An investment problem
in [Math]
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From: analyst41 on 24 Jul 2010 07:55 I would appreciate any suggestions for this problem. There are N investment opportunities, and the investor's starting wealth is B. At the start of each period, the investor is made aware of a subset ("none" and "all" included) of the opportunities that are available. He may split his starting wealth as he pleases among the available opportunities. Opportunity i has a single period average return r(i) > 0 (reasonable assumptions can be made as to the variance of the return - although the average return is positive, there will be periods when the return is negative). At the end of the period he closes out all investments that were chosen for that period and starts out all over again. What would be his best strategy to maximize his long-term wealth (any assumption can be made for returns from uninvested funds (zero or some riskless return)) ?
From: Chip Eastham on 24 Jul 2010 09:17 On Jul 24, 7:55 am, "analys...(a)hotmail.com" <analys...(a)hotmail.com> wrote: > I would appreciate any suggestions for this problem. > > There are N investment opportunities, and the investor's > starting wealth is B. > > At the start of each period, the investor is made aware > of a subset ("none" and "all" included) of the opportunities > that are available. > He may split his starting wealth as he pleases among the > available opportunities. Opportunity i has a single period > average return r(i)> 0 (reasonable assumptions can be made > as to the variance of the return - although the average > return is positive, there will be periods when the return > is negative). At the end of the period he closes out all > investments that were chosen for that period and starts > out all over again. > > What would be his best strategy to maximize his long-term > wealth (any assumption can be made for returns from > uninvested funds (zero or some riskless return)) ? I think you have to decide how "risk" will enter your formulation of this "portfolio control problem". Two possibilities are: 1) to maximize (the expected) rate of return, subject to some specified limit (upper bound) on "risk" (variance, if we model the investment opportunities as stochastic processes), and 2) to minimize the risk subject to some specified lower bound on the rate of return. In other words formulate the problem as a tradeoff between risk and (expected) rate of return. You are basically discretizing the time axis with these investment periods, so in this sense it is a discrete control problem. I'd look at a "finite horizon" formulation, in which there are only a fixed number of investment periods over which "planning" is to be done. In practice of course the difficulty is in obtaining information about parameters of risk and rate of return. Another search term you may want to look for is "utility of wealth". If (say) ten million dollars were exactly ten times as "useful" as one million dollars, then we'd call that a "risk neutral" utility function. If concave (down), it's called risk averse; if convex (concave up), risk-loving. regards, chip
From: Ray Vickson on 24 Jul 2010 13:36 On Jul 24, 4:55 am, "analys...(a)hotmail.com" <analys...(a)hotmail.com> wrote: > I would appreciate any suggestions for this problem. > > There are N investment opportunities, and the investor's starting > wealth is B. > > At the start of each period, the investor is made aware of a subset > ("none" and "all" included) of the opportunities that are available. > He may split his starting wealth as he pleases among the available > opportunities. Opportunity i has a single period average return r(i)> 0 (reasonable assumptions can be made as to the variance of the > > return - although the average return is positive, there will be > periods when the return is negative). There is not enough information here. You need to know the whole probability distribution of returns, not just the mean and variance. However, you might make a reasonable assumption, based on investment studies, such as a lognormal distribution; see http://en.wikipedia.org/wiki/Log-normal_distribution . In such problems a VERY important issue is "covariance" between the difference investments: when woolen underwear sells slowly, ice cream sells quickly, etc. Most investment planners who do mathematical modeling will take great care to include such covariance effects. Again, however, you might try to get by with a simpler "Beta index" type of model; see http://en.wikipedia.org/wiki/Beta_%28finance%29 . > At the end of the period he > closes out all investments that were chosen for that period and starts > out all over again. > > What would be his best strategy to maximize his long-term wealth (any > assumption can be made for returns from uninvested funds (zero or some > riskless return)) ? Have a look at the "Kelly criterion"; see http://en.wikipedia.org/wiki/Kelly_criterion . R.G. Vickson
From: achille on 24 Jul 2010 13:51 On Jul 24, 9:17 pm, Chip Eastham <hardm...(a)gmail.com> wrote: > On Jul 24, 7:55 am, "analys...(a)hotmail.com" <analys...(a)hotmail.com> > wrote: > > > > > I would appreciate any suggestions for this problem. > > > There are N investment opportunities, and the investor's > > starting wealth is B. > > > At the start of each period, the investor is made aware > > of a subset ("none" and "all" included) of the opportunities > > that are available. > > He may split his starting wealth as he pleases among the > > available opportunities. Opportunity i has a single period > > average return r(i)> 0 (reasonable assumptions can be made > > as to the variance of the return - although the average > > return is positive, there will be periods when the return > > is negative). At the end of the period he closes out all > > investments that were chosen for that period and starts > > out all over again. > > > What would be his best strategy to maximize his long-term > > wealth (any assumption can be made for returns from > > uninvested funds (zero or some riskless return)) ? > > I think you have to decide how "risk" will > enter your formulation of this "portfolio > control problem". > > Two possibilities are: 1) to maximize (the > expected) rate of return, subject to some > specified limit (upper bound) on "risk" > (variance, if we model the investment > opportunities as stochastic processes), > and 2) to minimize the risk subject to some > specified lower bound on the rate of return. > In other words formulate the problem as a > tradeoff between risk and (expected) rate > of return. > > You are basically discretizing the time > axis with these investment periods, so > in this sense it is a discrete control > problem. I'd look at a "finite horizon" > formulation, in which there are only a > fixed number of investment periods over > which "planning" is to be done. In > practice of course the difficulty is in > obtaining information about parameters > of risk and rate of return. > > Another search term you may want to look > for is "utility of wealth". If (say) ten > million dollars were exactly ten times as > "useful" as one million dollars, then we'd > call that a "risk neutral" utility function. > If concave (down), it's called risk averse; > if convex (concave up), risk-loving. > > regards, chip Third approach, optimize the Sharpe ratio, ie. expected return of your portfolio ------------------------------------ variances of the returns In a typical trading company, only investment strategies with Sharpe ratio > 1 are acceptable (when the returns and variances are expressed in annualized units). BTW, this and the two approaches suggested by OP are for rational investors. Many studies show that average human does not invest rationally. In particular, they tend to treat loss twice as powerful as gains. You might need to take this into consideration for designing any real life investment strategies (the keyword to google here is "loss aversion").
From: achille on 24 Jul 2010 20:19 On Jul 25, 1:51 am, achille <achille_...(a)yahoo.com.hk> wrote: > On Jul 24, 9:17 pm, Chip Eastham <hardm...(a)gmail.com> wrote: > > > > > On Jul 24, 7:55 am, "analys...(a)hotmail.com" <analys...(a)hotmail.com> > > wrote: > > > > I would appreciate any suggestions for this problem. > > > > There are N investment opportunities, and the investor's > > > starting wealth is B. > > > > At the start of each period, the investor is made aware > > > of a subset ("none" and "all" included) of the opportunities > > > that are available. > > > He may split his starting wealth as he pleases among the > > > available opportunities. Opportunity i has a single period > > > average return r(i)> 0 (reasonable assumptions can be made > > > as to the variance of the return - although the average > > > return is positive, there will be periods when the return > > > is negative). At the end of the period he closes out all > > > investments that were chosen for that period and starts > > > out all over again. > > > > What would be his best strategy to maximize his long-term > > > wealth (any assumption can be made for returns from > > > uninvested funds (zero or some riskless return)) ? > > > I think you have to decide how "risk" will > > enter your formulation of this "portfolio > > control problem". > > > Two possibilities are: 1) to maximize (the > > expected) rate of return, subject to some > > specified limit (upper bound) on "risk" > > (variance, if we model the investment > > opportunities as stochastic processes), > > and 2) to minimize the risk subject to some > > specified lower bound on the rate of return. > > In other words formulate the problem as a > > tradeoff between risk and (expected) rate > > of return. > > > You are basically discretizing the time > > axis with these investment periods, so > > in this sense it is a discrete control > > problem. I'd look at a "finite horizon" > > formulation, in which there are only a > > fixed number of investment periods over > > which "planning" is to be done. In > > practice of course the difficulty is in > > obtaining information about parameters > > of risk and rate of return. > > > Another search term you may want to look > > for is "utility of wealth". If (say) ten > > million dollars were exactly ten times as > > "useful" as one million dollars, then we'd > > call that a "risk neutral" utility function. > > If concave (down), it's called risk averse; > > if convex (concave up), risk-loving. > > > regards, chip > > Third approach, optimize the Sharpe ratio, ie. > > expected return of your portfolio > ------------------------------------ > variances of the returns > > In a typical trading company, only investment strategies > with Sharpe ratio > 1 are acceptable (when the returns > and variances are expressed in annualized units). > > BTW, this and the two approaches suggested by OP are > for rational investors. Many studies show that average > human does not invest rationally. In particular, they > tend to treat loss twice as powerful as gains. You might > need to take this into consideration for designing any > real life investment strategies (the keyword to google > here is "loss aversion"). Oops, should be expected return / standard derivations of returns not variances. sorry for the confusion :-p
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