"The Kelly criterion and optimal insurance of capital"
Anna Zambrzycka, E. W. Piotrowski
[email protected]
University of Silesia, University of Białystok
”The Kelly criterion and optimal insurance of capital” – p.1/17
Motivation
The goal of economic activity is a permanent profit maximisation, but not an average
income maximisation. It leads to analyse the expected values of the logarithms of forecast
prices for this purpose.
”The Kelly criterion and optimal insurance of capital” – p.2/17
Gambling and the securities market
•
The fundamental problem in gambling is to find positive expectation betting
opportunities. The analogous problem in the securities market is to find
investments with excess risk not adjusted expected rates of return. A gambler or
an investor must decide how much of his capital to bet.
”The Kelly criterion and optimal insurance of capital” – p.3/17
Gambling and the securities market
•
The fundamental problem in gambling is to find positive expectation betting
opportunities. The analogous problem in the securities market is to find
investments with excess risk not adjusted expected rates of return. A gambler or
an investor must decide how much of his capital to bet.
•
In both these settings, we explore the Kelly criterion, which is to maximise the
expected value of the logarithm of the capital (proportional betting, proportional
gambling).
”The Kelly criterion and optimal insurance of capital” – p.3/17
The Kelly criterion
If all bets paid even money, had positive expectation and were independent, the Kelly’s
criterion is a bet on each trial so as to maximise E(logV )- the expected value of the
logarithm of the capital V . Kelly’s betting recipe is simple: bet a fraction of your current
capital equal to your expectation.
”The Kelly criterion and optimal insurance of capital” – p.4/17
Example - Coin Tossing
Imagine that we are faced with an infinitely wealthy opponent who will wager constant
money on bets made on repeat independent trials of biased coin. Further, suppose that
on each trial our probability of success is p > 1/2 and the probability of losing is
q = 1 − p. Our initial capital is V0 .
•
Suppose we chose the goal of maximising the expected value of our capital E(Vn )
after n bets. How much should we bet Bk on the kth trial?
•
Letting Tk = 1 if the kth trial is a win and Tk = −1 if it is a loss, then
P
Vk = Vk−1 + Tk Bk , for k = 1, 2, ... and Vn = V0 + n
k=1 Tk Bk .
Pn
Pn
Then E(Vn ) = V0 + k=1 E(Tk Bk ) = V0 + k=1 (p − q)E(Bk ).
•
•
Since the game has a positive expectation, when p − q > 0 in this even payoff
situation, then in order to maximise E(Vn ) we would want to maximise E(Bk ) at
each trial. Thus, to maximise expected gain we should bet all of our capital at each
trial.
”The Kelly criterion and optimal insurance of capital” – p.5/17
Example - Coin Tossing
•
The probability of ruin is given by 1 − pn and with p < 1, limn→∞ [1 − pn ] = 1.
Ruin is almost sure. Thus the "bold" criterion of betting to maximise expected gain
is usually undesirable.
”The Kelly criterion and optimal insurance of capital” – p.6/17
Example - Coin Tossing
•
The probability of ruin is given by 1 − pn and with p < 1, limn→∞ [1 − pn ] = 1.
Ruin is almost sure. Thus the "bold" criterion of betting to maximise expected gain
is usually undesirable.
•
If we play to minimise the probability of eventual ruin by making a minimum bet on
each trial, we also minimise the expected gain. Thus ”timid” betting is also
unattractive.
”The Kelly criterion and optimal insurance of capital” – p.6/17
Example - Coin Tossing
•
The probability of ruin is given by 1 − pn and with p < 1, limn→∞ [1 − pn ] = 1.
Ruin is almost sure. Thus the "bold" criterion of betting to maximise expected gain
is usually undesirable.
•
If we play to minimise the probability of eventual ruin by making a minimum bet on
each trial, we also minimise the expected gain. Thus ”timid” betting is also
unattractive.
•
A optimal strategy was first proposed by J.L. Kelly, A New Interpretation of Information
Rate, Bell System Technical Journal, 35 (1956), 917-926.
”The Kelly criterion and optimal insurance of capital” – p.6/17
Example - Coin Tossing
•
The probability of ruin is given by 1 − pn and with p < 1, limn→∞ [1 − pn ] = 1.
Ruin is almost sure. Thus the "bold" criterion of betting to maximise expected gain
is usually undesirable.
•
If we play to minimise the probability of eventual ruin by making a minimum bet on
each trial, we also minimise the expected gain. Thus ”timid” betting is also
unattractive.
•
A optimal strategy was first proposed by J.L. Kelly, A New Interpretation of Information
Rate, Bell System Technical Journal, 35 (1956), 917-926.
•
Since the probabilities and payoffs for each bet are the same, an optimal strategy
will always involve wagering the same fraction f of our capital. This is possible
when we shall assume from here on that capital is infinitely divisible. We bet
according to Bi = f Vi , where 0 ≤ f ≤ 1, where S and F are the number of
successes and failures in n trials, then our capital after n trials is equal to
Vn = V0 (1 + f )S (1 − f )F , where F + S = n.
”The Kelly criterion and optimal insurance of capital” – p.6/17
Example - Coin Tossing
•
•
•
With f in the interval 0 ≤ f ≤ 1, Pr(Vn = 0) = 0. Ruin in the technical sense of
the gambler’s ruin problem cannot occur.
We note that since e
“
”1
n n
n ln V
V
The quantity gn (f ) = ln
=
0
“
Vn
V0
”1
n
Vn
,
V0
=
S
n
ln(1 + f ) +
F
n
ln(1 − f ) is the exponential
rate of growth of the gambler’s capital.
”The Kelly criterion and optimal insurance of capital” – p.7/17
Example - Coin Tossing
•
•
•
With f in the interval 0 ≤ f ≤ 1, Pr(Vn = 0) = 0. Ruin in the technical sense of
the gambler’s ruin problem cannot occur.
We note that since e
“
”1
n n
n ln V
V
The quantity gn (f ) = ln
=
0
“
Vn
V0
”1
n
Vn
,
V0
=
S
n
ln(1 + f ) +
F
n
ln(1 − f ) is the exponential
rate of growth of the gambler’s capital.
Kelly chooses to maximise the expected value of the exponential rate of growth
coefficient g(f ), where
g(f ) = E(gn (f )) = E
→
ln
„
Vn
V0
«1 !
n
=E
„
F
S
ln(1 + f ) +
ln(1 − f )
n
n
«
S
F
ln(1 + f ) +
ln(1 − f ) = p ln(1 + f ) + q ln(1 − f )
n
n
w.p. 1 as n → ∞.
”The Kelly criterion and optimal insurance of capital” – p.7/17
Example - Coin Tossing
•
Function g(f ) has a maximum at f = f ∗ = p − q , where optimal success rate is
g(f ∗ ) = p ln p + q ln q + ln 2 > 0. Thus f ∗ of current capital should be wagered on
each bet in order to cause Vn to grow at the fastest rate possible consistent with
zero probability of ever going broke.
”The Kelly criterion and optimal insurance of capital” – p.8/17
Proportional betting
The Kelly criterion can easily be extended to sequence of favourable games. If we bet x
on the nth game, then our return is xZn , where {Zn : n ≥ 1} is a sequence of i.i.d.
random variables with EZn > 0. Let Vn be our fortune after n bets, and let fn denote
the proportion of our wealth that we wager on the nth bet. Our fortune then evolves as
n
• Vn = Vn + (fn Vn−1 )Zn = Vn−1 (1 + fn Zn ) = V0 Q (1 + fi Zi ), n ≥ 1.
i=1
•
•
Of considerable interest is the special case where we always bet the same
constant proportion fn = f . In this situation the exponential rate of growth of our
strategy is
gn (f ) =
1
n
ln
“
Vn
V0
”
=
1
n
n
P
ln(1 + f Zi ).
i=1
”The Kelly criterion and optimal insurance of capital” – p.9/17
Proportional betting
The Kelly criterion can easily be extended to sequence of favourable games. If we bet x
on the nth game, then our return is xZn , where {Zn : n ≥ 1} is a sequence of i.i.d.
random variables with EZn > 0. Let Vn be our fortune after n bets, and let fn denote
the proportion of our wealth that we wager on the nth bet. Our fortune then evolves as
n
• Vn = Vn + (fn Vn−1 )Zn = Vn−1 (1 + fn Zn ) = V0 Q (1 + fi Zi ), n ≥ 1.
i=1
•
•
•
Of considerable interest is the special case where we always bet the same
constant proportion fn = f . In this situation the exponential rate of growth of our
strategy is
gn (f ) =
1
n
ln
“
Vn
V0
”
=
1
n
n
P
ln(1 + f Zi ).
i=1
The expected value of the growth rate is g(f ) = E(gn (f )) → E(ln(1 + f Zi ) w.p. 1
as n → ∞.
”The Kelly criterion and optimal insurance of capital” – p.9/17
Proportional betting
The Kelly criterion can easily be extended to sequence of favourable games. If we bet x
on the nth game, then our return is xZn , where {Zn : n ≥ 1} is a sequence of i.i.d.
random variables with EZn > 0. Let Vn be our fortune after n bets, and let fn denote
the proportion of our wealth that we wager on the nth bet. Our fortune then evolves as
n
• Vn = Vn + (fn Vn−1 )Zn = Vn−1 (1 + fn Zn ) = V0 Q (1 + fi Zi ), n ≥ 1.
i=1
•
•
Of considerable interest is the special case where we always bet the same
constant proportion fn = f . In this situation the exponential rate of growth of our
strategy is
gn (f ) =
1
n
ln
“
Vn
V0
”
=
1
n
n
P
ln(1 + f Zi ).
i=1
•
The expected value of the growth rate is g(f ) = E(gn (f )) → E(ln(1 + f Zi ) w.p. 1
as n → ∞.
•
We can optimise our long-ran growth rate by choosing f to maximise g(f ).
”The Kelly criterion and optimal insurance of capital” – p.9/17
Proportional betting
The Kelly criterion can easily be extended to sequence of favourable games. If we bet x
on the nth game, then our return is xZn , where {Zn : n ≥ 1} is a sequence of i.i.d.
random variables with EZn > 0. Let Vn be our fortune after n bets, and let fn denote
the proportion of our wealth that we wager on the nth bet. Our fortune then evolves as
n
• Vn = Vn + (fn Vn−1 )Zn = Vn−1 (1 + fn Zn ) = V0 Q (1 + fi Zi ), n ≥ 1.
i=1
•
•
Of considerable interest is the special case where we always bet the same
constant proportion fn = f . In this situation the exponential rate of growth of our
strategy is
gn (f ) =
1
n
ln
“
Vn
V0
”
=
1
n
n
P
ln(1 + f Zi ).
i=1
•
The expected value of the growth rate is g(f ) = E(gn (f )) → E(ln(1 + f Zi ) w.p. 1
as n → ∞.
•
•
We can optimise our long-ran growth rate by choosing f to maximise g(f ).
When we know an arbitrary distribution of Zn the optimal policy is still
f ∗ = arg supf E(ln(1 + f Zi ),
”The Kelly criterion and optimal insurance of capital” – p.9/17
Approximations for discrete-time models
It is known that proportional betting has many good properties, besides maximising the
growth rate for discrete-time models:
•
L. Breiman (1961), proved that f ∗ also asymptotically minimises the expected time
to reach a fixed fortune, and asymptotically dominates any other strategy.
•
Bell and Cover (1980) proved that this strategy is also optimal in a game theoretic
sense for finite horizons.
•
Bellman and Kalaba (1957) proved that this policy is optimal for the equivalent
problem of maximising the utility of terminal wealth at any fixed terminal time,
when the utility function is logarithmic.
”The Kelly criterion and optimal insurance of capital” – p.10/17
Approximations for continuous-time models
Proportional gambling and the Kelly criterion have also been considered in a
continuous-time models, when the underlying random walk is replaced by Brownian
motion (with positive drift). The problem of optimal gambling in repeated favourable
games is intimately related to the problem of optimal multi-period investment in financial
economics. The only essential difference in fact is the option of investing in a risk free
security that pays a nonstochastic interest rate r > 0.
”The Kelly criterion and optimal insurance of capital” – p.11/17
Approximations for continuous-time models
Proportional gambling and the Kelly criterion have also been considered in a
continuous-time models, when the underlying random walk is replaced by Brownian
motion (with positive drift). The problem of optimal gambling in repeated favourable
games is intimately related to the problem of optimal multi-period investment in financial
economics. The only essential difference in fact is the option of investing in a risk free
security that pays a nonstochastic interest rate r > 0.
Example
We consider the simple case in which there is only one risky stock available for
investment. In this case Zn denotes the return of the risky stock on day n, and if the
investor decides to invest a fraction fn of his capital in the risky stock on day n, with the
remainder of his capital invested in the riskless security, then his fortune evolves as
”The Kelly criterion and optimal insurance of capital” – p.11/17
Approximations for continuous-time models
Proportional gambling and the Kelly criterion have also been considered in a
continuous-time models, when the underlying random walk is replaced by Brownian
motion (with positive drift). The problem of optimal gambling in repeated favourable
games is intimately related to the problem of optimal multi-period investment in financial
economics. The only essential difference in fact is the option of investing in a risk free
security that pays a nonstochastic interest rate r > 0.
Example
We consider the simple case in which there is only one risky stock available for
investment. In this case Zn denotes the return of the risky stock on day n, and if the
investor decides to invest a fraction fn of his capital in the risky stock on day n, with the
remainder of his capital invested in the riskless security, then his fortune evolves as
Vn = Vn−1 [1 + r(1 − fn ) + fn Zn ] = V0
n
Y
[1 + r(1 − fi ) + fi Zi ]
i=1
.
”The Kelly criterion and optimal insurance of capital” – p.11/17
Approximations for continuous-time models
Thus, all our results, while stated mostly in the language of gambling, are in fact equally
applicable to investment problems. The Kelly criterion in this context, referred to as the
optimal-growth criterion was studied by Merton (1990) and Karatzas (1989). An adaptive
portfolio strategy that performs asymptotically as well as the best constant proportion
strategy, for an arbitrary sequence of gambles, was introduced by Cover (1991) in the
discrete-time and was extended to continuous-time by Jamishidan (1992).
”The Kelly criterion and optimal insurance of capital” – p.12/17
Optimisation of insurance
Kelly criterion asymptotically maximises the expected growth rate of the
by
“ capital
”
maximisation the expected value of the exponential rate of growth ln
Vn
V0
.
We employ this technique for optimisation of an insurance problem. We consider the
classical problem of the capital insurance, which proceeds according to the scheme
ω
1+α−
→ ez(ω) + f (ω),
•
where z(ω) and f (ω) are the random variables define on the probabilistic space
(Ω, F , P ),
•
•
•
α ≥ 0 is an insurance premium respondents to a unit of insurance capital,
ez(ω) is a price of unit of capital in result of an event ω,
f (ω) ≥ 0 is the amount paid by insurer as a result of an event ω,
”The Kelly criterion and optimal insurance of capital” – p.13/17
Optimisation of insurance
Kelly criterion asymptotically maximises the expected growth rate of the
by
“ capital
”
maximisation the expected value of the exponential rate of growth ln
Vn
V0
.
We employ this technique for optimisation of an insurance problem. We consider the
classical problem of the capital insurance, which proceeds according to the scheme
ω
1+α−
→ ez(ω) + f (ω),
•
where z(ω) and f (ω) are the random variables define on the probabilistic space
(Ω, F , P ),
•
•
•
α ≥ 0 is an insurance premium respondents to a unit of insurance capital,
ez(ω) is a price of unit of capital in result of an event ω,
f (ω) ≥ 0 is the amount paid by insurer as a result of an event ω,
We define this type of insurance as proper.
”The Kelly criterion and optimal insurance of capital” – p.13/17
Optimisation of insurance
We call the random variable z(ω) the logarithmic rate of increment from the uninsured
capital.
”The Kelly criterion and optimal insurance of capital” – p.14/17
Optimisation of insurance
We call the random variable z(ω) the logarithmic rate of increment from the uninsured
capital.
Assume that a profit of the insurer is not a random variable. Let us γ be the logarithmic
rate of increment of the insurer. Then the insurance premium is equal to
R
α = eγ ω f (ω)Pdω .
”The Kelly criterion and optimal insurance of capital” – p.14/17
Optimisation of insurance
We call the random variable z(ω) the logarithmic rate of increment from the uninsured
capital.
Assume that a profit of the insurer is not a random variable. Let us γ be the logarithmic
rate of increment of the insurer. Then the insurance premium is equal to
R
α = eγ ω f (ω)Pdω .
Let us ζ be the expected value of the exponential rate of increment from the insured
capital, we obtain
„ „
««
z(ω)
` z(ω)
´
R
e
+f (ω)
ζ(f, γ) = E ln
ln
e
+
f
(ω)
Pdω − ln(1 + α)
=
ω
1+α
”The Kelly criterion and optimal insurance of capital” – p.14/17
Optimisation of insurance
When will the inssured gain the biggest profit? This is the basic problem of underwriter.
Let us observe that, we can find the biggest profit and do not impoverish the client
simultaneously. So, we search for the function f ∗ (ω) that fulfils the condition
ζ(f ∗ , γ) = max ζ(f, γ)
f
and the highest admissible logarithmic rate γ ∗ for which ζ(f ∗ , γ ∗ ) = 0 .
”The Kelly criterion and optimal insurance of capital” – p.15/17
Optimisation of insurance
When will the inssured gain the biggest profit? This is the basic problem of underwriter.
Let us observe that, we can find the biggest profit and do not impoverish the client
simultaneously. So, we search for the function f ∗ (ω) that fulfils the condition
ζ(f ∗ , γ) = max ζ(f, γ)
f
and the highest admissible logarithmic rate γ ∗ for which ζ(f ∗ , γ ∗ ) = 0 .
We can find a maximum of the function ζ(f, γ) with initial conditions: α = eγ
R
and ω Pdω = 1 .
R
ω
f (ω)Pdω
”The Kelly criterion and optimal insurance of capital” – p.15/17
Optimisation of insurance
When will the inssured gain the biggest profit? This is the basic problem of underwriter.
Let us observe that, we can find the biggest profit and do not impoverish the client
simultaneously. So, we search for the function f ∗ (ω) that fulfils the condition
ζ(f ∗ , γ) = max ζ(f, γ)
f
and the highest admissible logarithmic rate γ ∗ for which ζ(f ∗ , γ ∗ ) = 0 .
We can find a maximum of the function ζ(f, γ) with initial conditions: α = eγ
R
and ω Pdω = 1 .
R
ω
f (ω)Pdω
The method of insurance provides the maximum possible rate of increment from the
insured capital is equal to
∗
“
f (ω) = max 0, e
−γ
(1 + α) − e
z(ω)
”
.
”The Kelly criterion and optimal insurance of capital” – p.15/17
Optimisation of insurance
The maximum rational rate of increment of the insurer is equal to
γ∗ =
Z
Pdω
f ∗ (ω)6=0
!−1 Z
(z(ω) − ln(1 + α)) Pdω .
f ∗ (ω)=0
”The Kelly criterion and optimal insurance of capital” – p.16/17
Plans on future
Employment of the Kelly criterion for pricing of option.
”The Kelly criterion and optimal insurance of capital” – p.17/17