The tradable content of this paper is a sliver. The three optimized frameworks put 0.47% (long-only), 1.01% (plus/minus 100% bounds) and 3.29% (unconstrained) of the book in crypto. The 4.14% at the top of the range is the equally weighted 1/N book. Those weights are fixed in advance. In the authors' words, "there is no optimization strategy put into practice." So the optimizers never wanted much crypto next to metals. Anyone reading this as a case for a 10% digital-asset sleeve alongside gold is reading past the weights column.
One housekeeping note before the argument. We could not trade what they traded. So we ran our own version on liquid US-listed metal ETFs, GLD and SLV plus others where coverage allows, against BTC and ETH. We report no number from that run here. Nothing below compares our results to the paper's 2.63% return or its 2.94% ratio.
What Jayawardhana and Colombage actually built
The mechanism is the plain diversifier argument. Crypto is a high-volatility asset, and on their evidence its price level shares no common long-run trend with precious metals. A small position should therefore raise return per unit of expected shortfall in a metals book. The money comes from crypto's realized return over the window. No hedging relationship is claimed or measured. The paper says as much about the sizing: the weights "further demonstrat[e] that the cryptocurrency is not a heavily weighted position in any of the portfolios."
The framing is global, and the paper leans on it. The title promises global evidence. The allocation section says "we adopt the view of a global investor." The BGCI is described as covering the top ten crypto assets traded globally, weighted mainly by market capitalization. Every series is then converted to USD at fixed August 2017 exchange rates, "to avoid exchange rate fluctuations." A strange step for assets already quoted in USD. It is never explained.
The data are daily Bloomberg closes, 1766 observations, August 2017 to November 2022. The universe is the BGCI plus Bitcoin, Ethereum, Bitcoin Cash, Ripple and Litecoin individually, against COMEX gold, silver, copper, platinum, palladium and nickel.
Two halves. The econometric block starts with ADF and KPSS unit root tests. Then a Gregory-Hansen structural-break cointegration test, which finds breaks in December 2017 and May 2020. Then ARDL bounds testing with BGCI as the dependent variable and lags (1,0,1,1,2,1,0). Then Toda-Yamamoto and Granger causality, plus an R-squared comovement matrix. The second half is a mean-CVaR allocation at 95% confidence, run in MATLAB. It uses 59 months of monthly data, a 12-month rolling estimation window and monthly rebalancing. Each of the four frameworks is run with and without crypto. The reported result: mean monthly return rises everywhere with crypto, and the mean monthly risk-return ratio improves in three of four frameworks.
Where the arithmetic stops closing
The headline metric is the risk-return ratio, and we could not reconstruct it from the two columns printed beside it. Unconstrained with crypto: return 2.63%, CVaR 1.15%, ratio 2.94%. The quotient is 2.29. Equally weighted with crypto: 1.26% over 0.80% is 1.58, against a reported 2.17%. The ratio does not divide.
Long-only is the sharpest case. Adding crypto raises the mean monthly return from 0.39% to 1.21% and lowers CVaR from 0.50% to 0.46%. The ratio is 0.54% in both rows. The paper's own text says "the mean-monthly risk return ratio improves consistently over all optimization frameworks when adding cryptocurrencies." Long-only gains nothing.
The likely reconciliation: the ratio is the mean of the monthly ratios, while return and CVaR are means of their own series. The three columns are then not a quotient. The paper says it modified the ratio from Campbell et al. (2001), and we did not find the modified expression written out. An implementer has to guess. The guess decides whether the long-only book improved at all.
The weights column needs the same guess. A 3.29% average allocation cannot move a monthly mean from 0.46% to 2.63%. Either the column is a per-asset average across the individual crypto series rather than the total crypto sleeve, or the unconstrained solution is running heavy gross leverage. The paper never names which crypto assets enter the eight portfolios. The allocation section says only that it builds books "including various broad indices for precious metals and top trading cryptocurrencies." The coin count is unstable too. The data paragraph names six individual coins: Bitcoin, Ethereum, Ripple, Bitcoin Cash, EOS, Litecoin. Table 1, Table 2 and the ARDL equation carry five. "BTC, ETH, BC, XRP and LTC denote the cryptocurrencies of the study." The authors say as much about that framework anyway: the enormous asset weights and frequent rebalancing "may render such a strategy impossible to implement."
Note also the plus/minus 100% book without crypto: 0.46%, 0.42%, 1.09%, identical to the unconstrained book without crypto. With crypto it reports 1.01% weight, 0.73% return, 0.50% CVaR and a 1.52% ratio, against 2.94% unconstrained. The bounds bind only once crypto enters.
We did not find a transaction cost or crypto borrow assumption in the portfolio section. Table 10 carries four columns with no Sharpe and no t-statistic on the return differences.
Does no cointegration buy you anything?
The ARDL F-statistic is 3.08 against a 5% lower bound of 3.14 and an upper bound of 5.69, so there is no long-run relationship. Fair enough. Though the paper's own decision rule as written says a statistic falling between the bounds implies no cointegration, and 3.08 sits below the lower one.
The step that matters is the next one. Absence of a shared stochastic trend in price levels does not deliver a stable hedge ratio. It does not deliver an out-of-sample allocation either. The paper's highlights claim the findings "favorably support the benefits of portfolio diversification in the long run." The claim is an inference from the absence of a long-run relationship rather than a measurement of one.
The return-side evidence is thinner than the levels-side story. Unconditional BGCI correlations with metals span -0.0084 (nickel) to 0.1058 (copper), with gold at 0.0232. Comovement R-squared within metals is high: gold-silver 0.7280, gold-copper 0.6796. It is low everywhere BGCI appears. Gold-BGCI sits at 0.0432, fourth of the six BGCI pairs. The conclusion states the study found "a higher level of comovement between the gold and cryptocurrency markets." The matrix they built does not show that.
The starred cell
The abstract's causality claim is that gold prices unidirectionally influence crypto prices. It rests on a Toda-Yamamoto statistic of 5.01 with p = 0.1534, marked significant at 5%. In the same column, silver to BGCI (p = 0.0224), platinum to BGCI (p = 0.0050) and bitcoin cash to BGCI (p = 0.0063) carry no star. The reverse direction, BGCI to gold, is 0.01 with p = 0.8572. So the direction is clean and the significance marking is not.
The ARDL short-run gold coefficient is more interesting than the causality test: -0.127728, t = -2.95, p = 0.003. It is negative, which cuts against the digital-gold comovement narrative. It also sits in a levels regression on eleven regressors with R-squared 0.953. Treat it as a co-trending artifact rather than a return relationship. Separately, the paper reads Durbin-Watson 1.8684 as positive serial correlation supporting diversification, citing Savin and White (1977). We did not follow that step.
Table 2 has to be set aside. Three of the five individual crypto rows report a minimum above the mean: Bitcoin Cash 744.00 against 113.00, Ripple 3.00 against 1.50, Litecoin 1506.13 against 350.00. Five crypto skewness values also duplicate metal values to four decimals. The sample statement is unstable in the same way. The data section says 1766 observations to November 2022. The ARDL section says 1218 observations and a three-year duration. The unit-root table is headed August 2017 to December 2020.
A backtest that lives inside two bubbles
The allocation exercise runs August 2017 to July 2022, five years. Table 10's own note says the figures are monthly means over that period. That window includes 2018 and 2019. Its return is nevertheless carried by the two booms the authors' own break dates flag, December 2017 and May 2020. Roughly 47 months are out of sample after the 12-month burn-in. Equal weighting is the exception, where the investment period is the full 59 months.
The authors concede both halves of the problem. "[T]he duration allocated for the evaluation was inadequate for assessing the long-term effect of cryptocurrencies on the portfolio of equity indices in the given region." And separately: "this potential may be reduced during periods of poor market sentiment, leading to a decline in portfolio performance." The reference to equity indices does not belong here. This paper holds no equity indices. The limitation passage looks carried over from a different study.
Our ETF version swaps the exposures. The low and unstable comovement should survive that swap, since the ETFs are built to track the metal exposure. Fees and tracking differences still mean the reported relationships will not come back exactly. On the allocation side we have written before that a mean-CVaR style optimum is only as good as the support it was estimated on (our note on Wasserstein-distance portfolio allocation). A 12-month rolling window over 2017 to 2022 is a thin support for a 95% tail.
The authors get to the obvious next test before any reader does. They ask for "[f]urther research" over "extended out-of-sample periods" to confirm the results. Rerunning the four frameworks from August 2022 forward is their idea. A 4% position that survives 2022 to 2025 with a stated cost model would be a real result. What is still missing is smaller and entirely within the authors' reach. Print the formula behind the risk-return ratio. Say whether the weights column is the total crypto sleeve or an average across coins. Until those two lines exist, nobody outside the paper can tell whether the long-only book improved at all.