# What is the risk-free rate of a precious metal, e.g. gold?

The risk-free rate of a x-currency denominated instrument can be determined from treasury bills, interbank borrowing rates (e.g. LIBOR), overnight index rates or interest rate swaps.

What instrument may be indicative for a risk-free rate of an instrument, which is denominated in a precious metal, specifically in gold? That is, the counterparties to such an instrument pay and receive payments directly in gold. The risk-free rate for gold is primarily needed for discounting reasons to discount an amount of gold from the future to the present date. For instance, 1 XAU (i.e. 1 ounce) in 3 months is actually 0.9998 XAU today for a positive discount rate. Something like that.

The current price of future access to any asset is its current forward price. This is true for any asset and true for whatever currency you use to measure price. Once you have the forward prices it is clear how to discount:

Call the period $T$ (e.g. 3 months) the forward price of you asset $P(T)$ and the current rate for exchanging your asset for currency i.e. the spot price $P(0)$. Then $P(T)$ is the current price of 1 unit of your asset at time $T$ measured in currency units. To get the price in terms of your asset you need to apply the spot rate, i.e. the discount factor is $\frac{P(T)}{P(0)}$.

Note that the actual currency which is used to quote the prices does not matter, since it cancels in the quotient. Hence this formula is always the same no matter what asset, currency or commodity you're interested in.

If you can rely on quoted forward prices you are pretty much done. If you want to mark to model and do not have forward prices available, then of course you need to worry about real world issues for your arbitrage pricing such as cost of carry, transaction costs and risk premia for financing or default.

• so this would suggest to "retriev" a discount rate from the quotes of forward and spot prices of gold? – Олег Бойко Feb 20 '17 at 12:16

In my opinion, settling your risk-free rate it is just a matter of convention. It is just what you consider to be safe.

Years ago, most people considered gold because most of the currencies where based on the gold pattern, but not any more.

In US usually paper (e.g. treasury bills) and in Europe usually German paper (e.g. German Bund).

Risk free rate is the rate of interest you earn of an asset which is virtually default free. Like the US dollar denominated bond guaranteed by the US govt. So essentially it can't fail and thus the person invested in such an instrument has no risk and earns the risk free rate. There is some debate at times as to the tenor, but I think if you want a 7.25 year rfr, you can find it from the US govt yield curve.

As far as gold is concerned, I dont think there is any entity which would backstop the metal. So I cant be sure that such a rate exists. But generally we use rfr for (relative) pricing of securities and in such cases you can use rfr off the yield curve and use it in your calculations.

• Thanks for your comment. Could you be more precise regarding which yield curve to use? There are US government bonds for the risk free rate in USD, Great Britain government bonds for risk free rate in GBP and so forth. – Олег Бойко Feb 17 '17 at 11:27
• I would use the one with the same currency. If you are pricing the instrument in USD, use the US govt bond yield curve and so on. – nimbus3000 Feb 17 '17 at 11:28
• The problem is that the instrument itself is denominated in gold, i.e XAU. The investor provides gold physically and then after some time the gold is returned to him plus some compensation in alternative real-life currency. – Олег Бойко Feb 17 '17 at 11:35
• Not an expert here, but can you not include the cost of hedgeing the gold? – nimbus3000 Feb 17 '17 at 11:36

The technical definition of a risk-free asset is an asset that pays out the same value in all states of nature. Obviously, none exist, if for no other reason that a meteor could hit the Earth tomorrow and there would be no one to pay out. There are a number of potential very low-risk assets you could use as a proxy, but there is no risk-free asset. Consequently, it doesn't matter which one you use as long as it is reasonable to believe that it will pay out the same amount of money in all states of nature. Indeed, it couldn't be a risk-free asset if it was contingent on the asset class you were using as the risk-free rate can have no contingencies.

i would say the standard way to pv such a cashflow is to assume settlement in usd (multiply by forward xau/usd rate) and then discount on usd curve thats appropriate for how its collateralised/funded eg 3m libor or ois

• might be. So this logic could be applied to any instrument in other than USD currency, e.g. CHF: convert CHF to USD and use ris-free rate for USD... Sounds as if we do not need risk-free rates in other currencies, to be honest. – Олег Бойко Feb 20 '17 at 11:29

If the question is about pricing an option on an asset denominated in gold, then the answer is that the asset denominated in gold will follow a martingale (by extensions to the first fundamental theorem of asset pricing). The price process of the asset denominated by gold will have to be modeled by some process and the choice of model will be key to pricing the option. However, the risk free rate need not be considered in such a problem.

If the question is about discounting some future cash flow then use a swap curve between the asset and gold if one exists. Depending on the terms of the swap some additional work may need to be done to tease a zero rate from the swap curve. The important thing is to try to ignore dollars completely unless the final payoff will be converted into dollars, though if the final payoff is in dollars then why bother with gold in the first place.

@g g: The forward price of gold will determine the zero rate for dollars, not for gold.

• There is a connection between the forward price of gold and the time value of money. But this is not relevant here. The OP wants to derive the current value in gold of a future amount of gold. My point is that you achieve this by using spot and forward prices. The zero rate for dollars (or any other currency) is implied (since some kind of financing is required for the arbitrage) but not explicitly necessary. The only thing you need are forward and spot price. No zero rate of currency required. – g g Feb 19 '17 at 10:46
• Two further remarks: Arbitrage pricing tells you that discounted(!) asset prices are martingales. You always need a notion of time value of money. So your statement "risk free rate need not be considered" seems incorrect to me. Second, if you have observed/quoted forward prices you do not need to model prices, since you know them already. – g g Feb 19 '17 at 10:59
• Current value of a future amount of gold...in what denomination? – user9403 Feb 19 '17 at 16:53
• Second point, the asset used for discounting can be anything. It is convenient to use a risk free asset but not required. The market's time value of money is already captured in the current value of all assets. – user9403 Feb 19 '17 at 16:54
• I don't know exactly what the OP was trying to ask, but I interpreted it as a question the risk free rate for gold (ie, I receive 100 gold bars today, and have to pay back 103 gold bars in a year). Dollars or any other currency is not involved in the equation. – user9403 Feb 19 '17 at 20:58