Exchange Rates And Gold's Zero Discount Value

Exchange Rates and Gold’s Zero Discount Value by Michael S. Rozeff This note provides a simple financial model of exchange rates – very simple. If it has merit, it is probably as a rough or long-term guide to their tendencies. I make no empirical attempt to explain the market’s ups and downs, with all the attendant noise, leads, lags, and manifold influences. The usual economics models look at such things as purchasing power parity, data on money supplies, prices, rates of price inflation, interest rates, and output. I focus on gold and the monetary base for precision of thought, but I recognize that in application, one probably has to use some measures of interest rates and other monetary aggregates to make sense of exchange rates. My approach is a balance sheet approach. In earlier articles, I introduced the notion of a Zero Discount Value (ZDV) for gold when there are bank notes issued against gold. The ZDV is the value of the notes in terms of their weight in gold. Here I relate the ZDV to exchange rates in three different cases. Case1 is the free and full convertibility of notes into gold. In this case, the ZDV is the market price of gold. It follows that the exchange rate between two different bank notes relates directly to the two ZDVs. In case 2, I assume that there is inconvertibility of notes and that the prices of gold are both discounted from the ZDVs by the same proportion. In this case, the exchange rate is the same as in the convertible case. In case 3, I assume inconvertibility but that the gold prices represent different discounts from the ZDVs. The exchange rate in this case is such that each note is worth the same market amount of gold after applying the exchange rate. However, both the exchange rate itself and the gold that backs each note diverge from the convertible case. A disequilibrium results. This produces pressure on the exchange rate that tends to move it toward the value that holds in the convertible case. If all central bank notes were freely convertible into gold, then they would all have to have the same gold-equivalent backing that is consistent with their exchange rates. If they did not, there would be arbitrage opportunities. Non-convertibility of bank notes into a common medium of redemption like gold abates or interrupts arbitrage. It allows disequilibria to persist. Other kinds of exchange restrictions do the same. But although arbitrage by individuals may not be possible, a tendency to achieve equilibrium still pressures exchange rates in a direction consistent with convertibility. That is the point argued in this note. Case 1. Suppose there are two bank notes D and E. There are 1000 D issued and 750 E issued. Let bank D hold 0.2 grains of gold per D, and bank E hold 0.33 grains of gold per E. Then 1D is equivalent to 0.6E. The arbitrage-free exchange rate between D and E is 1D = 0.6E.

Consider the totals. Bank D has issued 1000D bank notes worth the 200 grains of gold it carries as an asset, and bank E has issued 750E bank notes worth the 250 grains of gold it carries. Zero Discount Value (ZDV) is defined as the price of the bank notes in terms of the quantity of gold held by the bank. D’s ZDV is 5D per grain and E’s ZDV is 3E per grain. These values are consistent with 5D = 3E or an exchange rate of 1D = 0.6E. The market prices of gold in terms of D and E have to be the same as the ZDV values when there is convertibility. A grain of gold costs 5D and also 3E; 5D have to exchange for 3E to prevent arbitrage profits. Thus, if there is convertibility of bank notes into gold, the ratio of the Zero Discount Values of two different bank notes also gives the arbitrage-free exchange rate. Case 2. In a non-convertible system of banking, gold can sell at less than its ZDV. Suppose a grain of gold sells for 0.15 of D’s ZDV, i.e., at 0.75D per grain. Suppose that the discount of E’s notes to its ZDV is the same. That is, gold is 0.15 of E’s ZDV, i.e., 0.45E per grain. In this case, where the discounts are identical, the exchange rate is 1D = 0.6E, since 0.6 x 0.75 = 0.45. Either 1D or 0.6E buy the same number of grains of gold in the market, namely, 1.3333 grains. Furthermore, the backing of the notes at this exchange rate is the same. Both 1D and 0.6E have the same backing of 0.2 grains. The exchange rate in this non-convertible case remains the same as in the convertible case if and only if both bank notes sell at the same discount from their respective ZDVs. If they sell at different discounts to their ZDVs, then their exchange rate will no longer be at the rate indicated by the freely convertible benchmark. Case 3. Suppose now that gold priced in D and E notes sells at different discounts to ZDV. Suppose gold sells at 0.15 of its ZDV in D, and it sells at 0.30 of its ZDV in E. Gold is 5D per grain. At 0.15 of its ZDV, it is 0.75D per grain. If gold sells for 0.30 of E’s ZDV of 3E per grain, its price is 0.9E per grain. At the exchange rate of 0.75D to 0.9E, or 1D to1.2E, the same market amounts of gold are exchanged. This means that if we attempt an arbitrage at the observed exchange rates, it will not be profitable. Let us borrow 1D and exchange it for 1.2E. Sell 1.2E and get 1.3333 grains. Exchange that for 1D and repay the D loan. There is no profit using market rates. Gold prices are consistent with the exchange rates. It is the exchange rates that are out of equilibrium. This is because the notes are selling at different discounts to ZDV. This shows up in that the backing of the notes at this exchange rate is no longer the same. 1D has a backing of 0.2 grains, and 1.2E has a backing of 0.4 grains. If D and E are inconvertible and D sells at a larger discount to ZPV, D’s exchange rate moves higher compared to the convertible case. If the price of gold in D notes is relatively lower than the price of gold in E notes as reflected in the larger discount from ZDV, then the value of D notes is relatively larger as compared with the convertible case and the equal discount case. This means that the D notes appear overvalued and the E notes undervalued. This coincides with the lower gold backing of the D notes at the new exchange rate.

We uncover a relation that allows us to detect overvalued and undervalued currencies in a very simple way. Calculate a currency’s ZDV. Obtain the price of gold in that currency. Calculate the discount of gold’s market price from its ZDV. The larger is this discount, the greater is this currency overvalued relative to other currencies. All of them may be overvalued relative to gold, but that is a separate matter. Although 1D and 1.2E exchange for the same amount of gold in the market, the 1D has less gold behind it than the 1.2E. If a trader wanted gold and held both D and E, he might consider it better to sell his Ds rather than his Es to buy it. He could sell 1D and buy 1.333 grains, and he’d give up a backing of 0.2 grains. If he sold 1.2E and bought 1.333 grains, he’d give up a backing of 0.4 grains. He’d gain 0.2 grains backing by selling D rather than E. This can be seen in another way. The holders of both D and E notes have an incentive to sell them in order to get gold, since both notes are at a discount to ZDV. If a buyer can be found who will buy 1000 D notes and give up gold, he’d give up 1,333 grains at the price of 0.75D per grain. He’d get 1000 notes with a backing of 200 grains. This is obviously a very bad (unprofitable) deal for buyers, but that is what it means to say that arbitrage is possible. The same goes for the E notes. Someone who bought 600E notes would also have 200 grains of gold behind them, but he’d have given up 666.67 grains of gold at the price of 0.9E per grain to get them. A seller who had the choice of selling D or E notes does better to sell the D notes because he gets more actual gold as compared with the gold-equivalent in the notes he disposes of. There should be a tendency for those who want gold to hold on to the E notes and sell the D notes to get the gold. This should tend to lower the price of D in terms of E and move the exchange rate closer to the convertible benchmark. With this framework, we can now think about a sequence of events that changes exchange rates. 1. Assume a money system with central banks. It is not a free market system. It is a fiat money system in which the notes of the central banks receive partial backing from gold. 2. Assume that banks D and E have 1000 notes issued against 200 gold grains and 750 notes issued against 250 gold grains, and that both have gold prices that are at 30 percent of ZDV. Then D’s gold price is 1.5D per grain and E’s gold price is 0.9E per grain. The exchange rate of their notes is 1D to 0.6E. 3. The D central bank doubles its notes without adding any gold. This means that the ZDV rises to 10D per grain. If gold remains at 30 percent of its ZDV, its price becomes 3D per grain. The equilibrium exchange rate is 3D to 0.9E or 1D to 0.3E, showing the 50 percent devaluation. So does the doubling in the price of gold. 4. Assume that market reactions are not instantaneous or that expectations are slow to adjust, for which there are any number of possible reasons. Then when D’s ZDV rises to 10D, assume that gold in that currency stays temporarily at a price that is 1.5D per grain. Gold sells at 15 percent of ZDV rather than 30 percent.

5. This means that D’s bank notes appreciate relative to E’s notes. D’s notes are overvalued. Gold’s prices are 1.5D and 0.9E. The exchange rate is still 1D to 0.6E, but it should be 1D to 0.3E. 6. A tendency now appears for D’s notes to depreciate over time due to their overvaluation. 7. As market participants learn, the slow adjustment of expectations will dissipate. Reactions will quicken. If D inflates, the exchange rate falls more quickly. If traders come to anticipate greater inflation of D’s notes, then the exchange rate will decline and gold’s price rise before the central bank inflates. The actual behavior of the exchange rate will be that of a speculative price. Economic data will influence the exchange rate in a complex way both through actual spot trades in which currencies are exchanged and also futures trades in which expectations are involved. Unaccountable leads and lags will be present. Similar changes will be occurring in other currencies, complicating the observed exchange rates further. Through all of this, the ZDVs of the currencies and the discounts at which gold is selling from its ZDVs may provide some rough guidance as to market directions. The market prices of gold are always in equilibrium with the observed exchange rates. But when the discounts from the ZDVs are not all equal among central bank notes, then their backing varies. I hypothesize that currencies with lower backing are overvalued relative to currencies with greater backing. They all may be overvalued relative to gold, and they all may depreciate over time; but that is not the issue I address here as I addressed that in my earlier articles explaining ZDV. The point being made here is that the overvalued bank notes are the ones selling at the greatest discounts to their ZDVs, and they will tend to depreciate even against their sister bank notes. November 9, 2009