Transcript Analisi e Gestione del Rischio

Advanced Risk Management I
Lecture 2
Cash flow analysis and mapping
• Securities in a portfolio are collected and
analyzed one by one.
• Bonds are decomposed in their cash flows.
• Then, cash flows and other securities are
distributed in a limited set of reprsentative
exposures, for a more synthetic reporting of
risk.
Coupon bond cash flows
Let us define P(t,T;c) the price of a bond paying coupon c on a
schedule {t1, t2, …,tm=T}, with trepayment of capital in one sum at
maturity T. The cash flows of this bond can be replicated by a basket
of ZCB with nominal value equal to c corresponding to maturities ti
for i = 1, 2, …, m – 1 and a ZCB with a nominal value 1 + c iat
maturity T. The arbitrage operation consisting in the purchase/sale of
coupons of principal is called coupon stripping.
m
P(t , T ; c)   cv(t , tk )  v(t , tm )
k 1
Indexed (floating) coupons
• An indexed coupon is determined based on a
reference index, typically an interest rates,
observed at time , called the reset date.
• The typical case (known as natural time lag) is a
coupon with
– reference period from  to T
– reset date  and payment date T
– reference interest rate for determination of the coupon
i( ,T) (T –  ) = 1/v ( ,T) – 1
Replicating portfolio
• What is the replicating portfolio of an floating
coupon, indexed to a linear compounded interest
rate for one unit of nominal?
• Notice that at the reset date  the value of the
coupon, determined at time  and paid at time T,
will be given by
v ( ,T) i( ,T) (T –  ) = 1 – v ( ,T)
• The replicating portfolio is then given by
– A long position (investment) of one unit of nominal
available at time 
– A short position (financing) for one unit of nominal
available at time T
Cash flows of a floating coupon
• Notice that a floating coupon on a nominal
amount C corresponds to a position of debt
(leverage)
C
t

T
C
No arbitrage price:
indexed coupons
• The replicating portfolio enables to evaluate the coupon at time t
as:
indexed coupons = v(t,) – v(t,T)
At time  we know that the value of the position is:
1 – v(,T) = v(,T) [1/ v(,T) – 1]
= v(,T) i(,T)(T – )
= discount factor X indexed coupon
• At time t the coupon value can be written
v(t,) – v(t,T) = v(t,T)[v(t,) / v(t,T) – 1]
= v(t,T) f(t,,T)(T – )
= discount factor X forward rate
Indexed cash flows
• Let us consider the time schedule
t,t1,t2,…tm
where ti, i = 1,2,…,m – 1 are coupon reset times,
and each of them is paid at ti+1.
t is the valuation date.
• It is easy to verify that the value the series of
flows corresponds to
– A long position (investment) for one unit of nominal at
the reset date of the first coupon (t1)
– A short position (financing) for one unit of nominal at
the payment date of the last coupon (tm)
Floater
• A floater is a bond characterized by a schedule
t,t1,t2,…tm
– at t1 the current coupon c is paid (value cv(t,t1))
– ti, i = 1,2,…,m – 1 are the reset dates of the floating coupons
are paid at time ti+1 (value v(t,t1) – v(t,tm))
– principal is repaid in one sum tm.
• Value of coupons: cv(t,t1) + v(t,t1) – v(t,tm)
• Value of principal: v(t,tm)
• Value of the bond
Value of bond = Value of Coupons + Value of Principal
= [cv(t,t1) + v(t,t1) – v(t,tm)] + v(t,tm)
=(1 + c) v(t,t1)
• A floater is financially equivalent to a short term note.
From portfolios to exposures
• The securities collected and evaluated are
transformed into exposures to risk factors.
• The process to transform cash flows into
securities is called mapping.
• Aim of the mapping is to give a synthetic,
but informative, representation of the risks
to which a portfolio is exposed, and to
provide a guideline for risk management.
Fixed income exposures
• Fixed income exposure mapping requires the
definition of a set of reference maturities on which
to collect the cash flows. These reference
maturities are called bucket.
• Each and every cash flow of the replicating
portfolio of a bond that does not coincide exactly
with a bucket is split in two flows on the closest
buckets.
• The splitting is designed in such a way as to
preserve the financial features of the cash flow, as
closely as possible.
The art of reporting
How many bucket?
• The number of buckets should reflect the number of
risk factors characterizing the yield curve.
– A number of buckets too small can lead to ignore
important movements of the yield curve
– Too many buckets may induce too much “noise” in
the movements of the yield curve
• In most markets, three or four factors are sufficient to
represent the a huge percentage of the movements of
the yield curve, but in reality ten or twelve buckets are
used.
Why more buckets?
• Typically, more buckets than needed are
used for two reasons
• A limited number of buckets increase the
distance, in financial terms, between the
product and its mapped version.
• A less coarse representation in terms of
buckets provides a better guideline for
hedging strategies.
Cash flow-mapping
• Cash-flow mapping is based on the
following requirements
– The sign of the positions is preserved
– The value of the positions is preserved
– The risk of the positions is preserved
Cash flow mapping: two options
• Fisher & Weil: cash-flows are split
preserving i) sign ; ii) market to market
value; iii) duration
• Opzione RiskMetrics™ : cash-flows are
split preserving i) sign ; ii) market to market
value; iii) volatility
Fisher Weil option
• Let c be a nominal flow (positive or negative)
corrisponding to maturity . The cash-flow must be
decomposed in two flows of the same sign ci-1 and ci
on the closes maturities ti-1 and ti. The solution
requires that
ci-1v(t, ti-1) + civ(t, ti) = c v(t, )
(ti-1 - t)ci-1v(t, ti-1) + (ti - t) civ(t, ti) = ( - t) c v(t, )
• We used Macaulay duration, but we could do the
same with modified duration.
Fisher e Weil: solution
• Solving the system we get
ti 

t i  t i 1
ci 1v t , t i 1   c v t ,  ci v t , t i   1   c v t , 
…and the value of the item is split in proportion
to the ratio of duration differences.
RiskMetrics™ options
• Define: i-1,  and i the interest rate volatility of
maturities , ti-1 and ti respectively and i,i-1 their
correlation.
• We want back out , the share of value cv(t,)
that will be allocated to vertex ti-1
v(t, ti-1)ci-1 = cv(t,)
v(t, ti)ci = (1- ) cv(t,)
• The sign and value conditions are then verified for
0    1 and the duration condition is substituted
by
2Di-12i-12 + (1 - )2 Di2i2 +
+ 2 (1 - )  Di-1Dii,i-1 ii-1= D2 2
RiskMetrics™: solution
• The  value is obtained by solving the
second degree equation
a 2 + b  + c = 0 with
a = Di-12i-12 + Di2i2 - i,i-1 Di-1Di ii-1
b = 2(i,i-1 Di-1Di ii-1 - Di2i2)
c = Di2i2 – D2 2
• We choose the solution between 0 and 1,
assuming it exists!
Cash flow mapping: pros and cons
• Opzione Fisher & Weil: pros
– Independent of volatility estimation
– It is possible to estimate the degree of
approximation
• Opzione RiskMetrics™: pros
– If the volatility estimate is accurate, it gives a
better approximation.
– Consistent with many changes of the term
structure
A particular case
• Assume the volatility to be constant for all buckets and
perfect correlation
i) i-1=  = i = 
ii) i,i-1 = 1
• The RiskMetrics™ condition is
2Di-122 + (1 - )2 Di22 + 2 (1 - )  Di-1Di 2 = D2 2
from which
[ Di-1 + (1 - ) Di ]2 = D2
and the condition corresponds to Fisher and Weil.
30
25
20
15
10
5
0
1m
3m
6m
12m
2y
3y
4y
5y
7y
9y
10y
Fisher & Weil
Risk Metrics
15y
20y
30y
New RiskMetrics options
• RiskMetrics launched another proposal based
on linear inerpolation of interest rates
r(t,) =  r(t, ti-1) + (1 – ) r(t, ti)
• Mapping uses three funds instead of two: the
closest buckets and the cash bucket
• The mapping is built in such a way that the
sensitivity of the value with respect to the rate
of each bucket be the same as the sensitivity of
the mapped cash flow.
• The idea is to allow for the two rates to move
independently of each other.
The three equations
c v t ,   ci 1v t , t i 1   ci v t , t i   cash
c v t ,  c v t ,  r t , 

    t c v t ,  
r t , t i 1 
r t ,  r t , t i 1 
c i 1v t , t i 1 

 t i 1  t ci 1v t , t i 1 
r t , t i 1 
c v t ,  c v t ,  r t , 

 1     t c v t ,  
r t , t i 
r t ,  r t , t i 
c i v t , t i 

 t i  t c i 1v t , t i 1 
r t , t i 
The solutions
ti 

t i  t i 1
c i 1v t , t i 1  
c i v t , t i  
  t.
t i 1  t
  t.
ti  t
c v t , 
1   c v t , 

  t i 1 t i   
c v t , 
cash  
t i  t t i 1  t 
Reporting for equity risk
• The positions in the equity markets are
reported in appropriated buckets.
• Reporting choices:
– Market exposure = value of the position
– Market exposure = systematic risk
– Stock exposure = overall risk
Reporting for FX exposures
• For exposures denominated in foreign currency j,
with exchange rates ej with respect to the reference
currency, the exposure to exchange rate risk is
added to the market risk of the product.
• The percentage variation of every exposure i, from
the point of view of the reference currency, can be
decomposed as ri = r*i + re where r*i and re are
respectively the percentage change of the position
in domestic currency and that of the exchange
rate.
Exchange rate exposure
• From the decomposition of the percentage change
of value, the variance of the position is
i2 = *i2 + e2+2ie*ie
• Every exposure denominated in foreign currrency
then implies an exposure to exchange risk for the
same mark to market amount.
• Notice that in the extreme case ie = 1
i = *i + e