Market data is typically reported in a currency ($, €, ¥, £, pesos, lira ...) but economic actors are presumably interested in purchasing power or real values. The common method of deflating a nominal to obtain a real variable is to rid the nominal variable of its currency unit. If, for example, the average nominal market wage in Europe is denoted as W_europe and it has the dimensions of euros/labor-hour, we find the real or purchasing power of nominal wages by dividing by a measure of European prices. We often denote a real variable, such as real wages, with a small letter; thus, real European wages might be denoted as w_europe = W_europe/P_europe. If P is measured in euros/goods, then real wages are measured in goods/labor-hr.

Consider the price of a debt instrument that is actively traded on secondary markets and use Q for the nominal, market value of that instrument. In the US the variable Q would be measured in $/iou -- it's a nominal variable because it is measured in a currency. We can obviously deflate Q to obtain the real variable, Q/P, which we can denote as q and which has the dimension of goods/iou.

Just as the real wage rate, w, is the purchasing power of a unit of labor, q is the real purchasing power of a bond -- the number of goods that one debt instrument, when liquidated (when the IOU is converted into money and the money is converted in goods), will fetch at market prices.

Having finished with 101 macroeconomics, students typically know that nominal interest rates (i) are "deflated" to obtain real interest rates (r) by netting out inflation (P-hat). I.e., r = i - P-hat where P-hat signifies the rate of change of P or ΔP/P (or, more rigorously, (dP/dt)/P). So, an obvious question to ask is "when do we 'deflate' by division versus subtraction?" Or "when do we divide by P versus subtract P-hat?" This is neatly answered by considering a particular kind of debt instrument, viz., Treasury strips or corporate zeros (debt instruments w/o periodic interest payments).

Treasury Strip Yields: Debt (whether corporate or government) that offers no interest payments prior to the one-time "ballon" payment -- the principal. Such an instrument trades at a deep discount -- the current market value is invariably less than the face value or principal. In the case of government Treasury debt, such an instrument is known as a Treasury strip. At the date of maturity the instrument will pay the "face value," usually $1000 in the case of corporate bonds and $10,000 in the case of government debt. If the instrument in question matures one year in the future, then the yield can be calculated simply as (F-Q)/Q where F is the face value.

"The" interest rate on debt instruments reported in the media is the yield-to-maturity, ytm. The reported ytm for each debt instruments must include that instrument's capital gains and losses. Our focus on Treasury strips and corporate Zeros allows us to focus exclusively on capital gains to determine yield (ytm). Let Q-hat represent the rate of change of the price of such an instrument (Q-hat = ΔQ/Q). Q-hat fluctuates much more (in negative as well as positive values) on short-run (e.g., daily) data than, say, annual data. But, overall, it must rise from the current discounted value to its face value over the life of the instrument. The average value of Q-hat over the life of the instrument will be the instrument's yield-to-maturity as reported. The value of Q-hat for a particular day, or hour or week will be the rate of return for that day, or hour or week.

Real vs Nominal Yields: Q-hat is the nominal yield on strips and zeros but it is not measured in terms of a currency -- it has no currency dimension. Yet the currency matters and it is considered a nominal yield. We will denote the price of a European strip as Q_e and a US strip as Q_us and the real values of the two strips as q_e=Q_e/P_e and q_us=Q_us/P_us. The rate of change of the real or deflated values of debt instruments represents capital gains in real terms. Ie, q-hat = Q-hat - P-hat is the appreciation(+)/depreciation(-) of a debt instrument in terms of goods. q_e-hat can be used to denoted the real yield on European strips and q_us-hat denotes the real yield on US strips.

The point is that there are two kinds of nominal variables and two deflation procedures to convert nominal variables into real variables. If the initial nominal variable is measured with a currency unit, one must divide by another nominal variable (with the same currency unit) in order to obtain the a real measure of the original variable. If the initial nominal variable is the rate of change of a nominal variable, then one must subtract the rate of change of another nominal variable in order to obtain a real value for the initial nominal variable.

But the issue of whether one subtracts P-hat from Q-hat to obtain the "real interest rate" or divides Q by P to obtain the "real value of the instrument" are really two perspectives on one issue. We obviously obtained the real return, q-hat, by first constructing the real variable, Q/P. We "deflated" Q-hat by constructing q and taking its percent change. This intimate relation between the two procedures for "deflating" nominal variables to obtain real variables will facilitate our discussion of the neutrality proposition below.

Real Exchange Rates
The real/nominal distinction becomes quite interesting with regard to exchange rates since we have not one, but two, currency dimensions. To convert a nominal or market exchange rate, E, into a real rate, the nominal variable must be deflated twice to purge the nominal variable of both currency dimensions.

The notion of a "real exchange rate" became widely used when the period of generalized floating (of exchange rates) emerged with the fall of Bretton Woods in the early 'seventies.

The following exercise is valid for any currency -- "dollar," "peso" and "Mexico" are chosen as names to provide a greater sense of specificity to the exercise:

If the nominal exchange rate, E, is measured as the peso price of the dollar, then the real exchange rate (the nominal rate stripped of currency dimensions) is For the peso to appreciate, become worth more in dollars, the quoted, nominal exchange rate, E, must fall. Similarly, if Mexico's real exchange rate appreciates, the numerical value of RER must decline, ie, where X-hat is used to mean the percent change in X (as in dX/X).

Why do the numerical values of E and RER fall when the currency in question is said to "appreciate"? Because the market most often quotes currencies with numbers greater than unity. If we note that the British pound is quoted as dollars/pound, then the construction of the real exchange rate for the pound is

and, if the pound appreciates, RER_uk-hat   >   0.

The Neutrality Proposition Along with the assumption of rationality, a basic assumption or proposition of economics concerns neutrality. Very simply, the neutrality proposition states that real variables are independent of nominal (i.e., money) variables. On the face of it, this is rather curious since the values of most real variables are constructed from nominal variables.

Some real variables, such as the rate of unemployment, are measured independently of any nominal variables. The statement that this particular real variable is independent of all nominal variables (such as inflation) is known as the natural rate hypothesis.

One of the early (circa 1800) statements of neutrality concerns interest rates -- it states that real interest rates are independent of nominal variables such as the inflation rate. If r = i- P-hat is independent of P-hat, i must obviously change in a matter to compensate for changes in P-hat.

Another interesting neutrality proposition concerns the question of whether RER is influenced any of the three nominal variables used to construct RER. Most economists who have investigated this matter have concluded that neutrality may hold in the long run, but not in the short run.

The venerable argument for the gold standard can be seen as a statement that the real value of gold, ie, R_gold=P_gold/P_goods is independent of nominal variables. Countries on the gold standard made their currency convertible into gold at a fixed price, called Parity. If P_gold is rigidly fixed at Parity (eg, the US$ parity was $20.67/oz-gold for many decades before it was devalued in 1933) and if R_gold=Parity/P_goods is independent of a very important nominal variable, the quantity of money (M), then the price level (P=Parity/R_g) becomes invariant to the value of M. This has been, of course, the central point of gold-standard advocates -- that the gold standard will eliminate the ability of central banks to cause price inflation by printing money.