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Section 9.2 provides a rationale for the existence of interest. Howev(! there was little in that analysis that sheds light on the level of interest rates I exist at various points in time.

If we examine interest rates existing in the past, we discover that variation has occurred over time. In 1945 the average yield on Treasury 1 was 0.33%. In 1981 the same security had an average yield of 14.71%. August 1980 the prime rate (to be defined later in this section) was 11%. December 1980 it was 21.5%. What can explain variations of this magniti

In looking at the level of interest rates it is necessary to distinguish 1 two types of variation. The first type of variation is general and pertains tol overall level of interest rates prevailing in the market. The second typejj variation is specific and pertains to the level of the interest rate on a partic transaction.

Basic economic theory would suggest that rates of interest, like other prici are established by supply and demand. If the demand for funds to borrov strong in relation to the availability of funds, interest rates will Conversely, if the demand for funds to borrow is weak in relation to availability of funds, interest rates will fall. It sounds simple, but in prac

On the demand side, the primary issue is the productivity of capital. Virtually all business firms need capital on which to operate successfully. Some s of this capital generally comes from borrowing. In the long run, the firm will be successful only if the return on capital employed is greater than the cost of j borrowing. Of course, not all borrowing is done by firms, but much is also] done by individuals and the government. Admittedly, a portion of this borrowing is used to finance current consumption. However, a portion of it is] also in some sense being invested, e.g. individuals borrowing to purchase homes] and to finance college educations, and the government borrowing to build j infrastructure such as roads and ai orts. In the end, a healthy economy wiy j require that a substantial portion of money being borrowed is invested! productively.

Although these two major theories are quire different, they are in no wajl incompatible. In fact, quite the contrary, they serve to reinforce each other.

The above discussion barely scratches the surface of some of the economic or even psychological and philosophical, theories attempting to explain existence of interest. However, the above discussion is sufficient for pu Oses of this book. Curious readers who would like to explore this more completely are encouraged to refer to relevant literature in economics i finance.

there are a large num.ber of factors that come together in complex ways to determine rates of interest.

The following is a list of major factors which have an influence on the level of the rate of interest. The list is not exhaustive, but it does include most of the major determinants.

1. The underlying "pure" rate of interest Most economic and financial theorists believe that there is an underlying "pure" rate of interest as a base which is related to long-term productivity growth in the economy. This rate would prevail on a risk-free investment if there were no inflation. This rate has proven to be relatively stable over many decades. In the United States this rate has typically been in the range of 2% to 3%.

2. Inflation

Experience has shown that inflation has a significant effect on the rate of interest. This factor is discussed in more detail in Section 9.4. }. Risk and uncertainty

Experience has also shown that risk and uncertainty have a significant effect of the rate of interest. This factor is discussed in more detail in Section 9.5.

i. Length of investment

There will normally be differences in the market between the rates of interest on short-term and long-term loans and investments, all other things being equal. This phenomenon is discussed in more detail in Section 9.6. 5. Quality of information

In finance theory "efficient" markets are defined as those in which all buyers and sellers (in this context borrowers and lenders) possess the same J information. Aberrations in the rate of interest are more likely to exist in "inefficient" markets. In the modern computer-information age, markets tend to be more efficient than in die past. However, certain market rigidities remain which can affect the rate of interest. Legal restrictions

Some rates of interest are regulated by die government. In the United States diere has been a trend toward deregulation in recent years, so that this has become a less significant factor than in the past. Nevertheless some I rates of interest still are subject to some degree of regulation. Governmental policy

The federal government has a major influence, even control, on the overall I level of interest rates through its monetary and fiscal policy. The primary control is the ability of the Federal Reserve Board to adjust the supply of

300 The theory of mterest

1 + / i + r

- r 1 + r

(9.3a) (9.36)

According to the theory outlined above, / will be relatively stable over time. However, / and r will tend to move up and down together. The above relationships are not precise. The real rate / is not totally constant over time nor is the correlation between / and r exact either. Also, we have the dual problems of measurement and time lags, since r should be the expected rate of I inflation. Thus, these formulas should not be thought of as exact relationships, but rather as convenient rules of thumb.

Nevertheless, formula (9.3a) is quite useful in performing calculations! involving rates of inflation. For example, assume that we wish to find present value of a series of payments at the end of each period for n periods ml which the base payment amount at time 0 is R, but each payment is indexed tS reflect inflation. If r is the periodic rate of inflation and / is the periodic rate! of interest, then the present value of this series of payments is


(1+0 (1+0"

by applying formula (4.39).

= R{\+r)

1 + /

However, if we use formula (9.3a) the formulation of the problem beco: 1.1. 1

1 +/

= Ra


{l+iY (l+/)«

Formulas (9.4) and (9.5) produce the same numerical answer and have interesting verbal interpretation. Formula (9.4) represents the present value the payments including inflation computed at the nominal rate of inte. However, in formula (9.5) this is seen to equal the present value of payments excluding inflation computed at the real rate of interest.

The above verbal inte retation really provides more general guidance computing present values of ftiture payments in practical situations as folio

If future payments are not affected by inflation, dien discount at the nomb rate of interest.

If fiiture payments are adjusted to reflect the rate of inflation and adjustment is reflected in the payment amount, then also discount at nominal rate of interest.

More advanced financial analysis 301

3 If ftiture paym.ents are adjusted to reflect the rate of inflation but the adjustment is not reflected in the payment amount, the correct procedure is to discount at the real rate of interest.

The above discussion involves a present value analysis. It is also instructive to consider inflation in connection with accumulated values. Consider the common situation in which an investor invests A dollars for n periods at interest rate /. The value of this investment in "nominal dollars" at the end of n periods

A{\ + if. (9.6)

However, how much is this investment really worth at that fiiture date? If die rate of inflation is r, then the purchasing power of this investment at the end of n periods is

. (1 + 0"

= A{\ + /)


(1 + )"

Thus, the value of this investment in "real dollars" is lower, since i >/. investors would find it quite enlightening to analyze their investment iprograms in terms of "real" results as well as "nominal" resuhs, rather than

isidering only the latter. j In the above discussion, there has been the implicit assumption Idiat i > r, i.e. that the nominal rate of interest is greater than the rate of rinflation. In general, this relationship will hold, particularly over significant periods of time.

Ifi However, in some cases, for at least some investors, this may not be the llCase. Such a result is most likely to occur during periods of high inflation. For IBxample, in the United States during 1979-1981 the rate of inflation, as sured by the Consumer Price Index (CPI), which is probably the most widely used index of inflation, was in "double digits" (i.e. over 10%). Yet lluring this same period of time billions of dollars were invested in savings counts at rates of interest in the 7% range. For all those investors, the Inominal value" of their savings was increasing, but the "real value" was jleclining.

Readers who are interested in pursuing a classical analysis of the elationship between rates of interest and rates of inflation are referred to the by I. Fisher (1930) listed in the bibliography.

Example 9.1 An insurance company is making annual payments under fie settlement provisions of a personal injury lawsuit. A payment of $24,000 f just been made and ten more payments are due. Future payments are exed to the Consumer Price Index which is assumed to increase at 5% per

interest rate. In general, the greater the risk in an investment the higher the risk premium.

This computed yield rate is somewhat misleading, however. After the fact, for this one bond the yield rate will actually turn out to be 14.89% if no default occurs. However, if total default occurs, the actual realized yield rate will be -100%. If partial default occurs, the realized yield rate will be somewhere in between.

We might be interested in knowing the probability of default which is implicit in the purchase of this high-risk bond. We define the expected present value (EPV) of a future payment as its present value multiplied by the probability of payment. We can compute the implicit probability of payment, denoted by p, as


940 = p


which gives p = .94. Thus, we have an implicit probability of default equal to .06. Note that the present value is computed at the risk-free rate of 8%.

The above analysis is valid as far as it goes, but it needs refinement. It is quite unlikely that bond purchasers would be willing to pay $940 for this bond if they really thought that die probability of default is as high as .06. Why would investors buy a high-risk investment with an expected yield rate of only 8% when they could buy a risk-free investment with the same yield rate?

Thus, a more reasonable inte retation of a price like $940 for such a high-risk bond is that the $60 price differential partially represents the probability of default and partially represents a higher return to the purchaser as compensation for the assumption of risk.

Let us assume that investors think that assumption of this level of risk is worth an extra 3% in die yield rate, i.e. 11% instead of 8%. Then, the implicit probability of payment p can be determined from


940 = /7


which gives p = .9661. Thus, the implicit probability of default is .0339. It is obvious that tiiis answer is not unique and that other combinations of yield rates and probabilities of default would also produce a price equal to $940.

The above analysis considers one bond bought in isolation. Now consider a diversified portfolio of bonds all of which are similar to the above bond, so that the law of large numbers applies. If die actual rate of default on the portfolio turns out to be .0339, then tiie yield rate on die entire portfolio will be 11 %. Of course, in practice the rate of default will not be exactly .0339 but will follow a probability distribution of some type. The results of mathematical

statistics can then be applied to make probability statements about various possible yield rates to be expected on the overall portfolio. This will be illustrated in Example 9.3.

Now, let us turn this illustration around and look at it another way. Assume we want to quantify the risk factor separately from the interest rate. Thus, if the market price of the bond is $940 and the risk-free rate of interest is 8%, then the value at the end of the year would be 940(1.08) = $1015.20. This way of quantifying risk adjusts the expected payoff downward from $1080 to $1015.20 to reflect risk, but then computes present values at the risk-free rate of interest.

This second approach illustrates an error that is sometimes made in financial analysis. One way of quantifying risk is to adjust the interest rate. A second way of quantifying risk is to adjust the payment amount. These are two different, but both valid, ways of quantifying risk. The mistake that is sometimes made is to do both, i.e. lower the expected payoff and raise the interest rate. This procedure is flawed, since it "double counts" the risk involved.

We now generalize these results to more complex sitiiations involving multiple payments. Consider a series of future payments R, Rj, . . . ,R that are made at times 1,2, ... ,n. For example, on a 10-year $10( bond with 8% annual coupons we have Ri= Rj = , = 80 and , = 1080. Assume that the probabilities of payment are p, p, . . . , respectively. Then the expected present value of this series of payments is given by

EPV = E Rt(\+iyPt (9.8)

j. where i is an appropriate rate of interest reflecting the risk involved as discussed I above.

tlf Formula (9.8) involves three key values: (1) the expected present value EPV, (2) the yield rate of interest i, and (3) a set of probabilities of ayment /?, for / = 1, 2,. . . , n. Sitiiations arise n practice in which we know two of diese values and wish to determine the tliird.

One obvious complication is that there a© many patterns of Iprobabilities p, tiiat might be used. One assumptioj that is often made in [Oonnection with default risk is to assume that the probability of default is Konstant during each period. Let this constant probability of default during one period be denoted by q. The corresponding probability of non-default is tiien = I - q. Under these assumptions, the probability tlat the rth payment will made is

Pt P (9.9)

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