Forecasting a Price by Breaking It Down

Francis Ng
Department of Management Science and Engineering
Stanford University
June 2002

My research is a branch of applied mathematics that forecasts future prices. I make forecasts by breaking down an uncertain future price into a random private component and a random market component, which are then evaluated separately. The private component, such as machine maintenance cost, is independent of and therefore not affected by any assets traded in any markets, such as stocks and government bonds. On the other hand, the market component, such as revenue from sales, correlates with some of the market assets and therefore its worth can be derived from the price of market assets. Without a forecast, people negotiating the pricing structure of a contract cannot quantify a demand or a concession.

On a steaming day last summer, I walked into an antique shop in Shanghai, China, and an exquisite ivory incense holder caught my eye. I asked the shopkeeper how much it was and she said, "It was made in the 1920's and the price is 400 Renminbi." "That's 50 US dollars," I murmured to myself. I had no idea of what it should cost but I knew it would be foolish not to bargain. So I offered her 200 Renminbi and she happily accepted. I still do not know whether I paid a fair price or not but my gut feeling is that I did not. Multiply my predicament as a tourist a million times and you would have the predicament of a purchasing manager negotiating the price of a multi-million dollar long-term contract. She has no clue what the price will be five months from now in a volatile market.

It is often difficult to forecast a future price as it fluctuates too frequently and too much. For example, lumber prices have an annual standard deviation of 30%. This means that if the price is $180 per thousand square feet at the beginning of the year, there will be a 32% chance that it will end up higher than $240 or lower than $120 by the end of the year. The problem of forecasting is further aggravated by the lack of past price data. Therefore, when buyers and sellers negotiate the pricing structure of a contract, they often do not know how to quantify a demand or a concession. All they know is the more favorable the terms are the better.

My research is a branch of applied mathematics that forecasts future prices by projection.
I first use projection to break down an uncertain future price into a random private component and a random market component. The private component, such as machine repair cost in a factory, is independent of and therefore not affected by any market assets such as stocks and government bonds. On the other hand, the market component, such as revenue from sales, correlates with some of the market assets and therefore its worth can be derived from the price of those assets. The two components do not affect each other just as machine repairs do not affect revenue from sales. Therefore, I can evaluate them separately. I then quantify the private component by its expected value and estimate the market component by using market assets that correlate most closely with it.

Using the expected value to price the private component is like evaluating a bet in which one would get $10 for guessing the right side of a coin flip and $0 otherwise. The bet is worth $5 to most people. However, this argument is only valid as long as the contract makes up a small portion of the total portfolio of a company, because otherwise the private component should not be priced at the expected valued. To see this, one can ask if one would pay $5 million to play if one would get $10 million for guessing the right side of a coin flip but $0 otherwise. One would probably value the bet significantly less than $5 million. It is because when the stake becomes too high, one would be afraid to take the risk at its expected value.

On the other hand, pricing the market component by using the market prices of things that closely correlates with it is like pricing one's used car by checking out the sale prices of other used cars on the streets. Finally, I predict the future price by adding the private and market components together. By utilizing the abundance of the data of closely related market assets, my forecast is more accurate than prediction based solely on the historical pricing data of the product in the contract.