USD and Gold provide a more accurate insight into the true state of the US economy than the SP500

 

Why?

(1  -60-70% of trades in the US equities market are done by automated AI-driven algorithms, including high frequency micro-second trading. These algorithms make their decisions not on economic or corporate fundamentals but on momentum, arbitrage opportunity, pattern recognition etc.

(2 - Previously the USD was a safe-haven currency that investors flock to in periods of economic, political or military instability. But with the chaos that is now going on in the USA, the USD seems to be losing its safe-haven status. As for Gold, it is still a safe haven asset, and recently we have seen the Central Banks of China, Russia, Japan and even some of the EU countires load up on Gold while paring down the amount of US Treasuries they hold.

(3-The USD is the lynchpin that holds the US stock market and the Treasuries market together and a falling USD will make any rise in stock or bond prices unsustainable. Would you invest in US stocks if your profit is 5% but the USD falls more than 5%.  Would you buy US Treasuries if the yield is 5% but the fall in the USD is more than 5%.  In both cases when you buy USD to invest and convert back to SGD when you sell, you get less SGD when USD falls.

Entropy as a measure of Predictability

In Physics, the Second Law of Thermodynamics talks about Entropy, a measure of randomness and disorder. The Law states that all things in the Universe will go from a state of lower to higher Entropy i.e from Order to Chaos, Structure to Randomness, Hot to Cold and Predictability to Unpredictability.

So, our hypothesis is (1) If the downtrend in the USD value is fundamental then its Entropy should be decreasing. (2) If the uptrend in Gold is fundamental then its entropy should also be decreasing. In other words, the fall in the USD and the rise in Gold is more predictable and likely to continue if their Entropies keep falling. We use the prices from  DXY the USD Index ETF, and GLD the Gold ETF.

We use Clause Shannon’s (the pioneer of Theory of Information) measure of Spectral Entropy to attempt to establish whether the USD is in a secular downtrend and Gold is in a secular uptrend. Shannon entropy of the normalized Power Spectral Density detects shifts from noise-dominated (high Entropy) regimes to trend-dominated (low entropy) regimes. Its  formula calculates the average amount of information needed to describe the outcome of a random event, considering the probability of each possible outcome. It is normalized to lie between 0 and 1 with high values depicting randomness and unpredictability while low values indicate a structure and degree of predictability.

Formula for Shannon Spectral Entrop # Note: log base2 is used because Information as per Shannon's Theory of Information measure in Bits viz Binary.

Chart 1 (above): Gold vs USD prices standardized by Z-Score. It shows Gold rising and USD falling.

Chart 2: Entropy changes in DXY

You can see that from Jan 2025 to 30 June 2025 the USD has more frequent and steeper falls in Entropy than previously.  The period from May to December 2024 has only two points when the Entropy was below -0.005 while from Jan to June 2025, there are 5 points below -0.005. Thus, the downtrend is more predictable and likely to continue.

Chart 3: Entropy changes in GLD

The uptrend in Gold prices is not so established. There is only 1 point where its Entropy is well below -0.005 and that is in May 2025. Thus, it is less predictable, though there appears to be regular cycles between predictability and unpredictability. It remains to be seen whether the May 2025 downward swing in entropy will continue for Gold to have an established uptrend. 

Chart 4: Below you can see that when USD becomes more unpredictable, Gold becomes more predictable and vice versa.

Addendum and Clarification.

The original Shannon Spectral Entropy calculation used a direct Fast Fourier Transform (FFT) to extract the various frequencies and plot the Power Spectral Density (PSD). But especially because we have a 60-day rolling window we use the Welch PSD. Frequencies differ because the two grids are tied to their segment length (30 vs 60). Individual power levels jump around in the FFT estimate, whereas Welch’s are less erratic; averaging overlapping segments cuts variance of each bin. This smooths the sharp ups and downs of FFT. Thus the advantages of Welch vs FFT are: 

1 Variance reduction

Averaging several shorter, overlapping segments reduces the sampling noise in each power bin. That’s crucial when each window is only 60 observations long.

2 Bias/consistency trade-off

You sacrifice a bit of frequency resolution (coarser bins) in exchange for a materially lower estimator variance, which yields a stabler entropy series.

3 Rolling-window stability

In a rolling framework, small return spikes can make a single FFT periodogram jump wildly from one day to the next. Welch’s averaging dampens those day-to-day swings, giving a cleaner signal.

The chart below compares FFT with Welch. The two curves depict Power Spectral Density estimated over the same 60-day DXY return window:

• Blue circles = Welch (30-point segments, 50 % overlap, then averaged).
• Orange squares = single-window FFT periodogram (all 60 points in one shot).
• You can see that both describe the same overall shape, but Welch looks smoother—the whole purpose of the method.

Explanation on how the PSD is converted to estimated probabilities.

Refer to the two diagrams below:

1.We feed the 60-day window into Welch’s method.

–The stem plot on the left shows the power each discrete frequency contributes to the window.

2.   Convert power to probabilities

– Power values (PSD) are all positive, so we simply divide each by the total power.

– This yields a bona-fide probability mass function over frequencies.

– The green stem plot on the right is that distribution: the area (sum) equals 1.

3. -Each bar's height represents its probability