Macroshares currently offers two paired funds, Major Metro Housing Up (UMM) and Down (DMM), and promises to roll out similar ones every year. I refer to them as funds because they trade like ETFs. They are different from EFTs in many ways and there are volumes on this site arguing over their virtues and vices – thus, read the prospectus carefully before investing. Here, let me gloss over the operational differences and focus on the math for determining their values and other characteristics.

Let *U* denote the dollar value of the up fund, *D* that of the down fund and *H* the S&P/Case-Shiller Composite-10 Home Price Index. The primary relationships among them are as follows

In the prospectus, three other factors are considered: income distribution and management cost and interest rate on the treasuries. The first two factors partially cancel out the last. The combined effect of the three factors is negligible. The above two equations, neglecting other factors, generate calculated values that are agreeable with those in the prospectus with less than 1 cent error.

On any day, the values of these funds are determined by the home price index, absent premium or discount brought about by trading on the market. An interesting observation from here is the limits the relationships place on the range of the index. At 217, it pushes the up fund to 50 and the down fund to 0. At 109, the other extremes occur. In each of the extreme conditions, the funds may declare early termination.

The prospectus promises a leverage factor of 3. That is, if the index increases by 1%, the up fund will go up by 3%. On Page 17 of the prospectus of the down fund, I find:

*For example, a 5% increase in the Reference Value of the Index will always result in a 15% increase in the per share underlying value of the Down MacroShares (after applying the leverage factor). However, once the assets of the Down Trust (not taking into account treasury income) are no longer equal to the aggregate par amount of the outstanding Down MacroShares, a 2.1624 index point fluctuation in the Reference Value of the Index will result in a change in the per share underlying value of a Down MacroShare that is equal to some value that is less than $1.*

This is grossly inaccurate, and even outright wrong. In fact, from our first equation, we can obtain the derivatives of UMM and DMM with regard to the home price index

That is, for each point change in the index, the up and the down funds change by $0.46245 invariably.

The percentage change in *U* is 100*dU/U*, and in *H*, 100*dH/H*. The leverage factor for UMM, therefore, is

and for DMM

Therefore, their rates of change vary with the index value.

The chart below depicts the variations of the values as well as the leverage factors of UMM and DMM with regard to the value of the home price index. As the index increases from the lower limit to the upper limit, the value of UMM ascends, while that of DMM descends, linearly with it. The leverage factor varies from infinity to 2 for UMM and 2 to infinity for DMM. Only when their values are at 25, the leverage factors are 3.

The variations of the leverage factors have important implications for traders. When UMM and DMM are at 35 and 15, respectively, the leverage factor is twice as large as that for UMM. If a trader is bullish on UMM, instead of long UMM, he should be short DMM which gives him double the bang on the buck.

At this point, if other traders also heed this piece of advice and go short on DMM, they can drive it to a discount, say down to 10, while UMM remains 35. The arbitrageurs then buys up DMM for redemption. But he has to redeem UMM and DMM in pairs, which forces him to buy UMM also. In his attempt to narrow the discount on DMM, he must raise the premium on UMM. His actions would push the funds' values further from balance with relationship to the index. Since the funds don't track the value of fundamental assets, his source of profit is only from the deviation of the funds' combined unit price from 50. He has to ignore the funds' relationship with the home price index.

The long-term value investor should have the funds' terminal value in mind. If you believe the index will be at 180 at the end of 2014, you are expecting UMM 33 and DMM 17. If you are long UMM all the way to the end, that's a 5.2% annualized return.

If you don't care for the equations and the chart, let me summarize the key takeaway points in plain words:

- The theoretical values of the funds are calculated on the home price index, while other factors are negligible or fixed. Traders may bet them to premiums or discounts.
- The leverage factors for the up and down funds are different and can deviate away from 3 in opposite directions.
- Arbitrage will profit on deviation from
*U*+*D*=50. - Neither the Macroshares trusts nor the arbitrageurs have control over the prices of the individual funds. Nor can they assure that the prices will track the home price index.