Keywords: silver vs gold, beta of silver, is silver a good investment.

Summary:

1. Gold and silver, historically, have delivered statistically similar monthly returns.

2. Silver has a higher long-term variance, and thus is a riskier commodity, compared to gold.

3. For long-term risk reward relationship to hold, silver may see a significant price improvement.

4. Silver's beta in relation to gold is 1.425.

©Risk Concern. All rights reserved. Report (Commodities)

Gold (XAU) and silver (XAG) are considered as relatives in the precious metal space, and XAG’s price is expected to move with XAU. While the historical gold to silver ratio fluctuates, the most important question to commodities investors and speculators is:

Which precious metal yields a higher return? Is it gold, or silver?

Also, what about the risk profile: If one yields a higher return, shouldn’t it be a riskier investment, as higher risk = higher possible returns?

This report analyzes these questions, relying on data from the last four decades, for definitive answers.

For analysis of how much metals, energy & grains can fall in a market crash, see our report.

### So, which precious metal yields more, Silver or Gold?

The average growth rate (geometric mean) of silver for the last 50 years stands at 5.13%.

The monthly movement of silver price, from 1979, with 98% CL has been in the range of 0.771 ± 1.06 percent.

Source: Macrotrends

The average growth rate (geometric mean) of gold for the last 50 years stands at 8.11%.

The monthly movement of XAU price, from 1979, with 98% CL has been in the range of 0.5196 ± 0.553 percent.

Source: Macrotrends

Another important issue worth analyzing is the covariance of silver and gold. The beta of silver = 1.425, this theoretically means that ±1% change in gold should result in a ±1.425% change in silver:

Regression analysis

### So, is silver better, as when gold rises, silver should rise higher? It’s not that simple, and let’s not jump to conclusions yet.

We need further steps to clarify whether gold is a better investment? If not, then evaluate whether there is any evidence backing the case for XAG.

Hypothesis testing is applied to evaluate whether the difference between the returns of XAU and XAG has any statistical significance, then we move towards risk.

To answer this question with statistical significance, hypothesis testing must be employed. The hypothesis, hence, has been constructed as:

H0: μd - μ0 = 0 versus Ha: μd - μ0 ≠ 0.

, where

μd – the mean difference between the monthly returns of gold and silver.

μ0 – 0

Paired sample T-test, using T distribution (df=503) (two-tailed) (validation) 1. H0 hypothesis Since p-value > α, H0 cannot be rejected. The average of XAU-XAG's population is assumed to be equal to the μ0. In other words, the difference between the average of XAU-XAG's and the μ0 is not big enough to be statistically significant. 2. P-value The p-value equals 0.4254, ( p(x≤T) = 0.2127 ). It means that the chance of type I error, rejecting a correct H0, is too high: 0.4254 (42.54%). The larger the p-value the more it supports H0. 3. The statistics The test statistic T equals -0.7977, which is in the 98% region of acceptance: [-2.3338 : 2.3338]. x=-0.0025, is in the 98% region of acceptance: [-0.007345 : 0.007345]. The standard deviation of the difference, S' equals 0.00315, is used to calculate the statistic. 4. Effect size The observed effect size d is small, 0.036. This indicates that the magnitude of the difference between the average and μ0 is small.

The null is not rejected, therefore.

We now know that there is no statistically evidence that the monthly returns of gold and silver are different. On a monthly return scale, statistically, gold has not yielded a higher return than silver, and this is backed by statistical evidence.

What about the risk?

F test has been conducted to evaluate the difference in variance between Gold compared to silver. The hypothesis is constructed as:

H0: σ2XAU = σ2XAG versus Ha: σ2XAU ≠ σ2XAG,

, where

σ2XAU – the variance of the returns of gold; and

σ2XAG – the variance of the returns of silver.

Paired sample T-test, using T distribution (df=503) (two-tailed) (validation)

1. H0 hypothesis Since p-value < α, H0 is rejected. The average of XAU-XAG's population is considered to be not equal to the μ0. In other words, the difference between the average of XAU-XAG's and the μ0 is big enough to be statistically significant. 2. P-value The p-value equals 0, ( p(x≤T) = 1 ). It means that the chance of type I error (rejecting a correct H0) is small: 0 (0%). The smaller the p-value the more it supports H1. 3. The statistics The test statistic T equals 15.1981, which is not in the 98% region of acceptance: [-2.3338 : 2.3338]. x=0.048, is not in the 98% region of acceptance: [-0.007345 : 0.007345]. The standard deviation of the difference, S' equals 0.00315, is used to calculate the statistic. 4. Effect size The observed effect size d is large, 0.68. This indicates that the magnitude of the difference between the average and μ0 is large.

Test for XAG's variance being higher:

F test for variances, using F distribution (dfnum=503,dfdenom=503) (left-tailed) (validation)

1. H0 hypothesis Since p-value < α, H0 is rejected. The sample standard deviation (S) of XAU's population is considered to be less than the sample standard deviation (S) of XAG's population. 2. P-value The p-value equals -8.882e-16, ( p(x≤F) = -8.882e-16 ). It means that the chance of type I error (rejecting a correct H0) is small: -8.882e-16 (-8.9e-14%). The smaller the p-value the more it supports H1. 3. The statistics The test statistic F equals 0.2767, which is not in the 98% region of acceptance: [0.8326 : ∞]. S1/S2=0.53, is not in the 98% region of acceptance: [0.9125 : ∞]. The 98% confidence interval of σ12/σ22 is: [0 , 0.3323].

The nulls are rejected, therefore, and silver's higher variance is established.

### What does this mean, simplistically?

This means that the variance of silver returns is higher than gold; silver for the past four decades has been riskier than gold, and this is backed by statistical evidence. In the future, it would be logical to expect XAG to be a riskier investment, compared to XAU.

### Putting it all together

With investment in gold, data for the last four decades suggests that an investor should expect to earn 8.11%; with investment in silver, the expected return, as per the data, is 5.13%. Nonetheless, since 1979, monthly returns are very much comparable.

Silver, compared to gold, has been a riskier investment over the last 4 decades. Data, however, does not clearly identify gold as the winner, but does present a paradox: Silver has a higher risk but a comparable monthly return; this could mean that in the future, silver may see a sharp up-move to compensate for the long-term higher risk compared to gold, as this imbalance, as per the principle of no-arbitrage in the financial markets, should not stand long-term.

With almost double the risk, XAG should, theoretically, increase in value, so that its long-term returns reflect the relative risk, so the risk-return equilibrium holds.

### So how much can we expect silver to rise?

Theoretically, for the risk-return equilibrium to hold, with double the risk, three scenarios have been calculated: 1. Equivalence with gold price, 2. Increase as per the risk profile of silver, 3. Increase as per the covariance calculated earlier:

January 1979 silver price ($6.08) × long-term rate of return for gold (3.7%) = theoretical 2025 price of $32.34.

January 1979 silver price ($6.08) × 2X long-term rate of return for gold (as per risk) (7.4%) = theoretical 2025 price of $162.22.

January 1979 silver price ($6.08) × long-term rate of return, considering silver’s beta of 1.4, in relation to gold (3.7 ×1.4 =5.18%) = theoretical 2025 price of $59.

The mean of the three values = $84.5. Beta distribution value = $71.6

Nonetheless, the practical issues of supply and demand hold, which of course, are not theoretical. The observed difference, practically, also suggests that silver and gold are not comparable at a specific ratio, due to higher price fluctuations of silver.

More silver is mined each year, as it's relatively more abundant compared to gold. What justifies the observed condition is that gold, simply is in higher demand, and relatively limited supply. Silver is not as highly demanded as gold, and the supply is more abundant than gold. This explanation can partly justify the higher variance.

Gold’s relative stability justifies gold as a precious metal that is considered an investment and a store of value – a financial asset. Demand also includes the industrial requirements of gold. On the other hand, silver has not performed as a financial asset that could be classified as a store of value; its long-term growth rate is also less than the long-term inflation rate. Silver, arguably, resembles an industrial commodity more.

Presently, therefore, gold is a better investment. However, it is not irrational to expect a significant up move in silver that increases the price to a range between $32-$160, and most logically close to $59, as per the long-term covariance.

What’s the final verdict?

An individual investor would need to make a decision based on a cost-benefit analysis: are the stable returns and comparative risk worth more to her? If yes, then gold is the ideal choice. If, however, the investor is interested in speculation and willing to accept relatively higher risk for a significant upward move in the next few years, adding XAG to the portfolio makes more sense.

Data file for the analysis is attached below:

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