What is going on with interest rates and the yield curve in Civilization VI?

Now that the summer doldrums have set in, I’ve turned back to playing Civilization VI. Fully equipped with its two expansions, Gathering Storm and Rise and Fall, the game really is fantastic – probably the best in the series – and able to keep me in rapt attention for many, many hours.

Concomitantly, I’ve started watching the videos of prolific YouTuber PotatoMcWhiskey (PMW from here), who is very good at (1) playing the game, (2) talking about the game and (3) teaching viewers how to be better at it. One thing he said in a recent video, while he was playing Arabia, caught my eye.

He is trying to trade with Rome and offers them 20 diplomatic favor, to which they bid 7 gold/turn for 30 turns. McWhiskey would rather have the gold up front, and takes the viewer through his process for figuring out how much gold up front Rome will be willing to give based on their bid of 7 gold/turn for 30 turns. A natural first guess is that they’d be willing to pay 210 gold up front, but that turns out not to be the case, because they impose a discount. In the end, he gets them to pay 143 gold up front.

This, I noted, is evidence that Rome considered the time value of money in their calculation. And from there, I wondered, what is the term structure of interest rates facing Rome that led to that decision?

I’ll step back – for the uninitiated, Civilization VI (like all the Civilizations before it) is a turn-based strategy game not too dissimilar from something like Settlers of Catan. Each player plays the leader of a civilization, like Pachacuti of the Inca, or Trajan of the Romans, or Peter the Great of the Russians. The game begins in 5000 BC and usually takes about 500 turns, ending in 2050 AD.

The best 'Civilization VI' leaders for all four victory types
A set of leaders you can play as in Civ 6, including Pachacuti of the Inca, Victoria of the English, Qin Shi Huang of the Chinese, Teddy Roosevelt of the Americans, Cleopatra of the Egyptians, and Seondeok of the Koreans. [edit: had him as Sejong, not Seondeok, credit a kindly Reddit user]

Your goal is to expand your empire by founding new cities, putting down roads, building the optimal mix of infrastructure, creating great works of culture, expanding your religion, investing in science, and dealing with the other player civilizations, be that through war, trade, or diplomacy.

Interface - Sid Meier's Civilization VI Game Guide | gamepressure.com
Might look bewildering, but this is the main interface you see when playing Civ 6. In this case, the player is Pericles of the Greeks, and he is considering options for his city of Mycenae.

You gain resources by having citizens in your cities work specific tiles on the board. Some of those resources can be traded to other civilizations, either for gold or for other resources.

After the Review: The Frustration of Civilization VI's Diplomacy ...
An example of what it looks like to engage in diplomacy with another player civilization. In this case, the player is speaking with Tomyris of the Scythians. Options include “Declare Friendship,” “Declare Surprise War,” and “Make Deal,” which opens the trade interface.

Because it’s a computer game, when you’re playing alone, all the rest of the player civilizations will be controlled by AIs. The efficacy and ability of the AI to play the game skillfully, or rather the apparent lack thereof, has long been a bugbear of devoted Civilization fans. But the game offers different levels of difficulty for single-player games, and on the highest levels, the AIs receive large material benefits which do make them worthy rivals to even the sharpest human tilters.

In the video I referenced above, PMW was playing a single-player game on the highest difficulty level as Saladin of the Arabians, and encountered in the course of the game Trajan of the Romans. He wanted to sell a unique resource, “Diplomatic Favor,” to Trajan for cold, hard gold. Gold payments in Civilization can be structured in two ways: either as a lump sum, payable immediately, or as an annuity lasting 30 turns, the first payment of which is delivered immediately. Just like any other rational actor, the AIs are programmed to be able to switch between equivalent amounts of either structure.

Calculating Present and Future Value of Annuities
How an annuity works. If you thought we were here to talk about video games, buckle up.

Presented with the resource on offer, Trajan made a bid: 7 gold/turn for 30 turns. As in real life, the extension of credit in this way presents tradeoffs. In particular, if any two trading partners go to war in Civilization, all per-turn payments, either in gold or resources, between them are suspended. Lump sum transactions, however, are not reversed.

Cognizant of the risk that Arabia and Rome might go to war someday soon, owing mostly to geographic proximity and rising power, that old Thucydidean thorn, Arabia wanted the money up-front. PMW then had to guess the max lump sum Trajan would be willing to pay – he first guessed linearly, so 210 gold, but Trajan rejected the deal. Then PMW applied successive discounts until Trajan agreed at 143 gold.

The equivalence of these two payments – 7 gold/turn for 30 turns and 143 gold – can be used to get a sense for what the interest rate facing Trajan was. Classically, finance types like to say that “a dollar today is not worth a dollar tomorrow” – this notion is the time value of money, which gives rise to the notion of the present value of a future stream of payments.

Time Value of Money Formula | Calculator (Excel template)
Silly diagram, but basically all of finance is here.

The critical bit for switching between the present value and the sum of the value of the future payments is (usually) the risk-free rate of interest over the period in question. Different structures have different formulas for switching between the two values, so let’s take a look at the formula for the present value of an annuity, which is what we’d call a setup like paying 7 gold per turn for 30 turns.

Present Value of Annuity - Formula (with Calculator)
Some nice LaTeX for everyone today.

Using this funky little calculator, we can plug in the present value, 143 gold, the periodic payment, 7 gold, and the number of periods, 30 turns, and backsolve for the “rate per period”. In our case we get 2.683%.

We need to state precisely what this 2.683% figure means. In particular, it is the internal rate of return (IRR) of the annuity. It’s the number you get by assuming each future payment for this contract is to be discounted by the same risk-free rate of interest. The IRR wouldn’t be the same if each payment was 8 gold per turn, so it’s hard to use this figure to generalize. Moreover, risk-free rates can vary from year to year, so there’s no reason to expect that each payment would use the same rate!

Example of Calculations of the Forward Yield Curves - Risk ...
This little chart tells you a lot about what interest rates are today, but more importantly, what they’ll be in the future. Today, the 1-yr rate is 3.5%. In one year, the 1-yr rate will be 5%. In two years, the 1-yr rate will be 8%.

So unfortunately, we can’t use this one calculation to say anything interesting about the term structure of interest rates facing Trajan of the Romans as he decides to finance some resource acquisition. In particular, we’d need to have the ability to vary the length of the annuity to see what the IRR would be for a 5-turn contract, or a 10- or 20-turn contract. With that kind of data, we could start to build a yield curve.

But from there, we start to get into thornier theoretical considerations. These are the rates we can recover from the contracts entered into between Trajan and Saladin, but who’s to say those couldn’t differ from the ones between Trajan and Victoria, or Trajan and Teddy Roosevelt, or Victoria and Teddy Roosevelt?

Moreover, we know that this is the rate faced by Trajan early in the game, when neither of the two economies are very strong or advanced. Who’s to say those rates wouldn’t vary throughout the game? Or vary with strength of diplomatic relationship, or with level of technology achieved, or vary with difficulty level?

At any rate, I’m glad to see that even in 5000 BC, Trajan and his advisors have a solid grasp for the principles of banking, lending, and money markets. Hopefully Civilization VII will allow players to earn interest on excess reserves at their central bank.