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one way to hedge well-diversified equity portfolio is to reduce its beta or to neutralize its beta and here I'd like to show in two steps with a simple and then the more complicated example that follows John hole I'd like to try and show the intuition for how we determine the number of futures contracts that would reduce or neutralize the portfolio's beta and the key is to think about the dollar beta which is the product of the portfolio's value and its beta that's the dollar beta so here is the formula for the special case of changing the beta of a well-diversified equity portfolio and we know from experience that it can be hard to access the intuition of this formula and so for that reason I have this in I'm illustrating this in two steps the first step here is my simple case and then the second step generalizes by showing you John Hall's example and the key idea I think my opinion here is to focus on the dollar beta more than the beta so if we look at this formula right here if we just back this out and start at the first step what we have here on the right is the dollar beta of the underlying portfolio or the portfolio that we call the underlying exposure and dollar beta is simply the beta of the portfolio multiplied by the value of the portfolio very straightforward if we wanted to neutralize the beta of that portfolio all we would do is hedge or in this case use short futures contracts with a exactly offsetting dollar beta and the dollar beta of that of the futures contracts would simply be the number of futures contracts times the notional value per contract multiplied by the beta of each contract so very similar except that we're using here the S&P 500 index futures contracts where we can roughly assume that the beta is one so that drops out conveniently and so that dollar beta of the S&P 500 futures contract is going to be simply the number of contracts multiplied by the notional value per contract so if we wanted to neutralize the beta of the portfolio which is the underlying exposure we would simply short the number of contracts or assuming our portfolios a long position we'd short the number of contracts such that our dollar our dollar beta was exactly offsetting and you can see how easy this solves for n if we just if we are if we comprehend this equality of dollar beta idea solving for n then the number of contracts to neutralize the beta is simply the portfolio of this beta multiplied by the ratio of values that is the portfolio in the numerator and then the denominator the notional value of a single futures contract and so over here just some simple assumptions the sp500 index is currently higher than 2000 I'm rounding down try to keep it as...