Discounting Gold Cash Flows with Forward Prices and Funding Curves

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


# 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.

## Answer by g g (score 4)

https://quant.stackexchange.com/a/32520

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.

## Answer by arodrisa (score 1)

https://quant.stackexchange.com/a/32496

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).

## Answer by nimbus3000 (score 1)

https://quant.stackexchange.com/a/32497

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.

PS: I've never traded commodities.

## Answer by Dave Harris (score 1)

https://quant.stackexchange.com/a/32498

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.

## Answer by Randor (score 1)

https://quant.stackexchange.com/a/32500

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

## Answer by user9403 (score 1)

https://quant.stackexchange.com/a/32522

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.

## Answer by Johan Smith (score 1)

https://quant.stackexchange.com/a/79567

Gold and other precious metals don't have a 'risk-free rate' because they don't guarantee returns like government bonds do. Instead, gold is valued as a hedge against inflation and economic uncertainty. Its price is influenced by factors like market demand and geopolitical events. When analyzing precious metals in financial contexts, we often use government bond yields as a benchmark for risk assessment, but it's important to recognize that gold investment itself carries inherent risks and volatility.