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Asset Pricing Model

February 29, 2004

Using the asset-pricing model, analyse the effects of the following changes on the value of the home currency:

a domestic monetary expansion

a foreign monetary expansion

an expected home currency appreciation

Finding a definition of the asset-pricing model can be frustrating. To understand this, I turned to a paper by Markus Müller & Ulrich K. Schittko, "Transmission of Policy Shocks in a Monetary Asset-Pricing Model" (PDF).

The Monetary Asset Pricing Model they describe can be summarized thus:

Two countries, A & B, both must buy products from the other in the producer's currency
A makes x, B makes y. As it happens, y is used exclusively as a capital ("investment") good. Output of y in period t (y_t) is wholly arbitrary.
Output of x in period t (x_t) is dependent wholly on k_t and some exogenous shock ε_t—in other words, x_t = x (k_t, ε_t).
k_t, in turn, is wholly dependent upon y_t-1, i.e., upon the available supply of y in the previous period.
As always in models of this sort, ∂x_t/∂k_t is a concave function; formally, x_k > o and x_kk < 0.¹
A and B have the same population. In point of fact, this is not as bizarre an assumption as it sounds; while the OECD is outnumbered by the rest of the world (1:6.8), the vast majority of non-OECD citizens are not really involved in the commercial economy of their country. So for example the GDP of India does not really reflect closely on the level of productive activity of all Indians, or even so much as a fourth of them. The great majority are subsistence farmers who pay some rent and tax, conduct a small number of purchases in cash over their lifetime, but mainly have exchanges that are embedded in social relations
The populations of both countries have constant relative risk aversion (CRRA).²
Explanatory note: this model strikes me as a bit strange; most of the capital inputs of the US economy, for example, come from the US. However, my suspicion is that A represents the OECD nations (of which the US and the rest of the "1st World" are members); and country B represents the OPEC nations, or possibly nations whose exports are primarily raw materials.

While trade flows are very complex, the real decisive force is the cheapness of oil and other raw materials. So we are, in a sense, following up rather neatly on OE-3 and trying to assess why trade awards greater or lesser rewards to fundamentally different economic actors in the process. In theory, the OECD could devise industrial policies contrived to increase output per unit of fuel consumed; it could devise tax policies which favored saving and investment, which would also stimulate investment; but this is unlikely to happen. Instead, the evolution of such policies in the OECD is gradual, stochastic, and usually eclipsed by other determining factors, like the price of oil.³

Currencies and Money Supply

The model for the monetary system in the two countries is straightforward;

A and B each have their own currency, and hency, money supply. The money supply for A at t is M_t; for B, this is N_t. M grows at rate ω_t, while N_t grows at rate ω*_t.
Under the model, the monetary policy is stochastic
The price level for x (the output of country A) is denoted as P_t; for y, this is P*_t. The exchange rate between the two national currencies is f_t.
The purchasing power of A's currency is πM_t = (f_tP*_t)-1 and therefore, if f_t goes up, then that means A's currency has decreased in relative value; also, if P* increases and f_t does not decrease (or does at a slower rate) then of course the real purchasing power of A's currency will decline. Another way of understanding this is to imagine if Canada had inflation of 20% and the USA had inflation of 10%. At the same time, at the beginning of the year the exchange rate was 1USD=1CDN, but at the end of the year it was 1USD = 1.05CDN. Then the real purchasing power of the CDN in the USA would have risen (1.20/1.10)/1.05 = 1.0389. or 3.9%.

Government and Investment

Both A and B have one government and one firm apiece.

g_t represents government spending in A; g*_t represents that for B.
In A there is a company called x, inc. which manufactures x. Equity ownership in x is open citizens of both A and B; the same is true for y, inc., based in B.
The share of x® owned by A at time t is α_xt; for B, that share is α*_xt. Conversely, the share of y® owned by A at time t is α_yt; for B, that share is α*_yt. Actually, M&S actually imply that we're dealing with a few million identical households, each of which own α_x and α_y. The incentive to own stocks comes from dividends paid. However, M&S also ignore the existence of an actual firm buying y from y® and converting it into x. No, they imagine that each household does this instead. So it should be pointed out that household income comes from dividends δα_yt, or some income stream per share. From this is subtracted the expenditure on investment goods, α_ytk_t+1.
Government revenue comes from money creation and a lump sum tax.
Everyone begins with endowments of m_t and n_t of money from both countries.

After the goods markets open, there is a great swapping of currency for goods, including the obligatory purchase of y for investment purposes. Let x^d_τ be private consumption of home goods in A, and y^d_τ be private consumption of foreign goods (in other words, some of x consumed in Aand B will be consumed publicly by g).

Each citizen maximizes objective function

subject to constraints πM_tm_t≥ p_txd_t and πN_tn_t≥ yd_t + αx_tk_t+1 and a real budget constraint measured in foreign goods, which is spelled out in M&S's paper (Eq7) but much more ornate than I'd care to go into. On the expenditure side we have the purchases of goods, the tax payments, and the end-of-period demands for money and securities. On the revenue side we have the dividends paid out by firms (x® and y®). Dividends include, naturally, all of the payments the firms make—except for purchases of y_t.

The First Order Conditions (FOC) are listed (8-16) with a few notes on the optimization problems for the firms. Section 3 "The Equilibrium" treats market clearing conditions. The solution of the FOC would comprise a list of some 21 functions of k and s. A few of these are quite interesting.

The distribution of national currency would be the same in countries—namely, everyone would have half foreign, half domestic. To be more precise, the rate of money growth would be the same—i.e., every period people would hold an amount of money that increased by the absolute rate. Half of the net increase in the money supply of each country would wind up in the hands of the people of A and of B.
Surprising to me, in view of the sharply divergent nature of their economies, was that the rate of growth in both countries was the same. I had always assumed there was some fundamental aspect of trade relations that meant B's terms of trade would deteriorate under such an arrangement.
Holdings of the two companies were the same in both countries; since the taxes of both governments came from these holdings and 100% of income came from ownership of these firms, this explains the similarities in income. There would be no geographical advantage to residing in one place
The model includes a liquidity service (which accounts for "liquidity preference") that I had overlooked (Eq 35, p.8). It is embedded in the Lagrangian multiplier μ; if μ = 0, there is no liquidity preference. As it happens, this is one of the most important features in the operation of a healthy economy; if μ is low, or even 0, the economy's potential for growth is maximized.
Curiously, the earlier point about income distribution between A and B does not mean the value of the two companies and national output are identical. It's conceivable under this scenario—indeed, perfectly plausible—that B's economy could be far smaller, and it would have no impact on the incomes of the population of B. Under this scenario, all income is dividend payment, which comes from ownership of stocks in x® and y®. That has no correspondence with the actual size of the national economy. This latter is, I must concede, the least realistic aspect of the model.

The fascinating conclusion of this model is that nearly everything hinges up on the growth rate of the money supply. The authors ascertained a critical money supply growth rate ώ and ώ* (i.e., the growth in the money supply of A and B) such that all of the values depend on it. The value of ώ* is dependent upon the discounted expected marginal utility of the real quantity of B's currency;

so we can use our value for Ψ* to establish the critical value of ώ*:

It still seems to me some of this is circular. For example, Ψ* is a function of the total purchasing power of the money supply N.

Some Illustrations and Remarks on Exchange Rates

Common sense says there's going to be a few things that affect the relative exchange rate of two countries, A and B. The first is naturally the purchasing power of the two currencies. If the currency of A, the ^ (pron. "blerp") has the purchasing power parity (PPP) of Country B's ð (or "thorn"), then of course we would reasonably expect the ^:ð exchange rate to be 1. That it often isn't explains much of the reason why we even attempted to establish things like PPP. Why might the ð have an exchange rate of 1:1.2 against the ^ when they buy about the same? This is a thorny issue.

Part of the reason might be trade barriers; if a ð = ^1.20, it might be because Country B has some barrier to entry for Country A's products. If there were no barriers, of course, the merchants of A would be there in a minute, selling stuff in B at 16.6% off the usual B prices… unless… most stuff used to calculate the PPP for the two countries is actually not traded. An inexperienced merchant shows up and discovers it frankly costs more to set up shop in B; houses are costlier, there's a premium on potable water, the costs of services like education and medical care are more expensive... Moreover, the assumption of open markets is one that has to be relaxed, or abandoned entirely. Markets are so far from being untrammeled or open as to make this assumption intolerably silly for forex markets.

One compelling reason why this wouldn't be true anyhow is the fact that trade is really a small component of forex transactions—about 2%, roughly. That's an important 2%, of course, but clearly the largest component of exchange rate behavior is not directly related to the exchange of goods across international boundaries. (The reason why the 2% is important that, for net transactions, international trade is a very large share of the turnover. If the $1.5 trillion daily turnover were mostly in the same direction for the affected currencies, then international exchange rates would be as stable as the Fermium-258 isotope.) One important market for forex is foreign direct investment (FDI) and portfolio investment, both of which are elements of the capital account balance. The capital account balance can be described as the net investment flow, and it is extremely volatile. It includes foreign investment in the USA, net of American investments in the rest of the world.

The chart above uses US Treasury data to show monthly inflows and outflows of financial assets. Notice the extreme increase; all data has been adjusted for inflation. While real exports of the USA have not grown very much, capital flows have.

Here is the data since 1998. In this closeup, I've added the balance, which has been positive or very slightly negative every month. This means foreign purchases of US assets exceed American purchases of foreign assets. The little purple line looks trivial; it sums to approximately $400 billion annually! In addition, another 500 billion or so flows into the USA for rollover into other countries besides. A substantial share of American income from foreign service transactions comes from this.

Also with US Treasury data, I put together this chart showing the composition of foreign purchases of American assets. This is meant to show trends in US asset sales. The bulge in US corporate stocks occurred at a time when US Treasury bonds were "scarce," because the budget was in surplus.

In reference to a comment by Conrad in between initially posting this and adding some more, I thought I would furnish a graph to allow comparison with the relative magnitude of trade. This data is from the St. Louis Federal Reserve FRED II page. The nation continues to run a surplus in services that is eclipsed by the deficit in goods. As we can see, the ratio of capital flows to trade flows has increased stupendously, but if we sum the positive and negative movements over the course of the year, it's clear that net annual capital flows move in rough synchronicity with the balance of trade (this is not true on a monthly basis!). The effect of this on exchange rates needs to be explained below.

I've been using "trade balance" as an approximate synonym for "current account balance" (CAB); but the distinction between the two is more urgent than I realized. The CAB includes the trade balance (goods & services—for the USA, big negative), net income on factors abroad (returns to Americans minus returns from USA to foreigners; since 1985, increasingly negative); and "unilateral transfers" which includes foreign aid and military assistance programs (MAP). For some reason the Fed excludes military sales from the trade statistics; while the USA is a big exporter of military supplies, this is a surprisingly small component of trade.

Unlike the capital flows data above, where I hunted down monthly data to demonstrate the volatility, this is quarterly. Monthly trade data is much harder to locate, but since it involves underlying products, it is far more steady. However, the other components of the CAB do fluctuate dramatically, and if the data is averaged over the course of the year it smooths to a level similar to trade balances

See? My approximation isn't really so sloppy after all, unless it's forex we're talking about.

These, then, are some components of real forex over the long run, and over the course of a year they net out to something quite close to the annual value of trade. But as we shall presently see, it's the day-to-day business that drives currency exchange rates.

Some Illustrations and Remarks on Exchange Rates

As we've seen, capital flows for the USA net to something less than the current account balance does over the course of a year; however, over the course of a single day, it's an entirely different matter, where trade accounts for perhaps a 50th of forex turnover. This is easy to explain. I travel to work every day; and then I drive home. If I always drove the same direction I wouldn't be in California anymore. Conversely, my travels do have some effect; over the course of my life, I've moved a few times.

Let's start with the concept of "covered interest rate parity."

This is a really simple idea. Instead of insisting that purchasing power parities (PPP) should match (so that, for example, US$100 exchanged into Canadian money, should buy the same amount of goods in Canada), we look at rates of change. We expect that if the Canadian dollar is expected to decline in the forex market by (say) 2% against the US$, then this will be reflected in the forward markets, and in a 2% interest premium on Canadian money (actually, 1/98%, which is +2.041%) relative to US money. If this were not so—if the premium were 0%, for example—then it would make sense to contract to sell huge amounts of CDN at the forward rate 3 months ahead, then buy American 3-month T-bills worth the same amount on the spot market.⁴ In 30 days, one will make a handsome, zero-risk profit. Since "everyone" knows this, it will drive the CDN up or US T-bill rates down, and the differential would not survive a few seconds.

This is called "covered" interest rate parity. It can be risk free because the bond and the futures contract allow one to make absolutely certain judgments about future wealth now. If you buy a bond that matures in 3 months time, you know precisely how of the denominated currency you will have in the future. You can lock in the cheaper foreign money (interest times exchange rate movements) at the time of purchase because markets exist for bonds and futures. Now, what is odd about this is that it's difficult to imagine arbitrage opportunities lasting for very long, which means that this must be absolutely decisive in establishing the ratio of (USD*i_US$):(euro*i_€) to the same for very other currency. The European Central Bank can control the domestic interest rate, or it can control the exchange rate of the euro to the dollar. But it cannot control both at the same time.

Let e_€ be the euro expressed in US dollars (as of this writing, 1.2268). Also as of this writing, the 12-month forward rate for the USD is 1.2178, implying a 0.734% relative decline of the euro against the dollar, so bonds should have a 0.734%markup over equivalent bonds. As a matter of a fact, finding equivalent bonds is the rub. That is why financial publications speak of a "risk premium," the gap in the forward-spot ratio and the interest rate-ratio. It's the additional risk of default on bonds that is understood to exist. (The ECB's interbank rate is 1.00% higher than that of the "federal funds rate" of the US Federal Reserve, the rate at which reserves can be borrowed among member banks. Or try the "London interbank offer rate"—LIBOR—which explains exactly what the rate means!). However, it is an extremely reliable predictor of how exchange rates respond to changes in the interest rates. If e_€ goes up and the interest rate differential between the Fed and the ECB is unchanged, the forward rate must go up to keep the forward premium as it was.

Forward rates do have some influence on exchange rates, although trying to decide how much requires counterfactual analysis. If the forward rate of e_€ is high—i.e., if F[e_€] goes up, or is very high relative to the value of e_€, we can expect that interest rates will fall to reduce the forward premium, which tends over time to reduce demand for euros. At the same time, the expectation of a large demand at a known point in the future leads to the assurance that large stocks of euros will be available, perhaps by way of liquidating non-liquid assets.

Covered interest rate differentials do not last very long because the forward rates quickly rise to eliminate them. However, there are speculators who use detailed information about the present to anticipate the future. Earlier I referred to the F[e_€] as the forward rate of the euro; it can also be called, for absolute precision, the spot forward rate, meaning, "the precise forward rate at this instant." Even though the forward market is highly speculative, there is nothing ambiguous about the statement it makes about the future; once a futures contract is sold, the owner may be assured of the exchange rate it stipulates, no matter what the markets do afterwards (assuming, of course, that the seller honors the contract). Similarly, a bond holder may not be happy with the interest rate implicit in the discount bond she has bought, but she certainly knows what it is. Now I want to introduce E[e_€], which is entirely speculative. This is the expected rate, not the forward rate.

From the point of view of a nonspeculative⁵ user of these markets, we can imagine that choosing between currencies to hold—e.g., between USD and euros—is like a horserace, pitting E[e_€]*i_€ against E[e_USD]*i_US$. Although businesses may try to hedge against extreme currency movements by buying options, that can get expensive and it doesn't work forever. Still, this is what investors do, either by portfolio investment (buying shares in foreign companies) or FDI (taking a stake in a new venture, like a real estate development project in Poland).

Covered interest rate parity holds a lot better today than it used to. In earlier periods, such as the first term of Pres. François Mitterand (1981-1988), the franc suffered an enormous discount against the US dollar, sometimes as high as 11%. Over the same period, Germany and Japan experienced a premium of 2%. The reason for the gigantic forward franc discount was capital controls, a restriction on outflows which naturally prevented inflows. After '87, when the controls were abolished, the discount disappeared. This was captured by F[e_FFr], which was persistently low. Efforts to measure if uncovered interest rate parity applies requires someone's estimate of what the foreign currency x, E[e_x], will be. Apparently there is a periodical, Consensus Forecasts, which publishes one commonly used E[e_x] for many currencies; according to this forecast, uncovered interest parity does not apply. The evidence leads one to conclude that common predictions about forex are biased, although these biases vary over time.

Long Run Determinants of the Exchange Rate

Over the long run, PPP performs well at predicting the exchange rate, if the country is developed; Sweden, Canada, Belgium and Australia are big exceptions, with real exchange rates drifting behind, while Japan and Great Britain have pulled ahead. In the 3rd World, the preponderant tendency has been to drift behind. As everyone knows, the Chinese policy has been to stick to a peg that holds the renminbi artificially low. Moreover, while the changes from year to year are quite predictable, huge disparities may accumulate over time. Between 1975 and 1998, for example, the Indian rupee lost about 80% of its value relative to the dollar; yet over the same period, total CPI inflation was only 60% . This means that the real exchange rate of the USD in Indian rupees tripled over that period. In contrast, the PPP for the euro suggests that it should be somewhat cheaper than the dollar, not 1.22 time as much. And the Japanese yen is overvalued by perhaps 31% against the USD.

The standard theory has it that this is the result of the money supply and the comparative price levels, as well as some item k, which represents the link between real money supply and GDP; k is similar to the idea of velocity.

This equation simply means that a country's price level increases if its domestic money supply rises faster than the foreign money supply; if the demand for its currency increases at a slower rate than other countries; or if output increases at a lower rate than other countries. This is a tautological formula, since k is whatever a regression analysis says it is.

Now, we can answer the questions.

If there is (a) a domestic monetary expansion, we can assume the home currency will suffer a relative increase in prices; this would probably reduce the value of the home currency relative to that of other nations. Conversely, if (b) the main trading/investment partner increases their money supply, we can assume the home currency will rise. But what if (c) the home currency appreciates?

This seems like an odd question. Surely, my diligent reader will suppose, an expected increase in the home currency will lead to an increase in the home currency. Well, recall I was wondering about the impact of forward rates on the exchange rate? An increase in E[e_x] is interpreted here as an increase in F[e_x], the forward rate, which I pointed out would lead to a reduction in the bond rates of the home country. The demand for home-currency bonds would increase, since there was suddenly a covered intertest disparity.

Of course, this would have an immediate effect on the domestic spot rate, too. It rises to restore uncovered interest parity.

This chart seeks to illustrate this. Notice how the rise in the forward rate (red) causes the yield curve to have a lower grade. Over the affected time period (t_m), the yield curve falls by the amount that the forward rate rises. Of course, after the interest rate has fallen to protect covered interest rate parity, this new slope defines the new spot rate in order to maintain uncovered interest rate parity. The new spot rate will be something less than the new forward rate, but clearly more than the old spot rate.

NOTES: 1 In plain English, this means as k (tools and other forms of capital per unit of labor) increases, output increases too; but x_kk < 0 means that the contribution of ever more capital adds less and less to be used up in the production period. The output of the domestic good in period t, x_t, depends on an endogenously determined productivity.

Just imagine the function behaves like this. Please observe x_kk means "second derivative of x with respect to k and k." In HTML, this is by far the easiest notation to use.

2 Constant relative risk aversion: Used in economic models to forecast how the average consumer/worker will behave faced with a limited range of alternatives. When probability theory was first applied to economics, it was assumed by economists that a 0.50 chance of making $100 was worth exactly the same thing as $50. The problem with this was that it could lead to a situation where a person has no unique "tipping point" between rejecting a bet and accepting it; there's no logical reason why a 0.00000005 chance at winning $1,000,000,000 shouldn't also be worth $50; and if I were to offer lottery tickets under these conditions at $49, it would be exactly the same thing as selling a $50 bill for $49. To resolve this, Von Neumann and Morgenstern developed a formula:

where V(Y) is utility as a function of Y (income), V_Y represents the marginal utility of wealth, and V_YY represents the second derivative (the rate at which the marginal utility is diminishing). This is also called "elasticity of marginal utility." The function ρ(Y) is called the coefficient of relative risk aversion; usually we are not interested in mapping it out over different values of Y because it's used to anticipate how a household will assess its options at any given moment. The significance of a high ρ is that one is going to give a much lower weight to improbable windfalls; a .25 likelihood of winning one's monthly income in a casino will be worth far less that a quarter of one's monthly income; at very high levels it will be worth a tenth one's monthly income. My textbook (On Well-Being and Destitution, Dasgupta p.196) says that this is believed to be decreasing in Y, meaning that the wealthy are less risk averse.

CRRA is used very heavily in economic modeling. It has been proven mathematically that if consumers violate CRRA, they become money pumps—a set of "bets" can be contrived where they lose a large sum of money over the long run. As you might expect, real casino gamblers do violate CRRA and are immense money pumps. Other exceptions abound, but in an economy of efficient markets these exceptions would be exploited. A note to students of economics: these concepts are not consistently named.

3 Foreign good...used as an investment: someone might object that oil is not an investment (capital) good. Technically, it's a consumption good, but the price of oil has a drastic impact on the productivity of capital, and of course the price of energy inputs move in sympathy with crude oil.

4 "Forward Rates" or "spot forward" is financial lingo for the mean exchange rate predicted by the markets for futures and options. Let us describe this in terms of options. Suppose you are assigned the job of figuring out what the forward rate of the euro is—mathematically, F[e_€].

There is the "spot" rate, which is the rate for instant "delivery" of euros in exchange for US dollars, and this involves no prediction at all. In addition, one can buy an option for euros that can be "exercised" (or used) at some point in the future. For ease of exposition, I'm going to assume these are "European style" options that may only be exercised at a specified time in the future—say, 12 months hence. An option to buy—a "call option"—specifies the exact price at which one can buy the euros at a future date, and this price is the "strike price." An option to "sell" is known as a "put" option.

Now, dollar-denominated options for euros are available in many flavors; there are 3-month, 6-month, and 12-month terms. However, let's focus on the 12-month variety. There will be a range of strike prices: 1.2275 (what it is now); 1.2250 (a quarter-cent lower); 1.2225; 1.2200; and then, of course, a range of strike prices which are higher. Now, suppose the 12-month euro call with a 1.2275 strike price is selling at a very high markup. And the 12-euro put (strike 1.2275) is selling at a very low markup. In that case, the market is saying that the privilege of buying euros at $1.2275 each is expected to be worth a lot, while the privilege of selling them at such a low price is not worth very much at all; in other words, the forward rate is something more. We can use premia on options or futures contracts to find the rate at which puts and calls are marked up by the same (small) amount, or, if we are lazy, we can simply take the market value of any call option, add the price of the option to the strike price, subtract the usual fee from the option itself, and we will then know what the forward rate is. It's a little bit more complicated than that, of course, since options aren't directly comparable to future contracts, and we have American-style options in the USA, naturally (which have higher premia because you can exercise them anytime before they expire).

5 "Speculative" —participating in the markets as one's primary source of income. If you're trading currencies as your main job, I'd say you’re a speculator. If you're participating in the market to facilitate some other business, like international commerce and banking, then that's another matter.

Comments on this Post:

I am doing a group project on the topic of covered interest rate parity. I was wondering if a simple example could be sent to me as I am having trouble in understanding the concept.

Posted by: Laura Toye at October 27, 2004 07:57 PM

The concept of covered interest rate parity can be illustrated thus: bonds and currency futures are like rival predictions of the future.

Suppose you go to Mme Sosostris, fortune teller, who says, "I see you will buy an ounce of gold for $500." Then you go to Mme Tsiganis, who says "I see you will buy an ounce of gold for $550." So you buy an ounce of gold from Mme Sosostris, and pay with a check for $501, give the ounce of gold to a trusted friend, and have that friend sell it to Mme Tsiganis for $549. Then you deposit the $549 in your checking account and take your friend out to dinner. If Mme Tsiganis had merely talked to Mme Sosostris, then you would have no profit-making opportunity.

That's covered arbitrage. There's no risk (assuming the two fortune tellers believe what they tell you).

Posted by: James R MacLean at October 27, 2004 11:53 PM