than MBS (under "Other Products" title in the original reading), and mentiond "Convexity-free" and I did not understand what "Convexity-free" means, the paragraph text is as follows:

"Constant maturity mortgage (CMM) products allow investors to trade mortgage rates directly as a convexity-free alternative to trading prices of MBS that depend on the mortgage rate"

thanks Boss
Nabil

 

[QuantMan2318]

Dear Nabil

As you know, convexity of a Bond is the shape of the yield curve in response changes in interest rates or the degree of curvature of the Price/Yield curve, the price/yield or price rate relationship is not always linear as far as Bonds are concerned, therefore, x% increase in yield does not cause x% reduction in value of a Bond. In fact as the interest rates rise, the reduction in value of the Bond is at a different rate than the rise in the interest rate. This is captured by the second derivative of the Price Yield called the convexity. (The first derivative being the Duration). Here is a video by @David Harper CFA FRM on Bond Convexity

As you can see, the degree of fall or rise in the value of the Bond depends on where you are on the yield curve in the first place

Now, the second derivative of the Price-Yield or the derivative of the Duration wrt the Yield is the convexity. Mortgages as in the case of all the Bonds display this convexity effect. As far as CMM is concerned, the product is structured in such a way so as to eliminate this convexity and have a linear relationship to avoid compounding of risks. An added advantage of CMM is that the prepayment is not modeled as well, causing the Mortgage not to have a prepayment curve, wherein, the effect of people prepaying their mortgages by replacing their existing mortgage with a lower interest mortgage as interest rates fall is also eliminated.

The above was just a primer to these things. More advanced practitioners over here can explain the same in more detail.

Thanks

 

[hellohi]

Dear @QuantMan2318

thanks a lot for that great help, because of your help, I know now great information about duration and convexuty, but please may you give explain about this sentence : "the product is structured in such a way so as to eliminate this convexity and have a linear relationship to avoid compounding of risks"

thanks
Nabil

 

[Dr. Jayanthi Sankaran]

Hi @hellohi,

I would like to take a shot in explaining as to what @QuantMan2318 meant by "the product is structured in such a way so as to eliminate this convexity and have a linear relationship to avoid compounding of risks"

However, before I do this, I would like to delineate the difference between mortgage bonds and other fixed income bonds with no embedded optionality. While both are subject to market risk, due to fluctuations in interest rates, credit risk, due to homeowner default, only mortgage bonds are subject to prepayment risk. This is the risk that the principal will be repaid early.

Let's use an example: 30 year fixed rate mortgage, Principal = $100,000, Annual yield = 8%, Monthly payment C = $734 (use pg 39 of the BA II Plus Professional Manual)

We now compute the weighted average life W of the mortgage where,
W8% = (nL - P)/aP
P = Principal amount
a = annual interest rate
n = term of mortgage (months)
L = monthly payment

W8% = [(360*$734) - $100,000]/(8%*$100,000) = 20.53 years

If interest rates drop to 6%, the homeowners will prepay early to refinance the mortgage, since it is cheaper to pay 6% than 8%. The weighted average life of the mortgage bond is now:

W6% = [(360*$600) - $100,000]/6%*$100,000 = 19.33 years

Because the average life of the mortgage bond has been shortened from 20.53 years to 19.33 years, this is called contraction risk.

If interest rates rise to 10%, the homeowner will be less likely to refinance early, and prepayments will slow down. As a result, the average life of the mortgage bond is extended - this is called the extension risk.

W10% = [(360*$877) - $100,000]/10%*$100,000 = 21.59 years

This creates a negative convexity at 8%, which reflects the short position in an American call option granted to the homeowner to repay early. This creates an extension risk when rates increase or contraction risk when rates decrease. At 10%, it is unlikely that the homeowner will refinance early. As a result, the option is worthless and the mortgage bond behaves like a regular fixed-income bond with positive convexity.

We now compute effective duration and effective convexity with the following example:

Initial yield y0 = 7.50%, Initial Price P0 = $100.125
Initial yield + 25bps, Price = $98.75
Initial yield - 25bps, Price = $101.50

Effective Duration DE = [P(y0 - dy) - P(y0 + dy)]/(2*P0*dy)]
=($101.50 - $98.75)/(2*$100.125*0.0025) = 5.49 years

Effective Convexity CE = [(P(y0 - dy) - P0)/(P0*dy) - (P0 - P(y0 + dy))/((P0*dy)]/dy
= [$(101.50 - $100.125)/($100.125*0.0025) - ($100.125 - $98.75)/($100.125*0.0025)]/0.0025
=0

Summarizing, in the range between yield = 7.25% - 7.75%, we have zero convexity. I would love to illustrate this with the PSA, but I don't want to complicate it further.

This is what @QuantMan2318 meant when he said that "the CMM is structured in such a way so as to eliminate this convexity and have a linear relationship to avoid compounding of risks". The market-price yield relationship of the mortgage bond is linear below 7.25%.

Hope that helps!

 

[hellohi]

thanks @Dr. Jayanthi Sankaran

it is really helpful in spite I did not get the answer of my question you pushed me to know the difference between positive and negative convexity and this is great point,,thanks thanks thanks....
I try to study your answer another time and tell u the result.
page 39 does not show the same.
how did u get C = 600 or 877?
I did not get the idea of zero convexity?

thanks, I would give all my respect
Nabil

 

[Mkaim]

thanks @Dr. Jayanthi Sankaran

it is really helpful in spite I did not get the answer of my question you pushed me to know the difference between positive and negative convexity and this is great point,,thanks thanks thanks....
I try to study your answer another time and tell u the result.
page 39 does not show the same.
how did u get C = 600 or 877?
I did not get the idea of zero convexity?

thanks, I would give all my respect
Nabil

Nabil--

You've run into some very kind folks above who've taken the time to go over some very critical information in great detail. Good to know you're thankful. Why don't you summarize what you've understood from the information above and share it so that folks can better understand where you're at and address your further questions accordingly. I think you should first try to fully understand what Dr. Sankaran took the time to explain and not "another time" as they're considered key concepts.

 

[hellohi]

Nabil--

You've run into some very kind folks above who've taken the time to go over some very critical information in great detail. Good to know you're thankful. Why don't you summarize what you've understood from the information above and share it so that folks can better understand where you're at and address your further questions accordingly. I think you should first try to fully understand what Dr. Sankaran took the time to explain and not "another time" as they're considered key concepts.

thanks dear @Mkaim for great professional advise

 

[brian.field]

I second what the Great @Mkaim implies (or I think he is implying) above.

 

[Mkaim]

I second what the Great @Mkaim implies (or I think he is implying) above.

You got it boss!

 

[David Harper CFA FRM]

@hellohi

This can be difficult without some quantitative foundation (in particular calculus), convexity is among the harder concepts. Below is the classic bond price-yield plot for a 30-year zero coupon bond, in blue. The green line plots a tangent. Dollar duration, is the slope of the (green) tangent line; in this case, at 4.0% the dollar duration is -$904 because the Δy/Δx (i.e., rise/run) = -$904.0/1.0 = -$904.0/100% = -$904.0/10,000 bps = $9.04/100 bps = $9.04/1.0%; i.e., at the tangency point, a 1.0% yield (Δy) increase corresponds to a -9.04 price drop.

But the price/yield plot (in blue) isn't straight, it's curved; curvature represents the convexity. This illustrated bond exhibits entirely the typical "positive convexity;" convexity is a function of the second derivative. If we start at 4.0%, and consider a 0.10% shock to the yield, positive convexity is demonstrated by the lack of symmetry that produces a bigger price increase (for a -10 bps down shock) than the price decrease for the same +10 bps up shock. Specifically, dropping yield from 4.0% to 3.9% increases the bond price by $0.90 but increasing the yield to 4.1% increases decreases the bond price by only $0.88. Put simply, positive convexity here is when the same 10 bps yield shock has greater impact when the yield goes down, than up. You can probably visualize this. (I'm getting price with 100*e^[-30*y]).

If you follow so far, then probably you can imagine that zero convexity is visualized simply by any segment of the P/Y plot that is exactly linear (straight)! That's because zero convexity is when the slope of the tangent is not changing (i.e., the second derivative is the rate of change of the first derivative). For effective convexity, I prefer:
C = [P(+Δy) + P(-Δy) - 2*P] / [Δy]^2 * 1/P

@Dr. Jayanthi Sankaran 's example is highly interesting to me because it shows mathematically how we get a zero convexity. We get C = 0 if the numerator is zero; i.e.,
If P(+Δy) + P(-Δy) - 2*P = 0, then
P(+Δy) + P(-Δy) = 2*P, and
[P(+Δy) + P(-Δy)]/2 = P; and this shows mathematically the visual point i was making above. That is, if the initial price is exactly equal to the average prices given by the same shock, Δy, then we have zero convexity. As Dr. Jayanthi's example also implies, realistically, the zero convexity happens at lower yields, where the the negative convexity (at even lower yields) is shifting over into the typical positive convexity (at higher yields). This is, in the context of the FRM, relatively advanced material so I would suggest you work on the fundamentals first, I hope that's helpful!

 

[Dr. Jayanthi Sankaran]

Hi David,

For some strange reason, I get the dollar duration to be 903.70 as against your $869 above - could be a decimal problem:
Using the Modified Duration equation:

-D = (1/P0)*dP/dy
P0 = $100*e^-30*y

At y = 4%, P4% = $100*e^-30*.04 = $30.1194
At y = 4.1%, P4.1% = $100*e^30*.041 = $29.2293
At y = 3.9%, P3.9% = $100*e^30*.039 = $31.0367

-D = (1/$30.1194)*($29.2293 - $31.0367)/0.002 = -30.0039
Dollar Duration = D*P4% = -30.1194*30.0039 = 903.70

Also, there seems to be a small typo "but increasing the yield to 4.1% increases (decreases) the bond price by only $0.88. Put simply, positive convexity here is when the same 10 bps yield shock has greater impact on when the yield goes up, than down (goes down, than up)

Thanks!

 

[David Harper CFA FRM]

Hi @Dr. Jayanthi Sankaran

• Re dollar duration. Yes, thank you! I must have flubbed the calculation of effective duration. Although it now occurs to me that it's not even necessary as for the continuously compounded zero-coupon bond, exact (analytical) dollar duration is conveniently given by T*P = 30*100*exp(-0.04*30) = $903.5826, which is appropriately near your effective dollar duration
• Re directional typos. Yes, thank you again, my bad. Shows how easy it is to get confused by this I fixed those above to avoid confusion. Thank you twice!