Interest structure

The USD interest rate structure from 2000 to 2006: It can be clearly seen that short-term interest rates have risen over the years, while long-term interest rates have remained almost constant. The interest rate structure is currently flat to partially inverted.

Examples of normal ( 1973 ), flat ( 1990 ), inverse ( 1991 ) yield curves

The interest rate structure is the relationship between different interest rates. The graphical illustration of this is called the yield curve (also called the yield curve ). Often these terms are used synonymously.

Interest rates generally depend on factors such as the duration , risk , tax treatment and / or other characteristics of the financial instruments concerned . In the following, the temporal interest rate structure is considered, in which the dependence of the interest rate on the binding period of an investment ( bond or time deposit ) or on the term of an interest rate derivative is in the foreground.

There are different interest structures in different markets. The structures differ not only in terms of currency, but also in terms of the type of underlying asset (security or interest rate derivative). These can also be further subdivided if necessary, so the structures of interest rate swaps also differ according to the reference interest rate.

The short end denotes the term of up to one year and the long end the term of ten years or more.

meaning

A main application of yield curves is the valuation (calculation of the present value ) both of interest rate derivatives such as interest rate swaps and of fixed or variable interest rate bonds . The sensitivity of the present value (in the case of derivatives) or the price (in the case of interest-bearing securities) to changes in interest rates can thus be calculated.

In addition, the yield curve is also suitable for the calculation of implied forward yield principles and for scenario analysis .

The interest structure is also of great importance for economic researchers for estimating the future development of the financial markets and the economy.

Explanatory models for the existence of interest structures

There are three models to explain why the level of the interest rate depends on the period of commitment. These three interest rate structure hypotheses partly complement each other, partly they compete with each other.

The (pure) expectation hypothesis

The pure expectation hypothesis follows from the assumption of the complete information efficiency of the market and the assumption of the complete risk neutrality of the subjects trading in the market.

This results in the following picture:

If rising interest rates are expected on the market, investors prefer to invest in short-term investments, which means that demand at the so-called short end of the interest rate curve increases. This consequently reduces the returns for short-term securities and the yield curve rises ( normal yield structure ).
If interest rates are expected to fall on the market, the opposite occurs: Investors prefer to invest their capital at higher interest rates over the long term. The interplay of supply and demand then develops the inverse interest rate structure , which can often occur before recessions .

The expectation hypothesis provides the conceptual basis for calculating forward interest rates that correspond to the expected cash interest rates.

It has been known since Eugene Fama that forward rates predict the direction, but not the extent, of changes in interest rates.

The expectation hypothesis explains why the interest structure is often inverse in phases of high interest rates and why the interest structure is usually rising in phases of low interest rates. However, it does not explain why rising interest structures are the rule and inverse interest structures the exception. In addition, it neglects the fact that long-term investments have a higher interest rate risk than short-term investments.

The liquidity preference hypothesis

The liquidity preference hypothesis supplements the expectation hypothesis with the fact that investors do not exactly know their future plans and therefore prefer to invest their funds in the short term. This is justified by the fear that long-term funds can only be made liquid again under unfavorable conditions.

A liquidity premium is therefore paid to motivate investors to make long-term investments . This explains why the interest rate structure is usually rising. If you combine the statements of the expectation hypothesis and the liquidity preference hypothesis, you can derive the interest rate change expected by the market from the interest rate structure, for example:

A slightly rising interest rate structure means that only the liquidity premium is paid for long-term stocks and the market therefore does not expect any change in interest rates.
A sharply rising interest structure means that the market expects rising interest rates: more than the liquidity premium is paid for long-term securities compared to short-term bonds.

The liquidity preference hypothesis alone cannot explain inverse yield curves.

The market segmentation hypothesis

The market segmentation hypothesis is based on the experience that there is no single uniform investment market, but that market participants operate in one segment and rarely leave it. Thus there are supply / demand situations in each individual segment, which leads to different interest rates in the individual segments and thus a non-flat interest structure. Furthermore, it is assumed that due to a lack of foresight and the resulting risk aversion, the market behavior of lenders is characterized by liquidity preference. This explains the predominantly normal course of the yield curve. The market segmentation hypothesis basically excludes the influence of expectations about the development of interest rates on the yield curve.

The market segmentation hypothesis is thus able to explain why there are also (but rarely) irregular interest structures, e.g. B. with a hump. However, the model cannot provide an explanation as to why inverse yield curves occur more frequently at high short-term interest rates. It also follows from the model assumptions that securities with different maturities cannot be substituted intrasegmentally.

In addition to these interest structure theories or hypotheses, which aim to explain the course of the interest structure through factors that are basically outside the financial markets (expectations, preference for the most liquid assets possible and fixed, mostly institutionally determined preference for very specific terms), there are so-called interest structure models in the narrow sense. These have the much more modest claim to explain the relationships within the yield curve, that is, the relationships between interest rates of different remaining terms.

Characteristics of interest structures

A yield curve can have the following forms:

Normal (rising) yield curve

Normal yield curve

The yield curve is usually rising, i.e. H. higher interest is paid for longer commitment periods. This can mean that the market expects higher interest rates in the future; the longer commitment period is also compensated for with a liquidity premium and a risk premium .

As the name suggests, this is the most common type of yield curve.

Flat yield curve

This means that the interest is independent of the commitment period. Assuming that the market pays a liquidity premium and a risk premium, this means that interest rates are expected to fall.

Inverse (falling) yield curve

Inverse yield curve

Less interest is paid for long-term investments than for short-term investments. An inverse yield curve can be explained in different ways. The explanation depends on the respective interest rate theory (expectation theory, liquidity preference theory, market segmentation theory, preferred habitat theory). There is never just one possible theory to explain an inverse yield curve.

Irregular yield curve

Among the irregular yield curves, the "humped" one (as shown) is the most common.

Descriptions of interest structures

Four different descriptions of the interest rate structure, here based on the swap rates

There are several equivalent forms of description for a yield curve. This means that every interest structure can be clearly converted from one representation to another.

Swap or bond rates

The interest rate structure is a sequence of spot interest rates on, for example, bonds or swaps. The spot interest rate for a term applies from immediately to the corresponding term. Since the payment structure of interest rate swaps with current conditions has a great analogy to bonds quoted at par , these can be treated together.

This representation includes interest to be paid periodically. No accumulation is planned.

Discount factors (or zero coupon prices)

The yield curve is a sequence of discount factors . A discount factor is the factor by which a payment must be multiplied in the future in order to obtain the present value of this payment. Thus, a discount factor corresponds precisely to the bond price of a zero coupon bond with the same term.

Steady spot interest

The interest structure is a sequence of continuously calculated spot rates.

Spot rate and forward rate

Can from the yield forward rates (English rates forward are) calculated, these are rates which apply from a certain date in the future at a specified retention period.

A normal interest structure curve exists if and a normal interest structure does not have to mean rising single-period forward interest rates. ${\ displaystyle r (s, t)> y (s) \}$

Example: y (2) = 0.1 and r (2.3) = 0.16, then y (3) = 0.12 .., let y (3) <r (3.4) = 0, 14; so y (4) = 0.125. We have a normal interest structure y (2) = 0.1 y (3) = 0.12 and y (4) = 0.125, but not increasing forward interest rates r (2.3) = 0.16, r (3.4) = 0.14.

The interest rate structure is a snapshot of different remaining terms and does not allow any statements about the future. Only the implicit forward interest rates can be calculated. As a rule, however, these are not identical to the future spot rates .

Determination of the interest structure

The sources for the raw data differ depending on the interest rate curve considered. If necessary, if no original data is available for a yield curve for certain points, this is also taken from other yield curves.

Interest-bearing securities

An important source of raw data here are the yields of first-class zero - coupon bonds with various remaining terms, but also coupon bonds, e.g. B. the prices of government bonds are used. The coupon-bearing bonds bring with them the problem of coupon distortion (the coupon has a different maturity than the entire bond). Therefore, the calculation is very difficult. Basically, of course, all other variables, such as B. the creditworthiness of the debtor, be constant. If necessary, the yield curve is determined using the rates of the swap markets or market interest rates ( LIBOR , ...). However, the interest rates on swaps can empirically be 30 to 40 basis points higher.

Swap markets

This makes use of the fact that swap rates are identical to coupons for bonds that are quoted at par. With the help of so-called bootstrapping , the zero curve interest rates and the discount factors of the interest rate structure are then determined from the currently traded swap rates. Bootstrapping is a method for determining the spot rate structure curve from market data. The discount factors are determined successively, starting with the smallest period.

Since the following applies to a bond that is quoted at par:

{\ displaystyle 100 = cf_ {1} \ cdot df_ {1} + cf_ {2} \ cdot df_ {2} + \ cdots cf_ {n-1} \ cdot df_ {n-1} + (100 + cf_ {n }) \ cdot df_ {n}}

follows for the discount factor of the year :

{\ displaystyle \ n}

{\ displaystyle df_ {n} = {\ frac {100- \ sum _ {i = 1} ^ {n-1} cf_ {i} \ cdot df_ {i}} {100 + cf_ {n}}}}

or for the interest rate:

{\ displaystyle {\ sqrt [{n}] {\ frac {100 + cf_ {n}} {100- \ sum _ {i = 1} ^ {n-1} cf_ {i} \ cdot df_ {i}} }}-1}

where is the cash flow and discount factor for the year .

{\ displaystyle \ cf_ {i}}

{\ displaystyle \ df_ {i}}

{\ displaystyle \ i}

Special problems arise from the fact that yields for zero coupon bonds are only available at annual intervals. This means that the valuation of an old swap may not be possible. However, this can be solved by interpolation. In this way, for example, a fictitious return on the remaining term of T = ½ can be determined.

Another question is whether the bid or offer swap rate should be used. The mean value can be taken here.

In addition, the question of the yield curve in the intra-year area arises. The money market interest rates can be used for this , but this is unusual since it is spot market interest rates. Alternatively, money market futures are used, from which the yield curve in the sub-year range can be calculated using implicit forward rates.

Futurestrips

As values for the short end of the curve also those derived from suitable in circumstances Money Market - Futures are eliminated.

Interest rates from forward rate agreements

An alternative method of determining interest rates at the lower end of the curve is to choose this in such a way that the interest rates are matched by forward rate agreements .

Determination from forward prices

Determination from s-year forward prices on a coupon bond with x years remaining term. The forward price, discounted using the s-year interest rate, corresponds to the present value of the bond purchased in s (payments only taken into account from s).

Statistical procedures

A distinction is made between discrete and continuous processes. Continuous processes include the spline process , the Nelson Siegel process and the Svensson process used by the Bundesbank (also known as the extended Nelson seal ). The Bundesbank uses the average effective interest rate on current coupon bonds (especially Bunds) to determine yield curves.

arbitrage

Arbitrage possibility with constant inverse interest structure

If, after one year, an inverse interest structure is sure to change back into the same inverse interest structure, there is an arbitrage possibility.

There are two strategies:

Strategy A: rolling investment of € 1 over two years: ${\ displaystyle EV = (1 + y (1)) ^ {2}}$
Strategy B: Investment of € 1 with a two-year zero bond: ${\ displaystyle EV = (1 + y (2)) ^ {2}}$
Strategy C: Investment of € 1 with a three-year zero bond: ${\ displaystyle EV = (1 + y (3)) ^ {3}}$

Go strategy A long, B short and C middle-short

Then there is a payout of 0 today and after one year. After two years, the payout is the difference: due to the inverse interest structure. ${\ displaystyle (1 + y (1)) ^ {2} - (1 + y (2)) ^ {2}> 0}$

Other determination of arbitrage opportunities

Whether a yield curve offers arbitrage opportunities can be determined by creating an arbitrage table or by converting it into the forward curve / discount structure curve.

When forward rate sets there is an arbitrage opportunity if and only if there are negative forward rates (and you can hold cash "under the covers").
With discount structure curves there is an arbitrage option if and only if they are not falling over time.

literature

Jessica James, Nick Webber: Interest Rate Modeling . Wiley Finance, 2000. ISBN 0-471-97523-0 .
Riccardo Rebonato: Modern Pricing of Interest-Rate Derivatives . Princeton University Press, 2002. ISBN 0-691-08973-6 .
Andrew JG Cairns: Interest Rate Models An Introduction . Princeton University Press, 2004. ISBN 0-691-11894-9 .
Damiano Brigo, Fabio Mercurio: Interest Rate Models. Theory and Practice: With Smile, Inflation and Credit . Springer Finance ISBN 978-3-540-22149-4

Web links

Daily interest rate structure on the bond market at the Bundesbank
Explanations at riskglossary ( Memento from March 15, 2015 in the Internet Archive )
German yield curves on Bloomberg (Engl.)
FAZ newspaper article
Article on the estimation of yield curves at the Bundesbank