bonds, equities · United States · bonds, ETFs, futures · Monthly · United States Free

Gold to Oil Ratio Predicts Aggregate Stock Returns

Academic paper

Gold price ratios and aggregate stock returns

AuthorsTong Fang

Institute

Shandong University of Finance and Economics
?Shandong University - School of Economics

Paper linkhttps://papers.ssrn.com/sol3/papers.cfm?abstract_id=3950940

Strategy in a nutshell

The strategy dynamically allocates between the S&P 500 index and the one-month Treasury bill using the gold-oil price ratio (GO) as a predictor.

Process:

GO Predictor: Compute the natural log of the gold-to-oil price ratio.

Regression Forecasting: Regress the S&P 500’s excess return (vs. T-bill) on GO using a 240-month rolling window.

Forecasting Returns: At the end of month t, use the regression to forecast the S&P 500 excess return for t+1.

Portfolio Allocation Rule: \text{S&P 500 Allocation} = \frac{1}{\text{risk aversion}} \times \frac{\text{forecasted excess return}}{\text{forecasted variance}}

Variance forecast: 10-year rolling window of past returns.

Risk aversion coefficient = 3.

Allocation bounded between 0% and 150%.

Final Weights: S&P 500 weight determined by rule; remainder allocated to one-month T-bill.

Rebalancing: Monthly updates of regression, forecast, and weights.

Economic rationale

Asset prices reflect both expected cash flows and discount rates (Cochrane, 2011). GO’s predictive ability comes mainly from anticipating aggregate cash flow news.

GO also negatively predicts default spreads, financial stress, and uncertainty, making it a leading indicator of economic conditions.

A higher GO signals stronger economic outlooks, translating into higher expected equity returns

Backtest performance

Annualised return7.02%

Volatility9.7%

Beta0.498

Sharpe ratio0.72

Sortino ratio0.14

Win rate62%

Full Python code

1from AlgorithmImports import *
2import statsmodels.api as sm
3import data_tools
4# endregion
5
6class GoldToOilRatioPredictsAggregateStockReturns(QCAlgorithm):
7
8def Initialize(self):
9self.SetStartDate(2000, 1, 1)
10self.SetCash(100000)
11
12self.min_monthly_prices:int = 15
13self.regression_period:int = 5 * 12 + 1 # need n months of data
14
15self.min_market_alloc:float = 0.
16self.max_market_alloc:float = 1.5
17self.risk_aversion_coefficient:int = 3
18
19self.spy_daily_prices:list[float] = []
20self.latest_go_predictor:float = None
21
22self.regression_data:RegressionData = data_tools.RegressionData(self.regression_period)
23
24security = self.AddEquity('SPY', Resolution.Daily)
25security.SetLeverage(5)
26self.spy_symbol:Symbol = security.Symbol
27
28security = self.AddEquity('BIL', Resolution.Daily)
29security.SetLeverage(5)
30self.bil_symbol:Symbol = security.Symbol
31
32self.oil_symbol:Symbol = self.AddCfd('WTICOUSD', Resolution.Daily).Symbol
33self.gold_symbol:Symbol = self.AddCfd('XAUUSD', Resolution.Daily).Symbol
34
35self.recent_month:int = -1
36
37def OnData(self, data: Slice):
38# rebalance monthly
39if self.recent_month != self.Time.month:
40    self.recent_month = self.Time.month
41
42    if len(self.spy_daily_prices) >= self.min_monthly_prices and self.latest_go_predictor:
43        monthly_return:float = (self.spy_daily_prices[-1] - self.spy_daily_prices[0]) / self.spy_daily_prices[0]
44        self.regression_data.update(monthly_return, self.latest_go_predictor)
45
46        if self.regression_data.is_ready():
47            x_train, x_predict = self.regression_data.get_x_data()
48            y_train:list[float] = self.regression_data.get_y_data()
49
50            regression_model = self.MultipleLinearRegression(x_train, y_train)
51            market_return_prediction:float = regression_model.predict([1, x_predict])[0]
52            sse:float = np.sum(regression_model.resid ** 2) # regression_model.ssr
53            variance:float = sse / ((self.regression_period - 1) - 2)
54
55            market_allocation:float = (1 / self.risk_aversion_coefficient) * (market_return_prediction / variance)
56            market_allocation:float = max(self.min_market_alloc, min(market_allocation, self.max_market_alloc))
57
58            if self.bil_symbol in data and self.spy_symbol in data and data[self.bil_symbol] and data[self.spy_symbol]:
59                self.SetHoldings(self.spy_symbol, market_allocation)
60                self.SetHoldings(self.bil_symbol, 1 - market_allocation)
61    else:
62        # reset regresion data, because they stopped being consecutive
63        self.regression_data.reset_data()
64        self.Liquidate()
65
66    # reset
67    self.latest_go_predictor = None
68    self.spy_daily_prices.clear()
69
70# update GO predictor
71if self.oil_symbol in data and self.gold_symbol in data and data[self.oil_symbol] and data[self.gold_symbol]:
72    oil_price:float = data[self.oil_symbol].Value
73    gold_price:float = data[self.gold_symbol].Value
74
75    go_predictor:float = np.log(gold_price / oil_price)
76
77    self.latest_go_predictor = go_predictor
78
79# update spy daily prices
80if self.spy_symbol in data and data[self.spy_symbol]:
81    price:float = data[self.spy_symbol].Value
82    self.spy_daily_prices.append(price)
83
84def MultipleLinearRegression(self, x:list, y:list):
85x:np.array = np.array(x).T
86x = sm.add_constant(x)
87result = sm.OLS(endog=y, exog=x).fit()
88return result