Homework Assignment #1:Answer Key

Econ 497
Economics of the Financial Crisis
Professor Ickes
Spring 2012
Homework Assignment #1:Answer Key
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1. Consider the constant-growth dividend discount model, where = ,where − and
are prices and dividends in period and istherateofreturnonequity, is the growth rate
of dividends, and .Then
= 1+ . Provide an economic explanation for the effect of a
−
fall in on the price-dividend ratio.
brief answer If falls
rises. Effectively future dividends are discounted at a lower rate, so
they are worth more. Since equity is risky the rate of return on equity is equal to the risk
free rate plus a risk premium (that in the CAPM model depends on the covariance of the
return with the market return), so this could also be related to a fall in the risk premium.
If stocks are less risky that too could cause their prices to rise relative to earnings.
(a) If we observe a rise in the ratio of
relative to its historic average, how might this be
explained within the context of the model?
brief answer Arisein could occur because earnings are expected to grow faster. Faller
interest rates or a fall in the risk premium could cause to fall.
(b) How do these explanations relate to the ”this time is different” thesis? Explain.
brief answer ”Thistimeisdifferent” is used to explain why a high is not indicative
of a bubble. For example, during the dotcom period prices were very high relative to
earnings the story was earnings would eventually grow faster because the internet will
change everything. Or financial developments have so reduced risk that the return
to equity is lower so a higher price-earnings ratio is justified. Usually it is some
fantastic story about how earnings will grow faster.
(c) Can you relate your explanation to any historic examples of bubbles?
brief answer see above
2. Suppose we create an asset backed security (ABS) with five mortgages. These mortgages
either pay off or default, and the probability of a default is 1. Defaults are independent
across the mortgages. Now suppose that we create five tranches (senior1, senior2, senior3,
mezzanine, andequity). The senior1 tranche defaults only if all five mortgages default, and
the equity tranche defaults if any mortgage defaults.
(a) Calculate the probability of default for each of the five tranches. How does the likelihood
of a tranche defaulting compare with the risk of the underlying mortgages? [Note that
you need to calculate the probability that, say, any two (or three, or four, etc.) of five
mortgages default. This requires use of the binomial distribution, and you could use
Excel or a similar program to aid your computation.]. What does this say about the risk
of senior tranches?
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ief answer The senior tranche defaults only if all five mortgages default, the senior2
if there are four defaults, and so forth. So we use the cumulative binomial
distribution to calculate the ways that four mortgages default out of five experiments,
then how three can default, etc. For example, the senior1 defaults only if
all five default, so calculate the cumulative probability of only four defaults. Call this
BinomialDist(4; 51) Then 1 − BinomialDist(4; 51) gives the probability of five defaults.
We do this for each tranche. We obtain:
tranche probability of default
senior1
00001
senior2
00046
.
senior3
00856
mezzanine 08146
equity
40951
The senior tranche is obviously very secure. The probability of default is practically
non-existent. The same is true for the senior2 and senior 3, the probability of
default is still very low.
(b) Suppose we now form a new security made up of mezzanine tranches. That is, we combine
five securities with the same probability of default you calculated for the mezzanine
tranche in part a. Call this a Again tranche this new security into five parts with
the same pattern of seniority. Calculate the probability of default of the various tranches
of the
brief answer The probability of default of the mezzanine tranche is 08 So just imagine
we have five bonds with the default probability of this mezzanine trance and form a
new security. It is evident that the default probabilities should fall, since 08 1, that
is the default probability of the mezzanine tranche is less than that of the underlying
mortgage. So with uncorrelated risks, the CDO should be even less risky. That is
what we indeed see. If we tranche it just as in part (a), and perform the same calculation
we obtain:
Default Probability of the CDO of Mezzanine Tranches
tranche probability of default
senior1
000004
senior2
000205
senior3
004756
mezzanine .056117
equity
345918
We note that the senior tranches still show very low default rates.
(c) Suppose that the probability of default of the underlying mortgages is really 15. How
does this change the probability of the default of the tranches of the ? Howmuch
riskier (say, in percentage terms) does the mezzanine tranche of the get given this
50% increase in the default probability?
brief answer With a higher underlying default probability the default probabilities of
the tranches rise, and the lower tranches somewhat significantly. The default probability
of the mezzanine rises from 08146 to 1648. Sousing1648 in our CDO we get:
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Default Probability of the CDO of Mezzanine Tranches with .1648 probability of default
tranche probability of default
senior1
00012
senior2
00314
senior3
03397
mezzanine .19111
equity
59165
We see that the tranches of the CDO are now riskier that before, essentially by
an order of magnitude. For example, the senior3 tranche had a default probability of
.0047, now it has a default probability of .03397, almost ten times more likely. So the
CDO and CDO 2 are more sensitive to changes in the underlying default probability
than the original ABS is.
(d) What if there were 100 mortgages, 10 to a tranche, and the probability of default of
the underlying loans is 05. Consider tranche10, which defaults if 10 or more mortgages
default. What is the default probability of that tranche? What of the made up of
tranche10 securities? What happens if the underlying probability of default rises to 06?
brief answer The analysis is the same as before. We need to calculate the likelihood
that ten mortgages default out of 100 possibilities. So again use the binomial distribution.
We calculate 1 − BinomialDist(9; 10005) = 002818 8. Now if we form
a CDO made up to tranche10 securities we expect that these will have low default
probabilities, since the default probability of tranche10 is less than that of the underlying
mortgages (028 1). Andindeed,1 − BinomialDist(9; 10002818) = 00054.
But now here is the really interesting part. Suppose that the probability of default
was just a little higher, 06. This is a 20% increase in the risk of the underlying
mortgages. What happens to the default probability of these tranches? We calculate
1 − BinomialDist(9; 10006) = 0775. Notice that the percentage increase in risk of
this tranche is huge: 0775−0281 =%175. For the CDO it is even bigger. We calculate
0281
1 − BinomialDist(9; 1000775) = 02467 which is represents a 2467−00054 = 45339%
00054
increase in risk. That is quite a significant change, so we can conclude that the CDO
tranches are indeed quite sensitive to the underlying default probabilities.
3. Consider the limits to arbitrage model with three types of agents (noise, investors, and arbitragers)
and three periods. In period 3 the value of the asset is , and arbitragers know this
(so 3 = ). Noise traders receive a signal in periods 1 and 2 ( 1 2 ) that may cause them
to be pessimistic. Investors supply funds to arbitragers, and they base their decisions on past
performance. Let the demand for the asset on the part of noise traders be () = −
at
= 1 abitragers know 1 ,butnot 2 . In particular, there is some chance that 2 1 The
supply of the asset is one for simplicity. Let be the resources available to arbitragers in .
(a) Given that markets clear show that the 2 depends positively on and 2 and negatively
on 2 .Whatdoes 1 depend on? Explain.
brief answer Demand in period 2 from noise traders is − 2
2
, arbitragers want to purchase
as much as they can in period 2, 2
2
, and supply is unity, so 1= − 2
2
+ 2
2
and thus 2 = − 2 + 2 .Asfor 1 this depends on noise trader demand in period
one, − 1
1
and how much arbitragers will demand. But this is not necessarily equal
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to 1 , because arbitragers may wait to save some of their gunpowder till period two.
Hence, 1= − 1
1
+ 1
1
,whichimplies 1 = − 1 + 1 .
(b) If arbitragers had access to unlimited funds what will 2 equal?
brief answer If 2 −→ ∞ then arbitragers will offset all noise trader risk and we have
1 = 2 =
(c) Suppose that the supply function of funds to investors is given by
" µ1 2
2 = 1 + #
1 − 1
(1)
1 1 1
(1) = 1 0 ≥ 1 00 ≤ 0
where 1 is the amount invested by arbitragers in period 1, and 0 measures the sensitivity
of 2 to the term in the brackets. If 2 = 1 what happens to 2 ? What happens
if 2 1 ? Explain why this could be important.
brief answer If 2 = 1 then we have 2 = 1 h³ ´ i
1
1
+
1 − 1
1
=⇒ 2 = 1 ,that
is if there is no change in price then funds available are the same. If prices fall,
however, then clearly 2 1 . How much this falls depends on the size of 0 ,the
more sensitive is supply to recent returns the greater the fall. This could be important
because it makes traders reluctant to fully commit to arbitrage possibilities. A shortterm
fluctuation could wipe them out. This makes noise trader sentiment more
important. If fund supply is very sensitive then arbitragers may even have to sell out
in period two, which would move prices even farther from V.
(d) What does 0 0 imply about the nature of arbitrage? Explain. If 0 is larger are
arbitragers more likely to succeed or more likely to fail in offsetting noise traders signals
in period 2?
brief answer When we have PBA then agency intrudes into the arbitrage process. This
means that arbitragers must take into account the possibility that they will lose funds
at the most opportune times for investment. This makes them less willing to commit
their funds and makes it less likely that noise traders are offset.
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