Conference rusure::math

1656.0. "An optimization problem" by EPIK::FINNERTY (Sell high, buy low) Fri Aug 28 1992 13:28

    
    I have an optimization problem.
    
    Let o(t) be a random variable with value 0 or 1, let o(t) = 1 with
    constant probability p < 1, and let y1 and y2 be two positive real 
    numbers. Choose the constant R in the range (0..1) which maximizes 
    the expected value of V(t) for t >> 1/p.  
    
        V(0) = V0 (> 0)
    	V(t) = ((1-R) * V(t-1)) + (R * V(t-1) * (o(t)*y1 - (1-o(t))*y2))
    
    Or a more intuitive description: Given V(t) $, you choose to re-invest
    some fraction R into a risky investment.  The investment has probability
    p of yielding y1*investment, and has probability (1-p) of yielding
    -y2*investment (not really a "Yield", but you get the general idea).  
      
    What fraction R should be reinvested as a function of the known values
    y1, y2, and p?  The constraint t >> 1/p ensures that many 'failures' 
    will occur, and therefore V(t) will not be monotonically increasing
    for R > 0.
    
       /jim                    

    
    btw, t is a non-negative integer.  /jim
    

    V(t) = ((1-R) * V(t-1)) + (R * V(t-1) * (o(t)*y1 - (1-o(t))*y2)) =
         = V(t-1) * [ (1-R) + R * V(t-1) * (o(t)*y1 - (1-o(t))*y2) ] =
         = V(t-1) + V(t-1) * R * [ -1 + o(t)*y1 - (1-o(t))*y2 ] =
         = V(t-1) + V(t-1) * R * f(t)

            / -1 + y1  (probability p)
    f(t) = <
            \ -1 - y2  (probability 1-p)

It *seems* that if y1 is "big enough" we want R=1, else we want R=0.
Is there a possibility for any other R as a solution?

/Nitsan

    This looks like the "optimal f" problem from _Portfolio Management 
    Formulas_, by Ralph Vince.  ISBN 0-471-52756-4, 1-800-CALLWILEY, $50.
    Well worth it.
    
      John

    
    re: .2
    
        line 2 contained an extra V(t-1)
    
    re: .0
    
        viewed as a decision tree, this became pretty straightforward.  I
        can post some results if anyone is interested.  I suspect that .3
        has the complete analysis.
    
    /Jim