If I have two assets that both have the same returns, but one is liquid and the other is not, how do I calculate the difference in value between them?

That music track [for example] [which makes $1000 a year] wouldn't be quite worth $20,000, since it's not very liquid, right?

That's wrong.

It's really hard to price assets like that. It tends to come down to a judgement of how stable the income will be over time. This varies wildly by category and paradigm of the thing in question. Furthermore, in certain particular industries or fields, there is something of an "established" multiplier for a given asset^†; then again in other fields there is no such established norm. Furthermore, in certain fields there are very specific (say) laws that come in to play, such as copyright, or, things like long-term leases and laws regarding mineral extraction etc.

However, you absolutely, definitely, cannot say "well it's 20x - same as a theoretical stock" for some business or product entity you wish to sell. (Unfortunately!)

So in perhaps answer to your question,

• There is absolutely no relationship between "multiples seen on the stock market" and "multiples when selling assets or businesses"

To try to literally address the title:

If I have two assets that both have the same returns, but one is liquid and the other is not, how do I calculate the difference in value between them?

Superficially, one might think "sure, 'less liquid' assets are worth less". But that's just not true. Minecraft was both incredibly not liquid and incredibly valuable. If Paul McCartney sold his music publishing empire (MPL), it would be astoundingly not liquid but astoundingly expensive. And the same happens on lower price properties. (A cute example is domains, they are not liquid at all (often there's like "2 buyers") but the price can range from trivial to hilariously high. Same applies to patents, and even things like "investment gems", and art.)

On the other side, it's true that stocks (well, the popular stocks, not thin stocks) basically "define" the concept of liquidity. So, when you say "a restaurant is not very liquid" you are saying "compared to trading Apple and Ford public shares". But there's really nothing else like that, it's not a "comparison that can be made". So, if we asked to compare the "electricalness" of various cars, it would be pointless to observe that the tesla and volt are electric cars, and all the other models simply are not electric cars, it's just a different category. So, you and I could sit around and compare the liquidity of your song library, my programming practice, and our mate's restaurant. But it's meaningless to compare that to stock liquidity, it's just a different type of thing.

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^† Just a random example, if you have an app on the google/apple stores, and it is making $D per year, it's extremely lucky if you can sell it for 0.5x$D. Or, here's an article about the multiples when selling a restaurant https://sellingrestaurants.com/how-do-i-value-my-restaurant.html As I understand it with patents, it can vary greatly from really high multiples to low. Indeed, for any given field you name, you can indeed immediately find any number of articles on the typical multiple in that field - since obviously it's the whole raison d'etre of any business.

This article has a discussion of illiquidity discounts:

These studies generally flag up large discounts (25-50 percent) for illiquid assets – with 35 percent being given as a common median data point – and, as such, valuations of private assets have generally used such ‘off the shelf’ fixed estimates.

The illiquid asset only has less value if you plan on selling it. If you want to know the value of realizable cash on both assets in a sale, you have to include transaction costs in the NPV and profit calculations.

I am going to just take your assumptions on income stream and discount rate as given. Both are the exact same for each with no variability.

Your intuition is correct, a liquid asset is more valuable than an illiquid asset in a sale scenario given the same income stream and discount rate. You're missing the cost needed to acquire or dispose of that asset though.

In the case of a liquid stock, the cost to acquire/dispose of that asset on a liquid market is almost 0. You and the buyer will probably have to pay 1c per share in total to acquire/sell the stock. The buyer's costs needs to be subtracted from the PV of the asset to get to a "purchase price" the buyer will be willing to pay (ie NPV), and then the cost you have paid to sell the asset needs to be subtracted from the NPV to get your profit from the sale of the asset. Most people ignore this with a liquid asset given how low it is.

With an illiquid asset like the right to a royalty stream, this same principle applies. The PV may be the exact same as the stock, but the NPV and you're ultimate profit will be different because of the transaction costs involved. With something like a royalty stream, there's probably specialty brokers that charge a % of the sales price in order to market and ultimately sell your asset. There may also be some lawyer's fees involved in order to document the change in ownership. Lastly, the buyer may have to spend time (which has an associated $ value) in order to validate the royalty stream right is legitimate and has historically performed as represented. All of these items represent costs that ultimately make the profit to you in a sale scenario lower and/or make the purchase price lower than the liquid stock.

As a sidenote, you would not sell the royalty stream to another buyer if that buyer had the exact same discount rate as you. The transaction costs would make the trade negative for at least one of you since the asset is valued the exact same, $20K with a 5% discount rate. To be a profitable trade, you need to find a buyer with a lower discount rate. For example, let's assume your discount rate is 5%, a buyer has a 3% discount rate, and you have $2000 of transaction costs to sell the asset. The asset is worth $20K to you, but $33,333.33 to the buyer. They could offer you $25K and, even considering $2K of transaction costs, the trade would be beneficial for both involved.