The Determination of Interest Rates
Prof. P.V. Viswanath

Interest Rates

Interest rates primarily reflect the price that an investor is paid for the use of his money over time. Hence the interest rate will affect the price of assets, financial and real.

The Real Rate of interest versus the nominal rate

At a simple level of analysis, since money represents purchasing power, the interest rate represents the equilibrium reward at which people are willing to lend real resources from one period to another (machinery, units of land, corn etc.). Thus, if at the margin, a saver is willing to defer the use of one unit of resources this period, and instead transfer it to an agent (who wishes to invest in a project), in return for 1.05 units next period, we would say that the real interest rate is 5%. The rate is referred to as a real rate, since the payment next period is stipulated in real resource units.

Determinants of the real interest rate

The Nominal Interest Rate

The interaction of the demand for and supply of real resources determines the real interest rate. However, we do not normally borrow and lend in real consumption units, since this is only an abstraction. We must borrow and lend in money, that is, dollars. Now suppose the price level next period is expected to be 10% higher than the price level this period; i.e. things costing $1 today are expected to cost about $1.10 next period. Then, in order to receive this promised real return of 5%, our saver must demand a return of (1.05)(1.1) or $1.155 per dollar lent today. If he does receive 1.155 in dollars next period, then he can buy 1.155/1.10 or 1.05 times what he could have bought for the dollar that he gave up this period.

The nominal interest rate (i.e. the rate denominated in dollars) that would, on average, compensate an investor for expected inflation of p % and give him a real rate of interest of r % can be computed (approximately) by adding the inflation rate to the required real rate: p + r. However, if an investor faces inflation risk, he might want more than an average real return of r%. We may add a risk premium term as well to obtain the following augmented model of interest rate determination:

i = r + E(p ) + inflation risk premium

Note that the inflation risk premium compensates not for the fact of inflation (the E(p ) term does that), but for uncertainty regarding the future inflation rate.  However, the Fisher equation says that the required inflation risk premium is zero; the more common used form of the above equation is: 

i = r + E(p )

The Inflation Rate

On a simple level, inflation is the change in the price level. The total amount of money (not just currency) represents ownership of the totality of goods and assets in the economy. Hence the price level is simply the total money supply divided by the quantity of goods.

Price Level = Total Quantity of Money/Quantity of Goods

This is called the Quantity Theory of Money. Changes in the price level therefore reflect changes in the supply of goods, as well as changes in the supply of money. As we have shown above, the nominal interest rate will be affected by the market’s expectation of future inflation rates.

Using the Fisher Equation:

Note that the Fisher Equation simply relates real interest rates, nominal interest rates, and inflation rates. Hence, we could use it to predict nominal interest rates, by using inflation rate estimates and real interest rate estimates from outside the model. Alternatively, we could use it to predict inflation, but using real rate estimates from outside the model. Finally, we could use it to estimate real interest rates, by using nominal interest rate and inflation rate estimates from outside the model.

Let’s see how we would use this simple model to predict three-monthly interest rates. First, we would look at people’s preferences for immediate consumption versus delayed consumption. Even though these preferences might and do vary across time and across space, they are unlikely to vary much from one year to another in a given country. The average monthly inflation rate over the 1968-1997 period (using the urban CPI) was 0.435%. On an annualized basis, this works out to 5.35%. The average 3-monthly T-bill rate over that same period was about 7.4546%, annualized. This yields an ex-post real interest rate of about 2.1%. Assuming that people’s expectations of interest rates and inflation were correct on average, we can use this ex-post real rate average as an estimate of the ex-ante short-term real interest rate. Let us assume that this real interest rate does not change in the short run.

To predict the three month interest rate, we must predict average inflation over the next three months, and add it to our real interest rate estimate of 2.1%. The inflation rate over 1997 was about 1.7. Using the current inflation rate as a predictor of near-term future inflation rates, an analyst in early 1998 might come up with a 3-month T-bill rate estimate of 2.1+ 1.7 or 3.8%. (Alternatively, we could use a structural model to relate interest rates to variables such as monetary growth.  This structural model could then be used in conjunction with the current rate of monetary growth to estimate the inflation rate for the near future.  If we compare it to the actual 3-month T-bill rate on January 25, 1998, we see that it was 5.16%. These two numbers may be reconciled by positing either a higher short-term inflation rate estimate for the market or a higher short-term real rate.

Similarly, we can come up with a long-term T-bond rate estimate as well. Our estimated inflation rate using data from the last 20 years was 5.35%. The average 30-year T-bond rate over the same period was about 9.1%, giving us an estimated long-term real rate of about 3.75%. Adding our inflation rate estimate of 1.7%, we get an estimated nominal 30-year rate of 5.46%.

The Effect of the Fed’s Actions on Interest Rates:

An important point to keep in mind in analyzing the impact of changes in the money supply on prices and interest rates is the following.  In one sense, money is passive and nominal money prices simply reflect the relative valuation of different real goods and assets in the economy.  However, in addition to that, money has a real effect as well.  Money works like a lubricant; it facilitates transactions.  Hence a change in money supply can sometimes have real effects, as well as nominal effects.

Suppose the Federal Reserve increases the money supply, either by lowering the interest rate at which it stands ready to lend to member banks, or by buying back Treasury securities and releasing funds into the system.  The market may interpret this decision as an indication of freer availability of future money.  The pure nominal effect of this action would be to increase the price level, since the output of goods has not changed, while the supply of money has. This would lead, not just to a currently higher price level, but also to a higher inflation estimate.

However, there could also be a real effect.  Suppose there was not enough money in circulation, originally, to accommodate the transactions that people wish to undertake.  Then the Fed's actions might have the effect, not only of increasing the money supply, but also that of increasing the amount of goods by facilitating transactions.  The net effect could therefore be a zero change in the price level, but a higher level of the Gross Domestic Product.

However, this also depends upon the excess capacity in the real sector. If there is sufficient excess capacity, then the additional liquidity can provide the impetus for higher growth, thus restraining inflation. However, when there is little excess capacity, providing additional liquidity will lead to expectations of a heated economy and will lead to higher expectations of inflation, without any substantial corresponding increase in production. In general, since money is necessary as a lubricant for the economy to function and to grow, the Federal Reserve through its actions, can encourage growth as well as inhibit it. This, in turn, can affect the real interest rate.

In summary, the Fed's actions could have two different kinds of effects: one, increasing the money supply could lead to market expectations of higher future inflation, causing nominal interest rates to rise, but not necessarily affecting real rates; two, increasing the money supply could cause real interest rates to drop by stimulating growth. This, in turn, would decrease nominal interest rates. Decreasing the money supply would have the opposite effects.

Why does the Treasury market sometimes move with the stock market and sometimes not?

We will see first show that the simplest way of looking at the stock and bond markets leads us to expect that bond markets and stock markets will move in the same direction. We will then show that a consideration of indirect effects allows for the possibility of the markets moving in opposite directions as well.

Why stock and bond markets might be expected to move in the same direction:
Factors that directly affect stock market valuation

Let us first look at a simple model of equity valuation. Suppose we look at a firm i, which pays constant dividends of $D per share per year, its stock can be valued as D/ri, where ri is the required rate of return on the stock of firm i. Clearly, the stock price will go up if the market increases its estimate of D or if it decreases its estimate of ri. What determines ri?

A useful model for this is the Capital Asset Pricing Model (CAPM), which says that ri = rf + bi[rm - rf]. The symbol rf refers to the risk-free rate of return, which is essentially the same as the nominal interest rate, r, of which we have been speaking, in this note. rm is the required rate of return on the market portfolio, the portfolio of all risky assets in the economy. bi is the beta risk factor of the stock, which is a measure of the risk of a stock relative to all other assets; the b of the market portfolio as a whole is unity (see webnotes on CAPM). If we ignore movements in the rm for a second, it is clear that if the stock market rises, there are two possible reasons: either nominal interest rates have dropped and/or the market has increased its estimate of corporate earnings.

Finally, it is possible that the market has also changed its estimate of rm - rf, the market risk premium, or the excess of the required rate of return on the market portfolio over the risk-free rate. Again, we need a model of determination of the market risk premium. A simple model for this can be found in Bodie, Kane and Marcus' chapter on the CAPM, where we learn that, under appropriate assumptions, the market risk premium,  rm - rf, is directly proportional to As2, where A stands for the average risk aversion parameter of agents in the market, and s2 is the variance of returns on the market portfolio. Hence, the market risk premium, rm - rf, will increase if investors become more risk averse, or if market conditions become more uncertain. Now, investors' risk aversion reflects their taste for risk, and is fairly stable. Hence changes in the market risk premium are usually caused by changes in market uncertainty.

In summary, the following factors can, ceteris paribus, cause stock prices to increase:

At this point, looking only at the effect of one factor at a time, it would seem that bond markets and stock markets should move in the same direction, i.e. an increase in riskfree rates decreases bond prices, and also decreases stock prices through its impact on required rates of return on equities.

A look at factors that affect interest rates directly and indirectly
An attempt to explain more complicated combinations of stock and bond market movements

We have already seen above, the factors that directly affect nominal interest rates - the market's estimate of the inflation rate, and the relatively stable ex-ante real rate. However, there is also an indirect effect of market uncertainty on interest rates, due to a substitution effect. Because of this, if market uncertainty increases, it is most likely that there will be two effects: one, investors try to switch more of their wealth into risk-free assets, and end up lowering interest rates; two, investors try to switch more of their wealth away from risky assets, and end up increasing the market risk premium for risky assets. Supply adjustments are unlikely to affect the situation much, since the supplies of risk-free and risky assets are fairly constant in the short run. It is however, most likely that the drop in interest rates will be insufficient to offset the increase in the market risk premium. As a result, the required rates of return on risky assets as a whole go up.
Market uncertainy rises
  • Interest rates drop
  • The stock market drops
Market uncertainty falls
  • Interest rates rise
  • The stock market rises

Now we have a simple way of classifying the four possible scenarios -- the combination of the stock market's movement and the change in the level of interest rates. The following table gives some of the explanations for the four possible scenarios, assuming that investors' risk aversions don't change. 

  Stock Market Rises Stock Market Drops
Interest Rates Rise
  • The market has increased its estimate of corporate earnings more than enough to compensate for the increase in interest rates due to an increased inflation estimate.
  • Market uncertainty has decreased; in addition, the market may have increased inflation estimates.
  • The market has decreased its estimate of corporate earnings, and has increased inflation estimates.
  • If the market increases its estimate of corporate earnings, the effect is insufficient to compensate for the rise in interest rates due to an increased inflation estimate.
Interest Rates Drop
  • The market has either increased, or not changed its estimate of corporate earnings, and has lowered inflation estimates.
  • If it has decreased its estimate of corporate earnings, the decrease is not sufficient to offset the positive effect of lower interest rates, due to lowered inflation estimates.
  • The market has decreased its estimate of corporate earnings more than enough to compensate for the otherwise positive effect of lower interest rates, caused by lower inflation estimates.
  • Market uncertainty has increased

How to Interpret Current Events in the light of the Model:

Interpretation is valuable only if it reduces the content of the object being analyzed to a few variables, whose relationship is already well known.  The fundamental variables in this sort of analysis based on the discussion above are: expected rates of inflation, ex-ante real rates, market uncertainty, market risk premium and expected future corporate profits.  A fruitful approach is to look at the facts of the situation (as distinct from interpretations and opinions offered in news reports) and see how they impact the fundamental variables.  Opinions can then be looked at in the light of this analysis.

Case 1: On October 1, 1998, the yield on the 30 year T-bond dropped to a low of 4.95%, but the stock market also dropped. How do we explain this?

To begin with, let us consider that investors have three broadly defined options: one, to consume today; two, to invest in riskfree bonds and three, to invest in risky equity securities. If investors increase their preference for the first alternative over the second and the third, interest rates will increase and the bond and stock markets, confronted with higher required rates of return will fall. On the other hand, if investors maintain their current rate of preference for immediate consumption, but believe that the risk of equity markets have increased, then stock markets will drop with no concomitant movement in interest rates. Finally, it is possible to have what is often described as a flight to safe havens. This would correspond to the bond market rallying, with stock markets dropping. The initial impetus for this might be increased uncertainty in equity markets causing a fall in stock market values. The reduced wealth of investors might lead them to reduce current consumption and seek to invest more for the longer term. This then shows up in lower interest rates. This seems to be what happened in the market on October 1.

Case 2: If we had looked at the 30-year rate on Friday, Jan. 25, 1997, we would have found it to be 5.97%, compared to a rate of 5.87% as of late Thursday, Jan. 24, 1997. Using the augmented Fisher model, and assuming a real rate estimate of 4.2%, we would have inferred that the market’s expectation of the long-run inflation rate has jumped from 1.67% to 1.77%. What could have caused the market to spook? Was all of the rate change due to a revised inflation estimate? Or should we infer that the market was revising the real interest rate?

One, the White House scandal could lead to weaker leadership from the White House; this, in turn, could lead to a lack of control over Congress, potentially leading to inflationary legislation e.g. increased tax cuts, leading to higher money supply without necessarily higher real growth. This would point in the direction of a higher nominal domestic interest rate in the future, leading to higher long-term interest rates currently.

Two, the Associated Press reported the Japanese government was considering investing in top commercial banks.  The actions of the Japanese government could lead to increased optimism about Asian economies; this could generate increased demand for funds in those economies, which would raise the level of real interest rates everywhere (since the major world economies are integrated). However, the increased demand for limited productive resources could also lead to inflation.

Hence the domestic interest rate rise could be due to an increase in worldwide real interest rates, or due to a higher expected domestic inflation rate.

For Interpretations of market events on Feb. 10 and 11, 2000, look at the relevant WSJ articles in the Media Articles section under the heading "Interest Rates."

Go to P.V. Viswanath's Home Page
Go to FIN 652 Home Page

Go to FIN 351 Home Page