Question:

[source: 2009 sample exam II]  Your firm’s fixed‐income portfolio has interest‐only CMOs (IO),  callable corporate bonds,  inverse floaters,  noncallable corporate bonds.  Your boss wants to know which of the following securities can lose value as yields decline.

a.  callable corporate only
b.  inverse floater only
c.  IO and callable corporate bond
d.  IO and noncallable corporate bond

[my adds]

36b. What is unique about the duration of an interest only (IO) strip?
36c. What is unique about the duration of an inverse floater?
36d. How does the price of the callable corporate bond compare to the price of a noncallable corporate bond, for both low and high yields (assume otherwise identical features)?
36e. How and where (in the yield curve) does the call option induce negative convexity?
36f. Does negative convexity imply negative duration?

Answer:

CORRECT:  C

The IO decreases in value because a decline in rates implies an increase in mortgage prepayments, which decreases the notional principal upon which the IO pays its interest

INCORRECT:  A -  While the price of a callable corporate may decline as the call goes in the money,  the IO also decreases in value
INCORRECT:  B -  The IO decreases in value,  but so can the callable corporate
INCORRECT:  D -  The noncallable corporate bond increases in value as yields decline

Reference:  Tuckman,  Chapter 21

I disagree with this answer. I agree that an IO has negative duration but a callable bond exhibits negative convexity at low yields. As noted below, negative convexity does not imply negative duration. If someone can correct me, please give help? - David

36b. What is unique about the duration of an interest only (IO) strip?
It exhibit negative duration unlike most fixed income instruments.

36c. What is unique about the duration of an inverse floater?
Unlike most instruments, floaters tend to have a duration that is greater than their maturity!

36d. How does the price of the callable corporate bond compare to the price of a noncallable corporate bond, for both low and high yields (assume otherwise identical features)?

price of a non-callable = price of callable + value of embedded option, or
price callable = price non-callable - price of call option

At high yields, the bond price is low and the value of the call option is low: the price of the callable is nearer to the price of the non-callable
At low yields, bond price is high and the value of the call option is low: the price of the callable is significantly lower than the price of the non-callable
In all cases, the price of the callable < price non-callable where the difference equals the value of the call option

36e. How and where (in the yield curve) does the call option induce negative convexity?
At low yields, because the issuer will call the bond and this limits the upside price potential of the bond. Negative convexity exhibits at low yields for the callable bond
(per Tuckman, the prepayment option on the mortgage is the MBS equivalent to a call option)

36f. Does negative convexity imply negative duration?
No! That’s why the sample answer is incorrect.
Negative convexity implies the dollar duration is decelerating instead of accelerating
In a noncallable bond, as the yield decreases the dollar duration (the first derivative) is dropping (i.e., convexity or upward concavity); e.g., 4% to 3% giving dollar duration of, respectively, -600 and -700 would not be unusual.
In a callable bond, the negative convexity means a yield decrease is associated with an increase in dollar duration. However, the dollar duration would still generally be negative! For example, 4% with -600 dollar duration changing to 3% and -500 implies negative convexity (i.e., the first derivative is decelerating).

Or, to put intuitively, as yields drop, the value of the call option is increasing and the duration of the bond is decreasing. This is effectively “capping” the price reaction to further yield drops, but I don’t see where and why the price/yield curve would “bend back” such that the dollar duration (slope of the tangent) goes from negative to positive…

Callable bond [practice, market]
Hi David,
From this your graph above, let us replace non prepayable mortgage with non callable corporate bond, and pass through with wit callable corporate bond: (F. Fobozzi)  has a similar graph in Fixed income analysis for CFA program (level 2)

As the yield declines, the option embedded in the callable corporate bond becomes more and more in- the- money (ITM). At it becomes more and more obvious that   bond will be called, its price appreciation will be capped at a certain point after which it will start to exhibit negative convexity (will not appreciate as relatively as non callable bond)

Given this, it will be to see   from the options given by GARP on this question we can see that   it has to be (a) (CMO IO since we all know that when interest rate (yield) decline, homeowners will refinance/ prepay etc,( with the effect of reducing the interest cost or the principal outstanding)  ( b) callable corporate bond.
Callable bond in the corporate world is equivalent to the prepayable pass through (CMO, IO)  in the mortgage world because each has a an embedded option that the corporation /issuer/ homeowner will take advantage of

That is why I think   that C is correct.

Thanks,

Concepts

Hi Concepts,

I agree with your contention that the callable is like a prepayable mortgage, and that as yield is lower, its price appreciation must be capped. But please read my question 36f above. I am not certain, but I still don’t agree with the anwswer: if the embedded option caps the price at low yields, I understand that creates negative convexity, but this is not the same as negative duration. Negative convexity merely implies that as yield decrease, as the option increases in value, the negative duration has “reversed path” and is increasing instead of decreasing. But the question asks which instrument “can lose value as yields decline” and this requires *positive duration;*  a negative duration that is increasing is insufficient. (Mathwise, my point is: the second derivative changing signs doesn’t imply an immediate sign change for first derivative). Graphically, as yield decrease, the negative convexity implies the slope of tangent line has shifted and the price/yield is “arching back” but to exhibit postive duration, the price/yield curve must actually turn negative (i..e, lower yields—> lower price) and I don’t see why the embedded option can create this dynamic (my hunch is the question writer has mistakenly assumed that negative convexity implies negative duration)? ... i could be wrong…i just don’t get the counter-argument…

...or just to ask intuitively, we agree that the embedded option implies: progressively lower yields, due to capped price, limit the price appreciation of the bond (negative convexity). But why should we go further and say: the embedded option implies that, at some low yield, further yield declines will cause the bond price to decrease (not stay flat, that’s not enough. Lower yield has to actually decrease the price)?

David

Hi David,

I saw this question on FRM hand book by Jorion (CD ROM)

  Which of the following mortgage backed securities has a negative duration?
A)  Interest-Only strips (IO).
B)  Inverse floater.
C)  Mortgage pass-through.
D)  Principal-Only strips (PO)

Answer: A If interest rate falls, IO strips will decrease in value, the other 3 securities will increase in value This is due to increased mortgage prepayment s will cause the   outstanding principal to shrink that means decrease in the increase payments
Ref.  Fixed Income Securities: Bruce Tuckman Chapter 21 MBS page 475

The answer that Jorion gave here is similar to the one given above that we don’t agree on.

  My submission is:  It is either Jorion was the one who wrote GARP 2009 sample exams or ...

Thanks,

Concepts

hi concepts,

but this last question contradicts the earlier and agrees with me: it suggest C (mortgage pass through; clearly a mortgage pass through refers to a prepayable) has positive duration. I agree with the last question you just submitted (i.e., only the IO has negative duration) and I disagree with the sample 2009 question (i.e., the callable bond also has negative duration)  .. in any case, to be honest, I only partially trust the GARP questions (fyi, Jorion did not write the questions, if he had, they would not have so many errors. I submitted about 38 errors last year. GARP’s reply: we agree.). In the meantime, i’ll await logic or a concrete example (I checked my fabozzi sources and could only find postive duration for the callables) ... David