Yield curve

As yield is referred to each other the ratio of different interest rates. The graphical illustration of this is called the yield curve (also the yield curve ). Often these terms are used interchangeably.

Interest rates depend generally on factors such as maturity, risk, tax treatment and / or other properties of the relevant financial instruments. In the following, the temporal term structure is considered, in which the dependence of the rate of interest is the duration of a bond investment ( Interest-bearing securities or time deposits ) or the term of the interest rate derivative in the foreground.

In different markets, there are different interest rate structures. Thus, the structures not only differ by currency, but also on the nature of the underlying asset (securities or interest rate derivative). These may optionally be further subdivided, so the structures of interest rate swaps also differ according to the reference interest rate.

With short end is called the term of up to one year and with the long end of the term of ten years.

• 3.1 Normal (rising ) interest rate curve
• 3.3 Inverse (falling ) interest rate curve
• 3.4 Irregular yield curve

• 9.1 Income Securities
• 9.2 swap markets
• 9.3 Future Trips
• 9.4 Interest rates of forward rate agreements
• 9.5 Determination of forward prices
• 9.6 Statistical methods

Importance

A major application of yield curves is the assessment (calculation of the present value ) of both interest rate derivatives such as interest rate swaps as well as fixed and floating rate securities. The sensitivity of the present value ( in derivatives ) or the price ( interest-bearing securities ) against changes in interest rates can be calculated thus.

In addition, the yield curve is also suitable for the calculation of implied forward interest rates or scenario analysis.

The yield has also for economists is very important to estimate the future development of financial markets and the economy.

Explanations for the existence of interest rate structures

There are three explanations why the interest rate is dependent on the retention period. These three hypotheses yield complementary partly, partly competing with each other.

The (pure ) expectations hypothesis

The pure expectations hypothesis follows from the assumption of full information efficiency of the market and the assumption of complete risk neutrality of the acting in the market subjects.

This gives the following picture:

• Will rising interest rates are expected on the market, so investors preferably invest in " short-term", ie the so-called demand on the short end of the yield curve increases. This consequently reduces the returns for title short term and the interest rate curve rises (normal yield ).
• Become falling interest rates expected on the market, the opposite occurs: Investors prefer to invest their capital in the long term at higher interest rates. Through the interplay of supply and demand, then the inverted yield to which it can often come before recessions developed.

The expectations hypothesis provides the conceptual basis for the calculation of forward rates and implied forward interest rates that reflect the expected spot interest rates.

Since Fama is known that forward rates predict the direction, but not the extent of interest rate changes.

The expectations hypothesis explains why in high-yield phase, the yield is often inversely and why the yield is increasing generally in interest rates are low. It does not explain why rising interest rate structures are the rule and inverse interest rate structures are the exception. In addition, it ignores the fact that long -term investments a higher interest rate risk than short-term.

The liquidity preference hypothesis

The liquidity preference hypothesis adds to the expectations hypothesis the fact that investors their future plans do not exactly know and therefore prefer to invest their funds in the short term. This is justified by the fear that one can make long-term liquid funds only on unfavorable terms again.

To motivate investors to long-term investments, so a liquidity premium is paid. This explains why the yield in most cases is increasing. Combining the statements of expectation hypothesis and liquidity preference hypothesis, one can deduce from the yield expected by the market interest rate, for example:

• A weak rising interest rate thus means that for long-running title, only the liquidity premium is paid and the market therefore does not expect changes in interest rates.
• A strongly rising interest rate means that the market expects rising interest rates: It is paid more than the liquidity premium for long-running titles compared to short-term bonds.

The liquidity preference hypothesis alone can not explain inverted yield curves.

The market segmentation hypothesis

The market segmentation hypothesis is based on the experience that there is no single unified system market, but that market participants operate in one segment and leave it rare. Thus, there is supply / demand situation in each segment, resulting in different rates of interest in the individual segments and thus a non- flat yield. Furthermore, it is assumed that due to lack of foresight and the resulting justified risk aversion, the market behavior of lenders is characterized by liquidity preference. This explains the predominantly normal shape of the yield curve. The influence of expectations on the evolution of interest rates on the yield curve includes the market segmentation hypothesis of principle.

Thus, market segmentation hypothesis is to explain the situation, why it also (but rarely) often lead to irregular interest rate structures, for example, with a hump. However, an explanation of why inverted yield curves often occur at high short-term interest rates, the model can not give. From the model assumptions also follows that securities of different maturities are intrasegmental not substitutable.

In addition to these yield theories or hypotheses that attempt to explain the shape of the yield by factors that are basically outside the financial markets ( expectations, preference for liquid possible assets and fixed, usually institutionally specific preference for very specific times ), there is the so-called term structure models in the strict sense. These have the more modest claim that correlations within the yield curve, that is, to explain the relationship between interest rates of different maturities.

Expressions of interest structures

A yield curve can have the following formations:

Normal (rising ) interest rate curve

The yield curve is usually increasing, ie for longer retention periods higher interest rates are paid. This may be the expression that the market expects higher interest rates in the future; also the longer lock-in period will be paid on a liquidity premium and a risk premium.

As the name suggests this is the most frequently occurring form of a yield curve.

This means that the interest of the commitment period are independent. Assuming that the market has a liquidity premium and risk premium charged, it means that falling interest rates are expected.

Inverse (falling ) interest rate curve

For long-term investments less interest will be paid as a short-term investment. This means that the market for the future of sharply falling interest rates are expected (eg by rate cuts by the central bank ) and therefore prefer to invest the actors in long-term investments. The resulting demand " at the long end " pushes the corresponding interest rates.

This was in the past often a sign of economic crises. But there are also extraordinary economic circumstances that require an inverted yield curve. Shortly after the reunification of Germany, the yield curve was inversely example.

Irregular yield curve

Among the irregular yield curves the " hunchback " (as pictured) is the most common.

Description of methods of interest structures

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

Swap or bond interest rates

The interest rate structure is a sequence of spot rates, for example, bonds or swaps. The spot rate interest rate for a period is valid from now until the relevant maturity. Since the payment structure of interest rate swaps with current conditions can be treated have a great analogy at par Quotation bonds them together.

This representation includes periodic interest to be paid. There is no reinvestment provided.

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 in the future must be multiplied to obtain the present value of this payment. Thus, a discount factor is just the price of a zero- coupon bond with the same maturity.

Spot interest rates steady

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

Spot rates and forward rates

From the yield can be calculated forward rates, which are rates which apply from a specified date in the future at a specified retention period.

A normal yield curve is when and a normal yield curve does not necessarily mean increasing one-period forward rates.

Example: y ( 2) = 0.1 and R (2,3) = 0.16, then Y (3) = 0.12 .. Let y (3) < R (3,4 ) = 0, 14; then y (4) = 0.125. we have a normal yield curve y ( 2) = 0.1 y ( 3) = 0.12 y ( 4) = 0.125, but not rising forward rates r (2,3) = 0.16, r (3,4) = 0.14.

The yield is a snapshot with respect to different maturities and makes no statement about the future. It can only calculate the implied forward rates. But these are not the same as the future spot interest rates generally.

Determination of the yield

The sources for the raw data are different depending on the observed yield curve. Optionally, if no primary data for a yield curve are available for specific jobs, this is copied from other yield curves.

Income securities

An important source of raw data here are the returns of class zero coupon bonds with different maturities, as well as coupon bonds, for example, the prices of government bonds are used. The coupon-bearing bonds bring the problem of distortion coupon ( The coupon has a different duration than the total bond) with itself. Therefore, the calculation is very difficult. In principle, of course, all the other variables, such as the creditworthiness of the borrower, be constant. If necessary, the yield curve is determined by the rates of the swap markets or market interest rates (LIBOR, ...). However, the interest rates on swaps can be empirically 30 to 40 basis points higher.

Swap markets

This is made use of the fact that swap rates are identical with coupons from bonds that are at par. With the help of the so-called bootstrapping the Zero Curve interest rates and the discount factors of the yield is determined from the currently traded swap rates then. When bootstrapping, we describe a method for determining the spot rate curve from market data. The discount factors are successively, starting with the smallest period is determined.

Special problems arise from the fact that zero-coupon bond yields are available only at annual intervals. This could be the assessment of an existing swap is not possible. However, this can be solved by interpolation. In this way, a fictitious yield on the maturity of T = ½ for example, can be determined.

Another question is whether the bid or the Offerswapsatz to be used. Here, the mean value can be taken.

There is also the question of the yield curve in less than one year. Pre- pull can be sure the money market interest rates, but this is unusual in that it is cash market interest rates. Alternatively, money market futures are used, from which the yield curve can be computed in less than one year by means of implied forward rates.

Future Trips

The values ​​of the short end of the curves are may also interest rates, which are deducted from money market futures.

Rates from forward rate agreements

An alternative method to determine interest rates at the lower end of the curve is to select this so that the interest rates of forward rate agreements are made.

Determination of forward prices

Determination of s- year forward prices on a coupon bond with residual maturity of x years. The discounted with the s- year interest rate forward price equals the present value of the bond purchased in s ( payments not considered until after s ).

Statistical methods

There are discrete distinguished from continuous process. Continuous processes include spline method, the Nelson-Siegel method and used by the Bundesbank Svensson method ( also extended Nelson-Siegel ). The Federal Bank utilizes the average effective interest rate of current coupon bonds (especially government bonds) to determine yield curves.

Arbitrage

Arbitrage opportunity at constant inverse yield

If an inverted yield after one year safely back in the same inverted yield over, there is an arbitrage opportunity.

You'll probably choose two strategies:

• Strategy A: rolling investment of 1 € over two years:
• Strategy B: Investment of € 1 with a two-year zero coupon bond:

Go Strategy A long and short B

Then arises today and after a year a payoff of 0 After two years, the payout is the difference: due to the inverted yield.

Other identification of arbitrage opportunities

Whether a yield curve provides arbitrage opportunities, it can be established by an arbitration panel is made or done a conversion in the forward curve / discount curve.

• In case of forward interest rates, there is an arbitrage opportunity if and only if there are negative forward rates (and you can hold cash " under the covers ").
• With discount curves there is an arbitrage opportunity if and only if they are not decreasing with time.