Discount structure

The discount structure is the representation of zero bond prices in a homogeneous market segment depending on the remaining term . A homogeneous market segment can be federal bonds, government bonds, Pfandbriefe or corporate bonds.

By definition, zero bonds do not yield any interest; their return is the difference between the purchase price and the repayment price of i. d. R. 100%. The discount structure or its graphic representation, the discount structure curve, indicates at which rates (in%) zero bonds of a homogeneous market segment (for example the segment of zero bonds from impeccable debtors in a certain currency) with different remaining terms at the time of observation are traded.

Annual return

The annual return on zero bonds can be determined from the discount structure using the following formula:

{\ displaystyle r = {\ sqrt [{t}] {\ frac {R} {A}}} - 1}

with r = annual return , t = remaining term in years, R = repayment rate in% at the end of the term and A = purchase price in%.

properties

A stable discount structure must be free of arbitrage , i.e. the combination of the financial instruments represented in it ( duplication ) must not allow immediate, risk-free profits. Arbitrage-free discount structure curves fall monotonously in the remaining term, i.e. the longer the remaining term, the lower the price of the zero bond must be.

In general, the arbitrage strategy is: H. the zero bond in an earlier period is cheaper than in a later period: ${\ displaystyle B_ {0} (T_ {1}) <B_ {0} (T_ {2})}$

Buy the zero bond with a short remaining term
Sell the zero bond with a longer remaining term
0% investment (cash management)

Example of a discount structure that is not free of arbitrage

Remaining term	0	1 year	2 years	3 years	4 years
Zero bond course	100%	97%	93%	89%	90%

Here it would be possible to make a risk-free profit: If you sell four-year bonds for 90 EUR (i.e. take out a loan) and at the same time buy three-year bonds for 89 EUR, you immediately have one euro left, while you have no more open payment positions: After three years you receive 100 EUR from the three-year bond, with which you can repay the four-year bond one year later.

Connection to the interest rate structure

Coupon bonds can be synthetically generated (duplication) through a portfolio of zero bonds . To do this, the prices are simply taken from the discount structure curve. The yield curve can thus also be derived from a discount structure curve .

{\ displaystyle DSK (T) = {\ frac {NW} {(1 + y_ {T}) ^ {T}}}}

${\ displaystyle DSK (T)}$ is the price of a zero bond with the remaining term T (in% of the nominal value NW)
${\ displaystyle y_ {T}}$ is the value of the yield curve for the remaining term T (in%)

or alternatively:

In arbitrage-free markets, the current price of a zero bond corresponds to the fair market price. The Yield To Maturity (YtM) can be calculated from the current rate depending on the remaining term.

{\ displaystyle y_ {T} = \ left ({\ frac {NW} {K}} \ right) ^ {\ frac {1} {T}} - 1}

T is the remaining term
NW the face value
K the current rate corresponds to the discount structure DSK ( T ).

There is a clear (bijective, i.e. reversible) relationship between the YtM and the price, so they also contain identical information.

Discount structure

contents

Annual return

properties

Connection to the interest rate structure

See also