## Summary

- Simple relationships are shown for the dollar index and changes in the inflation rate versus the real gold price and annual changes in the real gold price.
- These variables are then included in a multiple regression with real interest rates and annual changes in the real interest rate for 1975-2013.
- Real interest rates and changes in those rates have a dominant influence on real gold prices and annual changes in real gold prices.
- Changes in the inflation rate have only a minor effect, because the Fed appears to be late in changing interest rates in response to changes in the inflation rate.

In Part I of this treatise, I found a strong relationship between the real price of gold (the July gold price divided by the CPI for the preceding month), the real interest rate as measured by the 3-month Treasury bill rate minus the inflation rate, and changes in that real interest rate. Annual changes in the real gold price were also closely related to these two variables. Here, in Part II, I will add two more variables that should influence the real price of gold.

## The Value of the Dollar

In Part I, I argued that we didn't have to worry about inflation in other countries affecting their demand for gold, because the value of their currencies should reflect differential geographic inflation rates. If a country has inflation higher than that in the US, its currency should be under pressure, and so its investors should be inclined to purchase US assets. In the case of buying US T-notes, they have a choice. They may buy the US dollar (Treasury bills) or gold. They should therefore respond just like rational US investors. They should buy T-bills if the real interest rates are positive. If I control for real interest rates, the value of the US dollar relative to other currencies shouldn't have an effect on the price of gold. That is, if real interest rates are high in the US, that should increase the value of the US dollar, and lower the price of gold at the same time. Let's look at the simple relationship first.

My independent variable is the adjusted major currencies dollar index, which I got from the Atlanta Federal Reserve. In the first scatter below, the linear correlation coefficient is -.32, although a negative exponential curve would describe the scatter much better. As the dollar index increased the real price of gold decreased, as anticipated. When the dollar is in demand, presumably people are responding to higher interest rates available in the US, rather than gold. We will see if real interest rates are the primary influence when both variables are tried in a multivariate model.

The next scatter has a correlation coefficient of r = -.40. When the dollar went down, the real price of gold went up. The pattern is there even if you remove the outliers. In fact, the outliers weaken the relationship. The outliers are 1986, when the dollar fell a lot and the real price of gold was unchanged, 1980 when the dollar was unchanged but gold rose a lot (the year of low real interest rates), and 1981 when the dollar rose a lot in association with a large increase in real rates and gold fell.

## Changes in the Inflation Rate

Inflation is a rate of change. It is the rate at which money loses value relative to the things that money can purchase. On average, the rate of increase in the price of gold should be correlated with the inflation rate, and Jeffrey Jones shows that it is, but with large deviations, that may last for many years. Erb and Harvey show this also. In addition to real interest rates, it should be expected that the rate of change in the real gold price will accelerate if the inflation rate accelerates; that is, if the rate of inflation is getting worse. Real gold prices should increase, in that case.

R = .25 in the above graph. The correlation coefficient indicates that when the annual inflation rate is high in any given year, the real gold price tends to be higher than average. But, the r-square is only .06, which is very weak, and much of the relationship is because of one year with very high inflation, 1980. The real price of gold was also above average in 2009, when we briefly had deflation.

The next graph looks at the effect of a change in the inflation rate.

R = .11. Curiously, this graph shows no relationship between the real gold price and whether inflation was higher or lower than the previous year. But, when I relate the annual change in the real gold price to the annual change in the inflation rate, I do get a positive relationship of r = .41 (below). The strength of this relationship is greatly influenced by just two years: 1980, when the inflation rate was already high and jumped some more, and real gold prices jumped, and 1981, when the inflation rate declined from a very high level to a still high level (10%), and the real price of gold plummeted. 1981 was also the year when real interest rates jumped from very negative to very positive.

The other notable year in the graph below is 2009, when inflation went from 5.6% to -2.1%, but the real price of gold was virtually unchanged from July 2008 to July 2009. Remember that the price of gold did plummet in 2008, and rallied in 2009.

## The Multivariate Models

These simple relationships have shown that certain critical years have muddied or highlighted the relationship between the real gold price and its primary determinants. In 1979-1980, the nominal and real price of gold were in a 4-year uptrend and jumped to exceptional levels in association with a high inflation rate, an increase in that inflation rate, a very negative real interest rate, a decline in that real interest rate, and a cheap dollar which was largely unchanged from the previous year. In 1981, gold collapsed, reversing its uptrend, in association with a high, but decreased inflation rate, a sharp increase in the real interest rate to a large positive level, and a sharp increase in the value of the US dollar. 1981 was also the year that the Hunt brothers' attempt to corner the silver market went awry. Did that affect the gold price, or did these other factors cause the price of silver to collapse?

Can we separate the role of the various factors from each other, and control for the fact that the contributing factors to the gold price are correlated with each other? The answer is, yes. Multiple regression attempts to do that. I explained it in Part I. In sum, multiple regression tries to find the co-variation between two variables, while the other variables do not vary.

## Model 1

My first model is the real price of gold regressed against last year's real gold price, the dollar index, the real interest rate of the 3-month Treasury bill, the spread between 3-month and 10-year T-bond rates, the inflation rate, and the annual change in the inflation rate.

The initial model returned an r-sq of .738, primarily because of the variable, last year's real gold price. The t-statistic is a good descriptor of the strength of the relationship between two variables in a particular data set, and so I used it to dump independent variables, which were adding little to the multiple r-sq for the regression.

My final version of model 1, as a result, was:

Predicted real price of gold = .922 + .81 (last year's real gold price) - .183 (the real 3 month interest rate) + .07 (change in the inflation rate).

The r-sq was .729; so deleting the other variables changed nothing, and the r-squared is little different from the model in Part I, which had only last year's real gold price and the real interest rate as the independent variables. So change in the inflation rate does not add much explanation to the variation in real gold prices.

The previous year's real gold price had the greatest effect on the real gold price by far, and the relationship with the real interest rate was strong. The change in the inflation rate had only a minor independent influence on the real gold price. It was the change in the real interest rate that mattered. Yes, if inflation accelerated by 1 percentage point (from 10% to 11% say), the independent effect was to increase the real price of gold by .07 (the plus .07 coefficient), but that's not a big effect.

Remember, the real price of gold is measured as the nominal price in dollars divided by the CPI index value for that month. So in July of 1980, the nominal gold price was $644, the CPI index was 82.70, and therefore, the real price of gold was 7.79, its all-time high in this 40-year time span. In July of 1981, the CPI was at 91.60, a 10.76% inflation rate, down some from the 13.13% rate of the previous 12 months. So the real price of gold should have gone down a little, but not by as much as it did, given that inflation was still very high. Volcker hiked interest rates from 8.62% to 14.87% to combat the inflation. The real rate of interest went from -4.51% to +4.11%, and this change overwhelmed the effect of still-high inflation on the real gold price. The nominal price fell to $409.28, and the real price dropped to 4.47.

The relationships in the multiple regression are quite dependent on just a few years, but I think those occasions matter. Most years, gold prices, real interest rates, and the inflation rate didn't vary much, but in a few years, interest rates and inflation moved a lot, and the response of gold prices was major and obvious. So I can explain variation in real gold prices to some extent, at least for the past 40 years, but can I predict the future movement in real gold prices?

By plugging the values for each year into the equation, I can get a predicted value (Yhat) for the real gold price for each year. I then subtract Yhat from the actual real gold price each year, and get the prediction error or residual. If the residual is positive, then the actual price was higher than the predicted price (i.e. gold was overpriced), and if the residual is negative, the actual price was less than the predicted price (i.e. underpriced). The question then is, if gold is overpriced, does it tend to be underpriced the next year, and vice-versa. Correlation of the residuals with next year's real gold price should generate an inverse correlation.

Dang. No inverse correlation. Actually, the correlation was positive (r = .47). Overpriced gold tended to stay overpriced, and vice-versa. There was also no correlation between the residuals and the real gold price 5 years in the future. So there was no indication of when the overpriced trend or underpriced trend would change.

The residuals were highly auto-correlated, reflecting the 3 major trends in gold prices during those 40 years. The residuals were generally negative from 1989 to 2005, meaning gold was underpriced according to the equation, but the real price of gold kept falling until 2001. After that, it rose every year, and the residuals became positive from 2005 until 2012. The average real price of gold over the 40 years was 3.31. In 2013, it was 5.51 ($1287 nominal price), down from 6.96 the previous year. The equation indicates the real price should have been 6.89, because a negative real interest rate became more negative. So the residual was negative. Gold is underpriced currently, but as I have just indicated, it tends to stay underpriced or overpriced for many years in a row.

Notice, however, that I didn't actually look at the annual change in price. Gold can remain overpriced, but the real price might still decrease. So I should correlate the residuals with next year's *change* in the real gold price, rather than the price itself. If gold is overpriced, according to the equation, then we should expect a drop in its real price, i.e. an inverse correlation. Well, the correlation coefficient between the residuals and the change in real price was minus .149, which essentially means no linear correlation. Was it Yogi Berra who said, prediction is very hard, especially when it's about the future?

## Model 2

The second model regressed the annual change in the real price of gold against last year's changes in the real price of gold, the 3-month real rate of interest preceding the annual change, the change in the 3-month real rate of interest, the change in the inflation rate, the change in the spread, and the change in the dollar index. The changes are not in percentages, but actual changes, and they are concurrent.

The initial r-sq was .443. Again, I used t-values to remove independent variables that seemed to have little independent effect on the real gold price changes. The previous year's change in the real gold price had no effect on the next year's change in real gold price. So, there was no auto-correlation in the changes from year to year, which surprised me.

I was also surprised to see that the independent effect of an increase in the inflation rate was to decrease the real gold price. It was a very weak effect, so the variable was deleted, but it appeared to make sense until I looked at the simple relationship between the change in the inflation rate and the change in the real interest rate. That correlation was -.705. Note that this is an inverse correlation. When inflation accelerated, real interest rates decreased (and went or were negative). My interpretation is that *the Fed was usually late in raising interest rates when the inflation rate was increasing, and late in lowering interest rates when there was disinflation*. What I was seeing was that in a few critical years, the market anticipated that the Fed would have to greatly increase rates to address inflation, and therefore sold gold, as inflation was still rising. This may be the reason why gold prices fell a lot in 2013. By the time real interest rates started to dramatically rise, gold was already well down, and so we see a good relationship between real interest rates, the change in real interest rates, and the change in real gold prices, but not with the change in inflation rates. And that is probably why other studies have found little relationship between annual inflation rates and gold price changes. They did not control for interest rates, and so they could not see that when inflation is high, gold will be sold in anticipation of higher real interest rates, and that when inflation is low, gold will be bought in anticipation of lower real interest rates, as in 2001. It wasn't the 9/11 event itself, it was the lower interest rates because of 9/11 which raised gold prices.

The final model was:

Predicted real price change = .202 - .2605 (change in the real interest rate) - .0309 (change in the dollar index) - .1286 (the 3-month real interest rate at the start of the time change).

The r-sq was .437, which is pretty awesome until you recognize that to predict the change in the real gold price, you need to be able to predict the change in the inflation rate, the change in interest rates, and the change in the dollar index, because these are all concurrent changes. Economists are renown for their lack of foresight. How many predicted that long-term interest rates would fall when the Fed started tapering QE this year? So don't rely on anyone's predictions; you are better off just using last year's values to predict next year's values.

I tried correlating the residuals against the next year's price change. The idea was that if the equation under-predicted the price change of gold, there would be reversion to the mean. If the gold price growth were much faster than predicted in one year, would it be lower in the next? No such luck. The correlation was very weak (r = -.22, r-sq = .05).

Would previous year changes in the dollar index and in the real interest rate help predict the gold price change the following year? Nope! Curiously, when I regressed the real gold price change against the concurrent changes in real interest rates and the dollar index, the level of real interest rates at the start of the change period had a strong inverse effect on the change in real gold prices. But, the same variable, when coupled with the previous year's changes in real interest rates and the dollar index, did not have a strong effect on the change in real gold prices, although its effect was still negative (high real interest rates were associated with smaller increases or decreases in real gold prices).

Returning to the equation above, what does it tell us? On average, if the real interest rate and the dollar index do not change (this is called ceteris paribus), a negative real interest rate at the start of the year (July to July in this study) was associated with a rise in the real price of gold. A positive real interest rate was associated with a decline in the real price of gold. The higher the real interest rate, the bigger the decline. If real interest rates rose during the one-year period, ceteris paribus, the real price of gold declined over that time period. So if the real rate was negative but became less negative, the gold price declined. If the real rate was positive but declined, the price of gold rose. The change in the real interest rate had the greatest individual effect on the change in the real gold price. If the real interest rate did not change, an increase in the value of the dollar was associated with a decline in the real price of gold, and a decline in the value of the dollar raised the price of gold. Presumably, if the dollar rose, that reflected a desire by foreigners to own the US dollar, rather than gold or their own currency. If the dollar fell, presumably people would rather own gold or a foreign currency in preference to the US currency. Of the three independent variables, the change in the dollar index had the weakest effect on gold prices.

An r-sq of .44 is pretty impressive. I found only 3 years where the model's error was somewhat large and got the direction of the real price change wrong. The worst was last year, 2012-13. The model predicted a small increase in the real price of gold because real rates were negative, and became more so as a result of a slight increase in inflation. The nominal price of gold fell $307 per oz. Perhaps it was a reaction to very high real prices of gold in the previous two years. Curiously, this was the year the Asians were supposedly accumulating gold. Clearly, some people/institutions were very eager to sell it to them.

In 1982-83, the model predicted a significant decline in real gold prices associated with a high real interest rate, which increased a lot. But the nominal price of gold rose from $339 to $423. Perhaps, this was an oversold bounce, because gold had declined from $644 in 1980 to $339 in 1982. Gold did continue its decline after that to $317 in 1985. A substantial decline for gold was predicted for 2008-9, because the brief deflation caused real interest rates to jump. In fact, nominal gold prices were largely unchanged.

So, what do we do with this model, which still requires that you predict inflation rates and interest rates to predict gold prices? Well, perhaps it could be used in conjunction with Jeffrey Jones' 12-month moving average model. For example, in 1983, Jeffrey's model probably had gold moving above its 12-month MA, but real interest rates were high and rising, and so the buy could have been ignored.

My thoughts on the next few years include the observation that the decline of the last two years is so large that we must entertain the possibility/probability that a new multi-year declining trend is underway. But, how can real interest rates possibly become positive again, especially since it will be very difficult for the Fed to raise interest rates given all those excess bank reserves. They have to start selling their Treasuries in great quantities to raise rates. If they did that, a recession would be swift and deflationary. If they don't sell their Treasury holdings, we might have deflation anyway, which would cause positive real interest rates. Unlike the last deflation of the 1930s, when gold rose because the government fixed the price, deflation in a free market should lower the price of gold substantially.

If inflation returns and the Fed doesn't raise rates to fix it, then gold prices will rise. But, I think the odds of rising inflation are lousy. You need much higher aggregate demand to increase inflation, and where is that demand going to come from with monetary stimulus and fiscal stimulus declining, populations aging, and workers not being rewarded (via pay increases) for their higher productivity.

As of the beginning of April (2014), the model for the real price of gold indicates it should be at 6.69 (nominal = $1581) versus an actual price of 5.42 ($1280 nominal). The model for the annual change in real price indicates that from April to April, the nominal price should have increased by $112, whereas it in fact, fell by $320 (-25%). So gold is underpriced. But as I showed above, whether the model indicated that gold was over- or underpriced, it didn't help predict the price one year later. My speculation is that you have to respect the trend; that the drop in price since 2012 is so large that it marks a change in the multi-year trend. Gold may not change much in price for a while, because we do have negative real interest rates, and they are likely to persist unless we have deflation. Then, gold prices should go down because the situation is different from 1934, when gold prices were fixed and had not been adjusted to reflect the inflation that had occurred since the early 1900s.

Remember, my regressions are for July to July. Using a different month will change the parameter values, but I would neither expect a change in the direction of the relationships nor a major change in the strength of the relationships. I welcome any attempt to test the stability of the parameters by testing the relationship on other months, but it's too much work for me, given that I expect little reward from the exercise.

I have not been long or short on gold or its derivatives this year. There are more remunerative investments, in my opinion.

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