What is log-utility?
I came across this problem today:
Calculate the log-utility optimal fraction of your capital to bet on a fair coin flip where you win $x$ on heads and lose $y$ on tails.
What is the meaning of log-utility in log-utility optimal fraction? Is this an portfolio management term or a statistics term? Apologies if this is in the wrong SE, please close this if it’s irrelevant!
probability statistics utility
asked Nov 26 at 1:23
user107224
405314
add a comment |
I came across this problem today:
Calculate the log-utility optimal fraction of your capital to bet on a fair coin flip where you win $x$ on heads and lose $y$ on tails.
What is the meaning of log-utility in log-utility optimal fraction? Is this an portfolio management term or a statistics term? Apologies if this is in the wrong SE, please close this if it’s irrelevant!
probability statistics utility
asked Nov 26 at 1:23
user107224
405314
Log utility on value $c$ is a function $u(c) = log c$. (It doesn't matter what base it is, since these are all equivalent up to a constant factor.) It represents constant relative risk aversion. It's a term from economics.
– Brian Tung
Nov 26 at 1:56
add a comment |
I came across this problem today:
Calculate the log-utility optimal fraction of your capital to bet on a fair coin flip where you win $x$ on heads and lose $y$ on tails.
What is the meaning of log-utility in log-utility optimal fraction? Is this an portfolio management term or a statistics term? Apologies if this is in the wrong SE, please close this if it’s irrelevant!
probability statistics utility
asked Nov 26 at 1:23
user107224
405314
I came across this problem today:
Calculate the log-utility optimal fraction of your capital to bet on a fair coin flip where you win $x$ on heads and lose $y$ on tails.
What is the meaning of log-utility in log-utility optimal fraction? Is this an portfolio management term or a statistics term? Apologies if this is in the wrong SE, please close this if it’s irrelevant!
probability statistics utility
probability statistics utility
asked Nov 26 at 1:23
user107224
405314
asked Nov 26 at 1:23
user107224
405314
asked Nov 26 at 1:23
user107224
405314
asked Nov 26 at 1:23
user107224
405314
asked Nov 26 at 1:23
user107224
405314
405314
Log utility on value $c$ is a function $u(c) = log c$. (It doesn't matter what base it is, since these are all equivalent up to a constant factor.) It represents constant relative risk aversion. It's a term from economics.
– Brian Tung
Nov 26 at 1:56
add a comment |
Log utility on value $c$ is a function $u(c) = log c$. (It doesn't matter what base it is, since these are all equivalent up to a constant factor.) It represents constant relative risk aversion. It's a term from economics.
– Brian Tung
Nov 26 at 1:56
Log utility on value $c$ is a function $u(c) = log c$. (It doesn't matter what base it is, since these are all equivalent up to a constant factor.) It represents constant relative risk aversion. It's a term from economics.
– Brian Tung
Nov 26 at 1:56
Log utility on value $c$ is a function $u(c) = log c$. (It doesn't matter what base it is, since these are all equivalent up to a constant factor.) It represents constant relative risk aversion. It's a term from economics.
– Brian Tung
Nov 26 at 1:56
add a comment |
1 Answer
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If the initial capital is $1$ unit of wealth and the fraction bet is $f$, then the capital after betting on the coin toss is
$$W_1 = 1 + fxi$$
where $xi$ is a binary random variable. Assuming a fair coin and WLOG $x,y > 0$ enumerated in wealth units , we have $$P(xi = x) = P(xi = -y) = 1/2$$
The expected capital is
$$E(W_1) = E(1 + fxi) = 1+ f(x-y)/2$$
If $x >y$, you have an edge and the optimal fraction is $f^* = 1$ to maximize expected capital. However, if you are risk-averse you might not play to avoid a $50 %$ chance of losing $y$.
A utility function is a construct that assigns preferences to random outcomes (gains and losses). Specifically log utility was introduced by Bernoulli to resolve the St. Petersburg paradox
The optimal fraction with log utility is obtained by maximizing
$$E( log W_1) = frac{1}{2} log(1 + fx) + frac{1}{2} log (1 - fy)$$
subject to the constraint $0 leqslant f leqslant 1$.
answered Nov 26 at 2:08
RRL
48.6k42573
add a comment |
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1 Answer
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1 Answer
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active
oldest
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oldest
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oldest
votes
If the initial capital is $1$ unit of wealth and the fraction bet is $f$, then the capital after betting on the coin toss is
$$W_1 = 1 + fxi$$
where $xi$ is a binary random variable. Assuming a fair coin and WLOG $x,y > 0$ enumerated in wealth units , we have $$P(xi = x) = P(xi = -y) = 1/2$$
The expected capital is
$$E(W_1) = E(1 + fxi) = 1+ f(x-y)/2$$
If $x >y$, you have an edge and the optimal fraction is $f^* = 1$ to maximize expected capital. However, if you are risk-averse you might not play to avoid a $50 %$ chance of losing $y$.
A utility function is a construct that assigns preferences to random outcomes (gains and losses). Specifically log utility was introduced by Bernoulli to resolve the St. Petersburg paradox
The optimal fraction with log utility is obtained by maximizing
$$E( log W_1) = frac{1}{2} log(1 + fx) + frac{1}{2} log (1 - fy)$$
subject to the constraint $0 leqslant f leqslant 1$.
answered Nov 26 at 2:08
RRL
48.6k42573
add a comment |
If the initial capital is $1$ unit of wealth and the fraction bet is $f$, then the capital after betting on the coin toss is
$$W_1 = 1 + fxi$$
where $xi$ is a binary random variable. Assuming a fair coin and WLOG $x,y > 0$ enumerated in wealth units , we have $$P(xi = x) = P(xi = -y) = 1/2$$
The expected capital is
$$E(W_1) = E(1 + fxi) = 1+ f(x-y)/2$$
If $x >y$, you have an edge and the optimal fraction is $f^* = 1$ to maximize expected capital. However, if you are risk-averse you might not play to avoid a $50 %$ chance of losing $y$.
A utility function is a construct that assigns preferences to random outcomes (gains and losses). Specifically log utility was introduced by Bernoulli to resolve the St. Petersburg paradox
The optimal fraction with log utility is obtained by maximizing
$$E( log W_1) = frac{1}{2} log(1 + fx) + frac{1}{2} log (1 - fy)$$
subject to the constraint $0 leqslant f leqslant 1$.
answered Nov 26 at 2:08
RRL
48.6k42573
add a comment |
If the initial capital is $1$ unit of wealth and the fraction bet is $f$, then the capital after betting on the coin toss is
$$W_1 = 1 + fxi$$
where $xi$ is a binary random variable. Assuming a fair coin and WLOG $x,y > 0$ enumerated in wealth units , we have $$P(xi = x) = P(xi = -y) = 1/2$$
The expected capital is
$$E(W_1) = E(1 + fxi) = 1+ f(x-y)/2$$
If $x >y$, you have an edge and the optimal fraction is $f^* = 1$ to maximize expected capital. However, if you are risk-averse you might not play to avoid a $50 %$ chance of losing $y$.
A utility function is a construct that assigns preferences to random outcomes (gains and losses). Specifically log utility was introduced by Bernoulli to resolve the St. Petersburg paradox
The optimal fraction with log utility is obtained by maximizing
$$E( log W_1) = frac{1}{2} log(1 + fx) + frac{1}{2} log (1 - fy)$$
subject to the constraint $0 leqslant f leqslant 1$.
answered Nov 26 at 2:08
RRL
48.6k42573
If the initial capital is $1$ unit of wealth and the fraction bet is $f$, then the capital after betting on the coin toss is
$$W_1 = 1 + fxi$$
where $xi$ is a binary random variable. Assuming a fair coin and WLOG $x,y > 0$ enumerated in wealth units , we have $$P(xi = x) = P(xi = -y) = 1/2$$
The expected capital is
$$E(W_1) = E(1 + fxi) = 1+ f(x-y)/2$$
If $x >y$, you have an edge and the optimal fraction is $f^* = 1$ to maximize expected capital. However, if you are risk-averse you might not play to avoid a $50 %$ chance of losing $y$.
A utility function is a construct that assigns preferences to random outcomes (gains and losses). Specifically log utility was introduced by Bernoulli to resolve the St. Petersburg paradox
The optimal fraction with log utility is obtained by maximizing
$$E( log W_1) = frac{1}{2} log(1 + fx) + frac{1}{2} log (1 - fy)$$
subject to the constraint $0 leqslant f leqslant 1$.
answered Nov 26 at 2:08
RRL
48.6k42573
edited Nov 26 at 2:17
answered Nov 26 at 2:08
RRL
48.6k42573
answered Nov 26 at 2:08
RRL
48.6k42573
answered Nov 26 at 2:08
RRL
48.6k42573
48.6k42573
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add a comment |
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Log utility on value $c$ is a function $u(c) = log c$. (It doesn't matter what base it is, since these are all equivalent up to a constant factor.) It represents constant relative risk aversion. It's a term from economics.
– Brian Tung
Nov 26 at 1:56