Computational Finance
Zvi Wiener
02-588-3049
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
CF-5
Bank Hapoalim
Jul-2001
Bonds and Interest Rates
Following T. Bjork, ch. 15
Arbitrage Theory in Continuous Time
CF-5
Bank Hapoalim
Jul-2001
Bonds and Interest Rates
Zero coupon bond = pure discount bond
T-bond, denote its price at time t by p(t,T).
principal = face value,
coupon bond - equidistant payments as a % of
the face value, fixed or floating coupons.
Zvi Wiener
CF5
slide 3
Assumptions
• There exists a frictionless market for Tbonds for every T > 0
• p(t, t) =1 for every t
• for every t the price p(t, T) is differentiable
with respect to T.
Zvi Wiener
CF5
slide 4
Interest Rates
Let t < S < T, what is IR for [S, T]?
• at time t sell one S-bond, get p(t, S)
• buy p(t, S)/p(t,T) units of T-bond
• cashflow at t is 0
• cashflow at S is -$1
• cashflow at T is p(t, S)/p(t,T)
the forward rate can be calculated ...
Zvi Wiener
CF5
slide 5
The simple forward rate LIBOR - L is the
solution of:
p(t , S )
1  (T  S ) L 
p(t , T )
The continuously compounded forward rate
R is the solution of:
e
Zvi Wiener
R (T  S )
p(t , S )

p(t , T )
CF5
slide 6
Definition 15.2
The simple forward rate for [S,T] contracted
at t (LIBOR forward rate) is
p(t , T )  p(t , S )
L(t; S , T )  
(T  S ) p(t , T )
The simple spot rate for [S,T] LIBOR spot
rate is (t=S):
p( S , T )  1
L( S , T )  
(T  S ) p( S , T )
Zvi Wiener
CF5
slide 7
Definition 15.2
The continuously compounded forward
rate for [S,T] contracted at t is
log p(t , T )  log p(t , S )
R(t ; S , T )  
T S
The continuously compounded spot rate for
[S,T] is (t=S)
log p ( S , T )
R( S , T )  
T S
Zvi Wiener
CF5
slide 8
Definition 15.2
The instantaneous forward rate with
maturity T contracted at t is
 log p(t , T )
f (t , T )  
T
The instantaneous short rate at time t is
r (t )  f (t , t )
Zvi Wiener
CF5
slide 9
Definition 15.3
The money market account process is


Bt  exp   r ( s )ds 
0

t
Note that here t means some time moment in
the future. This means
dB(t )  r (t ) B(t )dt

B(0)  1
Zvi Wiener
CF5
slide 10
Lemma 15.4
For t  s  T we have


p (t , T )  p (t , s ) exp    f (t , u )du 
 s

T
And in particular


p(t , T )  exp    f (t , u )du 
 t

T
Zvi Wiener
CF5
slide 11
Models of Bond Market
• Specify the dynamic of short rate
• Specify the dynamic of bond prices
• Specify the dynamic of forward rates
Zvi Wiener
CF5
slide 12
Important Relations
Short rate dynamics
dr(t)= a(t)dt + b(t)dW(t)
(15.1)
Bond Price dynamics
(15.2)
dp(t,T)=p(t,T)m(t,T)dt+p(t,T)v(t,T)dW(t)
Forward rate dynamics
df(t,T)= (t,T)dt + (t,T)dW(t)
W is vector valued
Zvi Wiener
CF5
(15.3)
slide 13
Proposition 15.5
We do NOT assume that there is no arbitrage!
If p(t,T) satisfies (15.2), then for the forward
rate dynamics
 (t , T )  vT (t , T )v(t , T )  mT (t , T )

 (t , T )  vT (t , T )
Zvi Wiener
CF5
slide 14
Proposition 15.5
We do NOT assume that there is no arbitrage!
If f(t,T) satisfies (15.3), then the short rate
dynamics
a(t )  fT (t , t )   (t , t )

b(t )   (t , t )
Zvi Wiener
CF5
slide 15
Proposition 15.5
If f(t,T) satisfies (15.3), then the bond price dynamics
1
2

dp(t , T )  p(t , T ) r (t )  A(t , T )  S (t , T ) dt 
2


p(t , T ) S (t , T )dW (t )

 A(t , T )     (t , s)ds

t

T
S (t , T )    (t , s)ds


tCF5

T
Zvi Wiener
slide 16
Proof of Proposition 15.5
Left as an exercise …
Zvi Wiener
CF5
slide 17
Fixed Coupon Bonds
n
p(t )  K p(t , Tn )   ci p (t , Ti )
i 1
Ti  T0   i
ci  ri Ti  Ti 1 K


p(t )  K  p(t , Tn )  r  p(t , Ti ) 
i 1


n
Zvi Wiener
CF5
slide 18
Floating Rate Bonds
ci  Ti  Ti 1 L(Ti 1 , Ti ) K
L(Ti-1,Ti) is known at Ti-1 but the coupon is
delivered at time Ti. Assume that K =1 and
payment dates are equally spaced.
Now it is t<T0. By definition of L we have
p(t , Ti )  p(t , Ti 1 )
L(t , Ti 1 , Ti )  
(Ti  Ti 1 ) p(t , Ti )
Zvi Wiener
CF5
slide 19
Floating Rate Bonds
ci  Ti  Ti 1 L(Ti 1 , Ti ) K
p(t , Ti )  p(t , Ti 1 )
L(t , Ti 1 , Ti )  
(Ti  Ti 1 ) p(t , Ti )
implies
ci 
Zvi Wiener
1
p(Ti 1 , Ti )
CF5
1
slide 20
ci 
1
p(Ti 1 , Ti )
1
This coupon will be paid at Ti. The value of -1
at time t is -p(t, Ti). The value of the first term
is p(t, Ti-1). Thus the present value of each
coupon is
PV ci   p(t , Ti 1 )  p(t , Ti )
The present value of the principal is p(t,Tn).
Zvi Wiener
CF5
slide 21
The value of a floater is
n
p(t )  p(t , Tn )    p(t , Ti 1 )  p(t , Ti )
i 1
Or after a simplification
p(t )  p(t , T0 )
Zvi Wiener
CF5
slide 22
Forward Swap Settled in Arrears
K - principal, R - swap rate,
rates are set at dates T0, T1, … Tn-1 and paid at
dates T1, … Tn.
T0
Zvi Wiener
T1
Tn-1
CF5
Tn
slide 23
Forward Swap Settled in Arrears
If you swap a fixed rate for a floating rate
(LIBOR), then at time Ti, you will receive
KL(Ti 1 , Ti )  Kci
where ci is a coupon of a floater. And at Ti
you will pay the amount
K R
Net cashflow
Zvi Wiener
K L(Ti 1 , Ti )  R
CF5
slide 24
Forward Swap Settled in Arrears
At t < T0 the value of this payment is
Kp(t , Ti 1 )  K (1  R) p(t , Ti )
The total value of the swap at time t is then
n
 (t )  K   p(t , Ti 1 )  (1  R) p(t , Ti )
i 1
Zvi Wiener
CF5
slide 25
Proposition 15.7
At time t=0, the swap rate is given by
R
p (0, T0 )  p (0, Tn )
n
  p (0, Ti )
i 1
Zvi Wiener
CF5
slide 26
Zero Coupon Yield
The continuously compounded zero coupon
yield y(t,T) is given by
log p(t , T )
y (t , T )  
T t
p(t , T )  e
 (T t ) y ( t ,T )
For a fixed t the function y(t,T) is called
the zero coupon yield curve.
Zvi Wiener
CF5
slide 27
The Yield to Maturity
The yield to maturity of a fixed coupon
bond y is given by
n
p (t )   ci e
 (Ti  t ) y
i 1
Zvi Wiener
CF5
slide 28
Macaulay Duration
Definition of duration, assuming t=0.
n
D
Zvi Wiener
T c e
i 1
Ti y
i i
p
CF5
slide 29
Macaulay Duration
T
T
CFt
1
D   t wt 
t

t
Bond Pr ice t 1 (1  y)
t 1
A weighted sum of times to maturities of each
coupon.
What is the duration of a zero coupon bond?
Zvi Wiener
CF5
slide 30
Meaning of Duration
dp d 
Ti y 

 ci e    Dp
dy dy  i 1

n
$
r
Zvi Wiener
CF5
slide 31
Proposition 15.12 TS of IR
With a term structure of IR (note yi), the
duration can be expressed as:
n
D
T c e
i 1
Ti yi
i i
p
d 
Ti ( yi  s ) 
 ci e
   Dp
ds  i 1
 s 0
n
Zvi Wiener
CF5
slide 32
Convexity
 p
C 2
y
2
$
r
Zvi Wiener
CF5
slide 33
FRA Forward Rate Agreement
A contract entered at t=0, where the parties (a
lender and a borrower) agree to let a certain
interest rate R*, act on a prespecified principal,
K, over some future time period [S,T].
Assuming continuous compounding we have
at time S: -K
at time T: KeR*(T-S)
Calculate the FRA rate R* which makes PV=0
hint: it is equal to forward rate
Zvi Wiener
CF5
slide 34
Exercise 15.7
Consider a consol bond, i.e. a bond which
will forever pay one unit of cash at t=1,2,…
Suppose that the market yield is y - flat.
Calculate the price of consol.
Find its duration.
Find an analytical formula for duration.
Compute the convexity of the consol.
Zvi Wiener
CF5
slide 35
Change of Numeraire
Following T. Bjork, ch. 19
Arbitrage Theory in Continuous Time
CF-5
Bank Hapoalim
Jul-2001
Change of Numeraire
P - the objective probability measure,
Q - the risk-neutral martingale measure,
We will introduce a new class of measures such
that Q is a member of this class.
Zvi Wiener
CF5
slide 37
Intuitive explanation
   r ( s ) ds 
Q

0
 (0; X )  E Xe




T
Assuming that X and r are independent under Q, we get
Q
X 
(0; X )  p(0, T ) E
In all realistic cases that X and r are not independent under Q.
However there exists a measure T (forward neutral) such that
(0; X )  p(0, T ) E X 
T
Zvi Wiener
CF5
slide 38
Risk Neutral Measure
Is such a measure Q that for every choice of
price process (t) of a traded asset the
following quotient is a Q-martingale.
 (t )
B (t )
Note that we have divided the asset
price (t) by a numeraire B(t).
Zvi Wiener
CF5
slide 39
Conjecture 19.1.1
For a given financial market and any asset
price process S0(t) there exists a probability
measure Q0 such that for any other asset
(t)/S0(t) is a Q0-martingale.
For example one can take p(t,T) (fixed T) as
S0(t) then there exists a probability measure
QT such that for any other asset (t)/p(t,T) is
a QT-martingale.
Zvi Wiener
CF5
slide 40
Using p(T,T)=1 we get
(0)
T   (T ) 
T
E 
 E (T )

p(0, T )
 p(T , T ) 
Using a derivative asset as (t,X) we get
 (0, X )
T
 E X 
p(0, T )
Zvi Wiener
CF5
slide 41
Assumption 19.2.1
Denote an observable k+1 dimensional process
X=(X1, …, Xk, Xk+1) where
Xk+1(t)=r(t) (short term IR)
Denote by Q a fixed martingale measure under
which the dynamics is:
dXi(t)=i(t,X(t))dt + i(t,X(t))dW(t), i=1,…,k+1
A risk free asset (money market account):
dB(t)=r(t)B(t)dt
Zvi Wiener
CF5
slide 42
Proposition 19.1
The price process for a given simple claim
Y=(X(T)) is given by (t,Y)=F(t,X(t)),
where F is defined by
   r ( s ) ds 
Q 

F (t , x)  Et , x Ye t




T
Zvi Wiener
CF5
slide 43
Practical Numeraire Approach
Zvi Wiener
02-588-3049
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
CF-5
Bank Hapoalim
Jul-2001
Options with uncertain strike
Stock option with strike fixed in foreign currency.
How it can be priced?
Margarbe 78 or Numeraire approach
1. Price it using this currency as a numeraire.
foreign interest rate
foreign current price
foreign volatility!
2. Translate the resulting price into SHEKELS using the
current exchange rate.
Zvi Wiener
CF5
slide 45
Options with uncertain strike
Endowment warrants
strike is increasing with short term IR.
strike is decreasing when a dividend is paid
What is an appropriate numeraire?
A closed Money Market account.
Result – price by standard BS but with 0
dividends and 0 IR.
Zvi Wiener
CF5
slide 46
Options with uncertain strike
An option to choose by some date between dollar
and CPI indexing (may be with some interest).
Margrabe can be used or one can price a simple
CPI option in terms of an American investor and
then translate it to SHEKELS.
Zvi Wiener
CF5
slide 47
Convertible Bonds
A convertible bond typically includes an
option to convert it into some amount of
ordinary shares.
This can be seen as a package of a regular
bond and an option to exchange this regular
bond to shares of the company.
If the company does not have traded debt
there is a problem of pricing this option.
Zvi Wiener
CF5
slide 48
Convertible Bonds
This is an option to exchange one asset to
another and can be priced with Margrabe
approach.
However in order to use this approach one
need to know the correlation between the two
assets (stock and regular bond).
When there is no market for regular bonds this
might be a problem.
Zvi Wiener
CF5
slide 49
Convertible Bonds
An alternative approach is with a numeraire.
Denote by
St stock price at time t,
Bt price at time t of a regular bond (may be
not observable).
CBt price of a convertible bond.
C - value of the conversion option, so that
CB = C(B) + B at any time
Zvi Wiener
CF5
slide 50
Convertible Bonds
Note that C is a decreasing function of B (the
higher the strike price, the lower is the
option’s value).
This means that as soon as
CBt < St = C(B=0) the right hand side of the
following equation (B - an unknown)
CBt = C(Bt) + Bt
has a unique solution.
Zvi Wiener
CF5
slide 51
Convertible Bonds
The left hand side is a known constant, the
right hand side is a sum of two variables.
The first one is decreasing in B, but its
derivative is strictly less than one and
approaches zero for large B.
The second one is linear with slope one.
This means that as soon as CB>C(B=0)+0=S
there exists a unique solution.
Zvi Wiener
CF5
slide 52
Uniqueness of a solution
CB
S
B
Zvi Wiener
CF5
slide 53
Pricing with known volatility
Let’s use Bt as a numeraire, then the stochastic
variable is St/Bt.
Assume that St/Bt has a constant volatility .
Then this option has a fixed strike (in terms of
B) and is equivalent to a standard option, which
can be priced with BS equation.
Call(St/Bt, T, 1, , r) (in terms of Bt),
the dollar value is then BtCall(St/Bt, T, 1, , r).
Zvi Wiener
CF5
slide 54
Pricing with known volatility
This means that when  is known the option can
be priced easily and consequently the straight
bond.
However  that we need can not be observed.
The solution is in the following procedure.
Zvi Wiener
CF5
slide 55
Pricing with known volatility
Assume that  is stable but unknown. For any
value of  we can easily price the option at any
date, and hence we can also derive the value of Bt.
Take a sequence of historical data (meaning St and
CBt). For any value of  we can construct the
implied Bt().
Then using these sequence of observations we can
check whether the volatility of St/Bt is indeed .
If our guess of  was correct this is true.
Zvi Wiener
CF5
slide 56
Pricing with known volatility
However there is no reason why some value of 
will give the same implied historical volatility.
This means that we have to solve for  such that
the implied volatility is equal .
Numerically this can be done easily.
Why there exists a unique solution???
Check monotonicity!!
Zvi Wiener
CF5
slide 57
Solution for 
Implied volatitity
1
0.8
0.6
0.4
0.2
0.2
Zvi Wiener
0.4
CF5
0.6
0.8
1
slide 58
MMA implementation
FindRoot[CB == B + bsCallFX[s, ttm, B, sg, 0, 0], {B,CB}]
ConvertibleBondHistorical[StockHistory_, CBHistory_, ttm_] :=
Module[{sg, len, ff, BusinessDaysYear = 250, sgg, t1, t2},
len = Length[StockHistory];
ff[sg_] := Log[StockHistory/
MapThread[StraightBond[#1, #2, ttm, sg] &,
{CBHistory, StockHistory}]];
FindRoot[sg ==
StandardDeviation[Rest[ff[sg]] - Drop[ff[sg], -1]]*
Sqrt[BusinessDaysYear], {sg, 0.001, 1}][[1, 2]]
];
Zvi Wiener
CF5
slide 59
Example 1
CB
1.15
S
1.1
1.05
20
40
60
80
100
0.95
0.9
B
Zvi Wiener
CF5
slide 60
Example 2
CB
0.9
S
0.85
20
40
60
80
100
0.75
B
Zvi Wiener
CF5
slide 61
Value of Value-at-Risk
Zvi Wiener
The Hebrew University of Jerusalem
[email protected]
CF-5
Bank Hapoalim
Jul-2001
VaR
1 day
1% probability 1w
1% probability 1d
1 week
P&L
Zvi Wiener
CF5
slide 63
Model
• Bank’s choice of an optimal system
• Depends on the available capital
• Current and potential capital needs
• Queuing model as a base
Zvi Wiener
CF5
slide 64
Required Capital
Let A be total assets
C – capital of a bank
 - percentage of qualified assets
k – capital required for traded assets
C  A(1   )0.08  Ak
Zvi Wiener
CF5
slide 65
Maximal Risk (Assets)
Amax
C

(1   )0.08  k
The coefficient k varies among systems, but a better
(more expensive) system provides more precise risk
measurement, thus lower k.
Cost of a system is p, paid as a rent (pdt during dt).
Amax is a function of C and p.
Zvi Wiener
CF5
slide 66
Risky Projects
Deposits arrive and are withdrawn randomly.
All deposits are of the same size.
Invested according to bank’s policy.
Can not be used if capital requirements are not
satisfied.
Zvi Wiener
CF5
slide 67
Arrival of Risky Projects
We assume that risky projects arrive randomly
(as a Poisson process with density ).
This means that there is a probability dt that
during dt one new project arrives.
Zvi Wiener
CF5
slide 68
Arrival of Risky Projects
A new project is undertaken if the bank has
enough capital (according to the existing risk
measuring system).
We assume that one can NOT raise capital or
change systems quickly.
Zvi Wiener
CF5
slide 69
Termination of Risky Projects
We assume that each risky project disappears
randomly (as a Poisson process with density ).
Zvi Wiener
CF5
slide 70
Termination of Risky Projects
We assume that each risky project disappears
randomly (as a Poisson process with density ).
This means that there is a probability ndt that
during dt one out of n existing projects
terminates.
With probability (1-ndt) all existing projects
will be active after dt.
Zvi Wiener
CF5
slide 71
Profit
We assume that each existing risky project gives
a profit of dt during dt.
Thus when there are n active projects the bank
has instantaneous profit (n-p)dt.
Zvi Wiener
CF5
slide 72
States
After C and p are chosen, the maximal number
of active projects is given by s=Amax(C,p).




2
s
0
1
2
s-1
s
0
1
2
s-1
s
Zvi Wiener
CF5
slide 73
States




2
s
0
1
2
s-1
s
0
1
2
s-1
s
Stable distribution:
0  = 1 
1  = 2 2
…
s-1  = s s
Zvi Wiener
n
n

n!  n
n
!
 n  s i  s i , where  





i
i  0 i! 
i  0 i!
CF5
slide 74
Probabilities
n

e  (1  s)
n  s i 
(1  n)(1  s, )


i  0 i!
n!
n
• Probability of losing a new project due to
capital requirements is equal to the probability
of being in state s, i. e. s.
• Termination of projects does not have to be
Poissonian, only mean and variance matter.
Zvi Wiener
CF5
slide 75
Expected Profit
s( s, )
E ( profit )   1     s ( p ) p  p  
p
(1  s, )
An optimal p (risk measurement system) can
be found by maximizing the expected profit
stream.
Zvi Wiener
CF5
slide 76
Example
k ( p)  0.015  (0.08  0.015)e
p

q
• Capital requirement as a function of p
(price) and q (scaling factor), varies between
1.5% and 8%.
Zvi Wiener
CF5
slide 77
Example
k ( p)  0.015  (0.08  0.015)e
Amax14.25

p
q
q=0.5
14
13.75
q=1
13.5
q=3
13.25
1
2
3
4
p
12.75
12.5
Zvi Wiener
CF5
slide 78
Example of a bank
• Capital $200M
• Average project is $20K
• On average 200 new projects arrive each day
• Average life of a project is 2 years
• 15% of assets are traded and q=1
• spread =1.25%
Zvi Wiener
CF5
slide 79
Expected profit
33
32.5
32
31.5
31
30.5
1
2
3
4
5
6
rent p
29.5
Bank’s profit as a function of cost p.
C=$200M, arrival rate 200/d,
size $20K, average life 2 yr.,
spread 1.25%, q=1, 15% of assets are traded.
Zvi Wiener
CF5
slide 80
Expected profit
25
24
23
22
21
1
2
3
4
5
6
rent p
Bank’s profit as a function of cost p.
C=$200M, arrival rate 200/d,
size $20K, average life 2 yr.,
spread 1%, q=1, 5% of assets are traded.
Zvi Wiener
CF5
slide 81
Conclusion
• Expensive systems are appropriate for banks
with
• low capitalization
• operating in an unstable environment
• Cheaper methods (like the standard approach)
should be appropriate for banks with
• high capitalization
• small trading book
• operating in a stable environment
• many small uncorrelated, long living projects
Zvi Wiener
CF5
slide 82
A simple intuitive and flexible model of
optimal choice of risk measuring method.
Zvi Wiener
CF5
slide 83
DAC
Zvi Wiener
02-588-3049
http://pluto.mscc.huji.ac.il/~mswiener/zvi.html
CF-5
Bank Hapoalim
Jul-2001
Life Insurance
• yearly contribution 10,000 NIS
• yearly risk premium 2,000 NIS
• first year agent’s commission 3,000 NIS
• promised accumulation rate 8,000 NIS/yr
• After the first payment there is a problem of
insufficient funds. 8,000 NIS are promised
(with all profits) and only 5,000 NIS arrived.
Zvi Wiener
CF5
slide 85
10,000 NIS
Risk
2,000 NIS
Client’s
8,000 NIS
Agent
3,000 NIS
• insufficient funds if the client leaves
• insufficient profits
Zvi Wiener
CF5
slide 86
Risk measurement
• The reason to enter this transaction is
because of the expected future profits.
• Assume that the program is for 15 years and
the probability of leaving such a program is .
• Fees are
• 0.6% of the portfolio value each year
• 15% real profit participation
Zvi Wiener
CF5
slide 87
Obligations
• The most important question is what are the
obligations?
• The Ministry of Finance should decide
• Transparent to a client
• Accounted as a loan
Zvi Wiener
CF5
slide 88
One year example
Assume that the program is for one year only
and there is no possibility to stop payments
before the end.
Initial payment P0, fees lost L0, fixed fee a%
of the final value P1, participation fee b% of
real profits (we ignore real).
Investment policy TA-25 (MAOF).
Zvi Wiener
CF5
slide 89
Liabilities (no actual loan)
P0 (1  a)
X1
P0 (1  a)
b
Call ( X 1 , X 0 ,1)
X0
X0
Assets (no actual loan)
X1
P0  L0 
X0
Zvi Wiener
CF5
slide 90
Total=Assets-Liabilities
P0 (1  a)
X1
aP0  L0   b
Call ( X 1 , X 0 ,1)
X0
X0
Fair value
Xt
P0 (1  a)
aP0  L0   b
Call ( X t , X 0 ,1)
X0
X0
Zvi Wiener
CF5
slide 91
Liabilities (actual loan)
P0 (1  a)
X1
Rt
P0 (1  a)
b
Call ( X 1 , X 0 ,1)  L0 e
X0
X0
Assets (actual loan)
X1
P0
X0
Zvi Wiener
CF5
slide 92
Total=Assets-Liabilities (loan)
Xt
P0 (1  a)
Rt
aP0
b
Call ( X t , X 0 ,1)  L0 e
X0
X0
Zvi Wiener
CF5
slide 93
2 years liabilities (no actual loan)
P0 (1  a)
X2
P0 (1  a)
b
Call ( X 2 , X 0 ,2)
X0
X0
2
2
2 years assets (no actual loan)
X2
P0  L0 
X0
In reality the situation is even better for the
insurer, since profit participation fees once
taken are never returned (path dependence).
Zvi Wiener
CF5
slide 94
2 years fair value, no loan
 


X2
P0 1  (1  a )  L0

X0
2
P0 (1  a )
b
Call ( X 2 , X 0 ,2)
X0
2
Zvi Wiener
CF5
slide 95
2 years liabilities (with a loan)
P0 (1  a)
X2
2R
P0 (1  a)
b
Call ( X 2 , X 0 ,2)  L0e
X0
X0
2
2
2 years assets (with a loan)
X2
P0
X0
Zvi Wiener
CF5
slide 96
10 years, L0=7%
With a loan
No loan
Profit
0.3
0.2
0.1
0.5
1
1.5
2.5
3
3.5
Stock index
-0.1
Zvi Wiener
2
CF5
slide 97
Partial loan - portion q
 


Xn
P0 1  (1  a )  (1  q ) L0

X0
n
P0 (1  a )
nR
b
Call ( X n , X 0 , n)  qLe
X0
n
Theoretically q can be negative.
Zvi Wiener
CF5
slide 98
Mixed portfolio
When the investment portfolio is a mix one
should analyze it in a similar manner.
Important: an option on a portfolio is less
valuable than a portfolio of options.
Another risk factor - leaving rate should be
accounted for by taking actuarial tables as
leaving rate.
Zvi Wiener
CF5
slide 99
Conclusions
It is a reasonable risk management policy not
to take a loan against DAC.
Up to some optimal point it creates a useful
hedge to other assets (call options and shares)
of the firm.
Intuitively DAC is good when the stock
market performs badly and profit participation
is valueless. DAC performs bad when the
market performs well.
Zvi Wiener
CF5
slide 100