Transcript Assicurazioni vita e mercato del risparmio gestito

Financial Products & Markets
Lecture 1
Markets and financial products
• Financial products allow to tranfer consumption from the
current period to the future
• In this course we show the tools that can be used (and how
they can be used) to achieve that goal
• One can decide to carry consumption to the future by
simply investing in the risk-free rate and then collecting
the amount of consumption at expiration.
• One can decide to carry consumption to the future by
taking some risk in exchange for a higher consumption.
• One can decide to consume in the future, contingent on the
state of nature. Consume more if some particular event
takes place.
Financial products
• Fixed income, i.e. bonds. The pay-off is not contingent on
the project that is funded by the loan, apart for the case in
which the project goes bust (credit risk).
• Variable income, i.e. equity. The pay-off is defined from
the cash flow of the project, as the residual after all other
claims have been satisfied (residual claim)
• Derivatives or contingent claims. Products whose pay-off
depends on the state of nature (but also prioduts that
include debt, that is leverage).
• Structure finance products. Include both standard equity or
bond and a derivative contract.
Risk transfer tools
• Derivatives are used to transfer risk from one
institution to the other, without any unwinding of
the portfolio.
• Risk transfer can be designed both for single
positions and for portfolios.
• Risk transfer can be achieved either using
regulated markets or by re-insurance, that is on the
so-called Over-the-Counter (OTC) market
(Metalgesellschaft case).
Financial actors and strategies
• Non financial firms may raise funding by
–
–
–
–
Issuing equity or relying on debt
Debt may be through loans or bonds
Equity and bonds may be listed or not in a market
Bonds may be designed for the retail market or
institutional investors
– New sources: P2P loans, micro-credit, crowdfunding
• Investors may provide funds by:
– Individual investment on the market
– Collective investment in funds and other intermediaries
Financial intermediaries
• Banking system: regulated, issue deposits and bonds and
provide loans, or invest in the market. They also provide
services to manage risk by designing derivative deals for
clients (dicussion about the dual capacity, Volcker-VickersLiitkanen rule)
• Insurance market: invest in bonds and stock to provide
long term consumption (life insurance) and provide reinsurance on risk to the financial system (cases: CDS in
AIG and the surety bonds in the Enron case).
• Shadow banking system: hedge funds, SIV, SPV and in
general all un-regulated intermediaries. Shadow banking
system is good or bad? Is it here to stay, or it is a residual
from the crisis
Regulators
• ECB (and NCBs) supervise the major banks
(about 130) of the Euro Area, with the task of
ensuring financial stability
• EBA (for banks) and EIOPA (for insurance) are
the European agencies for enforcement of the
regulation rules
• Transparency regulation (ESMA and local
regulators, such as CONSOB, Basfin, and so on)
• Accounting regulators (IASB): define the rules for
the transparent evaluation of assets and liabilities.
Markets
• Organized markets: open-outcry vs screen based, quotedriven vs order-driven, anonymous vs non-anonymous.
• OTC markets: collateral-based or with counter party risk
• Microstructure theory studies organization of exchanges
and the effectiveness in processing the information flows
(price discovery)
• Exchange venues and markets are places where agents
exchange information. Information is then backed out from
liquid and transparent markets in order to evaluate
products not traded in the market.
Brand new problems
• Negative interest rates: carrying consumption to the future
now is costly, and this increases the value of 1 euro
tomorrow above the value of it today
• Insurance companies getting negative returns must take
more risk to be able to face future payments (case of
German insurance companies in the stress test)
• Negative interest rates, due to loss aversion (prospect
theory), push investors towards taking more risk.
• QE blues: under the Quantitative Easing structure negative
yields are excluded from the assets that can be bought from
ECB, so leaving only long term maturities as possible
“ammunition” for the QE weapon.
A general template for
financial products
• Time schedule: {t0, t1, …,tn}
– Market holidays and day-count and calendar conventions
(following, preceding, mod following or preceding)
– Day-count conventions
• Coupon/dividend plans: {c0, c1, …,cn}
– Deterministic
– Indexed (interest rate, inflation, equity, credit, commodities,
longevity)
• Repayment plan {k0, k1, …,kn}
– Deterministic
– Stochastic (callable, putable, exchangeable, convertible)
The arbitrage principle
• No arbitrage, or no free lunch means
• Build a position whose value is zero at time
t and in the future took non-negative value
for sure, with possibly a positive value in
some of them
• Build a position whose value be negative at
time t and in future get non-negative value
for sure.
Arbitrage and replicating portfolio
• The replicating portfolio or a replicating strategy of a
financial product is a set of positions whose value at
some future time is equal to that of the financial product
in all possible states of nature
• If it is possible to build a replicating portfolio of a
financial product for a price different from that of the
financial product, one can exploit infinite profits selling
the portfolio (if it is cheaper than the financial product
to be replicated) and buying the financial product,
followed by repurchase of the portfolio and sale of the
bond when they have the same value.
Replicating portfolio:
pricing and hedging
• Assuming that no arbitrage profits are
possible means requiring that the value of
each financial product be equal to that of its
replicating portfolio (pricing)
• Buying a financial product and selling the
corresponding replicating portfolio means
building an immunized position or hedging
the position.
Structured finance
building block approach
• Structured finance products are built by aggregating
financial products and derivatives.
• Duty of the structurer is to put these products
together so that they could be useful to a set of
customers
• Duty of the pricer is to disentangle the financial
product in the elementary components in order to
evaluate them on prices consistent with the market.
• Duty of the risk manager is to evaluate the risk of the
different components and to decide what and how to
hedge.
Zero-coupon-bond
• Define P(t,tk,xk) the value at time t of a zero-coupon bond
(ZCB). It is a security that does not pay coupons before
maturity and that gives right to receive a quantity xk at a
futurre date tk
• Define v(t,tk) the discount funtion, that is the value at time
t of a unit of cash available in tk
• Assuming infinite divisibility of each bond, down to the
bond paying one unit at maturity, we obtain that
P(t,tk,xk) = xk v(t,tk)
Coupon bond evaluation
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
Bond prices and discount factors
• Based on zero-coupon bond prices and the prices
of coupon bonds observed on the market it is
possible to retrieve the discount function.
• The technique to retrieve the discount factor is
based on the no-arbitrage principle and is called
bootstrapping
• The discount function establishes a financial
equivalence relationship between a unit amount
of cash available at a future date tk and an amount
v(t,tk) available in t.
• Notice that the equivalence holds for each issuer.
Bootstrapping procedure
Assume that at time t the market is structured on m periods
with maturities tk = t + k, k=1....m, and assume to observe
zero-coupon-bond P(t,tk) prices or coupon bond prices
P(t,tk;ck). The bootstrapping procedure enables to recover
discount factors of each maturity from the previous ones.
k 1
vt , t k  
Pt , t k ; ck   ck  vt , ti 
i 1
1  ck
The term structure of interest rates
The term structure is a way to represent the discount function.
It may be represented in terms of discrete compounding
1
v(t , t k ) 
t k  t 
1  i(t , tk )
i (t , t k )  v(t , t k )
1 / t k t 
1
The term structure of interest rates
The term structure is a way to represent the discount function.
It may be represented in terms of continuous compounding
v(t , t k )  exp i t , t k t k  t 
lnv(t , t k )
i (t , t k )  
tk  t
The term structure of interest rates
The term structure is a way to represent the discount function.
It may be represented in terms of discrete compounding
1
v (t , t k ) 
1  t k  t i (t , t k )
1
i (t , t k ) 
tk  t
 1

 1

 v (t , t k ) 
Term (forward) contracts
• A forward contract is the exchange of an amount
v(t,,T) fixed at time t and paid at time  ≥ t in
exchange for one unit of cash available at T.
• A spot contract is a specific instance in which
 = t, so that v(t,,T) = v(t,T).
• v(t,,T) is defined as the (forward price)
established in t of an investment starting at  ≥ t
and giving back a unit of cash in T.
Spot and forward prices
•
•
•
Consider the following strategies
1. Buy a nominal amount v(t,,T) availlable at  on the
spot market and buy a forward contract for settlement
at time , giving a unit of cash available on T
2. Issue debt on the spot market for repayment of a unit
of cash at time T.
It is easy to see that this strategy yields a zero
pay-off at time both at time  and at time T.
If the value of the strategy at time t is different
from zero, there exists an arbitrage opportunity
for one of the two parties.
Arbitrage example
– v(t,) v(t,,T)
v(t,,T)
–
–
– v(t,,T)
1
v(t, T)
–
–1
Total
v(t, T) – v(t,) v(t,,T)
0
0
Spot and forward prices
• Spot and forward prices are then linked by a
relationship that rules out the arbitrage opportunity
described above
v(t,T)=v(t,) v(t,,T)
• All the information on forward contracts is then
completely contained in the spot discount factor
curve.
• Caveat. This is textbook paradigm that is under
question today. Can you guess why?
The forward term structure
Forward term structure is a way of representing the forward
discount function.
It may be represented with discrete compounding.
f (t , , T )  v(t , , T )
 v(t , ) 


 v(t , T ) 
1 / T  
1
1 / T  
1
The forward term structure
Forward term structure is a way of representing the forward
discount function.
It may be represented with continuous compounding.
lnv(t , , T )
f (t , , T )  
T 
lnv(t , )  lnv(t , T )

T 
i (t , T )(T  t )  i (t , )(  t )

T 
The forward term structure
Forward term structure is a way of representing the forward
discount function.
It may be represented with linear compounding.
1
f (t , , T ) 
T 
1

T 


1
 v(t , , T )  1


 v(t , ) 
 v(t , T )  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 coupons: some caveat
• It is wrong to state that expected future coupons
are represented by forward rates, or that forward
rates are unbiased forecasts of future forward rates
• The evaluation of expected coupons by forward
rates is NOT linked to any future scenario of
interest rates, but only to the current interest rate
curve.
• The forward term structure changes with the spot
term structure, and so both expected coupons and
the discount factor change at the same time (in
opposite directions)
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.
Forward rate agreement (FRA)
• A FRA is the exchange, decided in t, between a floating
coupon and a fixed rate coupon k, for an investment period
from  to T.
• Assuming that coupons are determined at time , and set
equal to interest rate i(,T), and paid, at time T,
FRA(t) = v(t,) – v(t,T) – v(t,T)k
= v(t,T) [v(t,)/ v(t,T) –1 – k]
= v(t,T) [f(t,,T) – k]
• At origination we have FRA(0) = 0, giving k = f(t,,T)
• Notice that market practice is that payment occurs at time 
(in arrears) instead of T (in advance)
Natural lag
• In this analysis we have assumed (natural lag)
– Coupon reset at the beginning of the coupon period
– Payment of the coupon at the end of the period
– Indexation rate is referred to a tenor of the same length
as the coupon period (example, semiannual coupon
indexed to six-month rate)
• A more general representation
Expected coupon = forward rate
+ convexity adjustment + timing adjustment
• It may be proved that only in the “ natural lag” case
convexity adjustment + timing adjustment = 0
Esercise
Reverse floater
• A reverse floater is characterized by a time
schedule
t,t1,t2,…tj, …tm
– From a reset date tj coupons are determined on the
formula
rMax –  i(ti,ti+1)
where  is a leverage parameter.
– Principal is repaid in a single sum at maturity
Swap contracts
• The standard tool for transferring risk is the swap
contract: two parties exchange cash flows in a
contract
• Each one of the two flows is called leg
• Examples of swap
– Fixed-floating plus spread (plain vanilla swap)
– Cash-flows in different currencies (currency swap)
– Floating cash flows indexed to yields of different
countries (quanto swap)
– Asset swap, total return swap, credit default swap…
Swap: parameters to be determined
• The value of a swap contract can be expressed as:
– Net-present-value (NPV); the difference between the
present value of flows
– Fixed rate coupon (swap rate): the value of fixed rate
payment such that the fixed leg be equal to the floating
leg
– Spread: the value of a periodic fixed payment that
added to to a flow of floating payments equals the fixed
leg of the contract.
Plain vanilla swap (fixed-floating)
• In a fixed-floating swap
– the long party pays a flow of
fixed sums equal to a
percentage c, defined on a year
basis
– the short party pays a flow of
floating payments indexed to a
market rate
• Value of fixed leg:
m
c ti  ti 1 vt , ti 
i 1
• Value of floating leg:
m
1  vt , tm    vt , ti ti  ti 1  f t , ti 1 , ti 
i 1
Swap rate
• In a fixed-floating swap at origin
Value fixed leg = Value floating leg
m
swap rate ti  ti 1 vt , ti   1  vt , t m 
i 1
swap rate
1  vt , t m 
m
 t
i 1
i
 ti 1 vt , ti 
Swap rate
• Representing a floating cash flow in terms of forward
rates, a swap rate can be seen as a weghted average of
forward rates
m
m
i 1
i 1
swap rat e ti  ti 1 vt , ti    vt , ti ti  ti 1  f t , ti 1 , ti 
m
swap rat e
 vt , t t
i
i 1
m
 t
i 1
i
i
 ti 1  f t , ti 1 , ti 
 ti 1 vt , ti 
Swap rate
• If we assume ot add the repayment of principal to both
legs we have that swap rate is the so called par yield
(i.e. the coupon rate of a fixed coupon bond trading at
par)
m
swap rate ti  ti 1 vt , ti   1  vt , t m 
i 1
m
swap rate ti  ti 1 vt , ti   vt , t m   1
i 1
Bootstrapping procedure
Assume that at time t the market is structured on m periods
with maturities tk = t + k, k=1....m, and assume to observe
swap rates on such maturities. The bootstrapping procedure
enables to recover discount factors of each maturity from the
previous ones.
k 1
vt , t k  
1  swap ratet,tk  vt , ti 
i 1
1  swap ratet,tk 
Forward swap rate
• In a forward start swap the exchange of flows
determined at t begins at tj.
Value fixed leg = Value floating leg
forwardswap rate ti  ti 1 vt , ti   vt , t j   vt , tm 
m
i j
forwardswap rate
vt , t j   vt , t m 
m
 t
i  j1
i
 ti 1 vt , ti 
Swap rate: summary
The swap rate can be defined as:
1. A fixed rate payment, on a running basis,
financially equivalent to a flow of indexed
payments
2. A weighted average of forward rates with
weights given by the discount factors
3. The internal rate of return, or the coupon, of a
fixed rate bond quoting at par (par yield curve)
Structuring choices
• Natural lag:
– Reference period of payment is equal to the tenor of the
reference rate
– Reset date at the beginning of the period (in advance)
• “In arrears”:
– Coupons reset and paid at the same date
• CBM/CMS: coupon indexed to long term interest
rates and swap rates.
The relevance of tenor: basis swaps
• From what we have seen, the value of a stream of indexed
payments is the same both if the reference rate is on 3m,
6m, or other.
• After the crisis this is no more true, and payments based on
different tenors have different values.
• Basis swaps is the exchange of indexed payments with
respect to interest rates of different tenors (i.e. 3m vs 6m).
• Basis swaps emerged during the crisis mainly because of
the appearance of credit risk in the banking market. In fact,
the interest rates on monthly tenors were much higher than
the swap rate on the daily tenor (EONIA).
Forward contracts
• The long party in a forward contract defines
at time t the price F at which a unit of the
security S will be purchased for delivery at
time T
• At time T the value of the contract for the
long party will be S(T) - F
Contratti forward: ingredienti
•
•
•
•
Date of the deal 16/03/2005
Spot price ENEL 7,269
Discount factor 16/05/2005: 99,66
Enel forward price:
7,269/0,9966 = 7,293799 ≈ 7,2938
• Long position (purchase) in a forward for 10000 Enel
forward for delivery on May 16 2005 for price 7,2938.
• Value of the forward contract at expiration date
16/05/2005
10000 ENEL(15/09/2005) – 72938
Derivatives and leverage
• Derivative contracts imply leverage
• Alternative 1
Forward 10000 ENEL at 7,2938 €, 2 months
2 m. later: Value 10000 ENEL – 72938
• Alternative 2
Long 10000 ENEL spot with debt 72938 for
repayment in 2 months.
2 m. later: Value 10000 ENEL – 72938
Syntetic forward
• A long/short position in a linear contract (forward)
is equivalent to a position of the same sign and
same amount and a debt/credit position for an
amount equal to the forward price
• In our case we have that, at the origin of the deal,
16/03/2005, the value of the forward contract
CF(t) is
CF(t) = 10000 x 7,269 – 0,9966 x 72938 ≈ 0
• Notice tha at the origin of the contract the forward
contract is worth zero, and the price is set at the
forward price.
Futures
• Asssume that a forward contract is closed and
settled every day (mark-to-market)
• You have obtained a futures market
– Margin (buyers and sellers post a deposit to guarantee
their performance on the contract
– Prices are marked-to-market every day and profits and
losses are settled on the margin (margin call)
– Products traded are standardized, and in some cases are
adjusted for “grade”. The seller has a “delivery option”
(if the contract is for “physical delivery”)