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Výpočet Macaulayovy durace krok za krokem

V minulém článku byl představen základní koncept durace dluhopisu podle svého tvůrce F. Macaulaye. Pojďme nyní projít výpočtem Macaulayovy durace u konkrétního vybraného existujícího dluhopisu. Duraci není nezbytně nutné vypočítávat tímto způsobem, existují například internetové kalkulačky, které tuto funkci zastanou. Funkce pro tyto výpočty jsou obsaženy též v různých tabulkových procesorech (např. MS Excel, Numbers, atd.). Cílem je však poodkrýt samotný postup a logiku. Pro podpůrné výpočty je vhodné použití například Excelu nebo některého z jiných tabulkových softwarů. Obrázek 1: výčet očekávaných příjmů do splatnosti dluhopisu Vybraným dluhopisem je státní dluhopis České republiky s poměrně dlouhou dobou do splatnosti, který má označení CZGB 2.0/33 . Tento reálný dluhopis má ISIN CZ0001005243 . Dluhopis má splatnost v roce 2033, přesně tedy 13.10.2033 . Dluhopis vyplácí jednou ročně kupon ve výši 2 % . Kupon je splatný vždy k 13.10. daného roku. Pro účely výpočtu je t

Bond Duration (Durace dluhopisu)

The term duration is often used in connection with bonds. The use of this term is sometimes confusing, because it has two different meanings, which have their specific conceptual distinctions. The first definition of the word duration is the so-called Macaulay duration , which got its name from its creator, the Canadian economist Frederick Macaulay. Macaulay duration measures the average time, which is essential for the investor to obtain all the future income from the bond. We can as well use the explanation, that the duration is the average time necessary for recovering the invested amount by the income from the investment. The result of the calculation of the Macaulay duration provides number of years. The second used definition of the duration is the so-called modified duration , which measures the sensitivity of the price of a bond on a change of the market interest rate. Modified duration provides a number of percentage points by which price of a bond changes as a result of a

Bond Valuation: Part 2: Market Interest Rates (Ocenění dluhopisů: Část 2: Tržní úrokové sazby)

The first part of the series on Bond Valuation included the effects of creditworthiness of the issuer on the pricing of a bond. Let us now review how market interest rates influence the bond prices. Generally, market interest rates do have a bigger impact on bonds of the “risk free” category, i.e. government bonds of countries with the highest investment grade ratings. These are, for example, the United States of America, Germany, the United Kingdom, Japan, and so on. The generic rule says that increasing market interest rates do cause the prices of bonds to decrease, i.e. their yield to maturity is growing, and vice versa: decreasing market interests are causing increasing prices of bonds, i.e. their yield to maturity is decreasing. It is therefore obvious that the yield to maturity tends to correlate with the market interest rates. The correlation is obvious throughout the whole time range, i.e. the yield to maturity of a two year bond tends to be in line with two year market rat

Bond Valuation: Part 1 (Ocenění dluhopisů: Část 1)

The recent article presented the concept of the yield to maturity as an universal indicator, which provides the real annual yield offered by a given bond. The universality of this indicator enables to compare various bonds between each other. The yield to maturity derives from the price, for which the given bond can be bought. If the price of the bond grows, its yield to maturity decreases and vice versa. Which factors influence the bond price, respectively its yield to maturity over time? First, it is the creditworthiness of the issuer , and second, it is the development of market interest rates . Market interest rates will be reviewed in the next article. Below we are going to look at the issuer’s creditworthiness. Creditworthiness of the Issuer The more is the issuer creditworthy the lower are his borrowing costs. Among the most reliable issuers are being considered states and their governments, and that is why their bonds are usually more expensive, respectively they offe

Yield to Maturity (Výnos do splatnosti)

In the recent article it is mentioned that issued bonds are being traded in a so-called secondary market. Bonds are traded for a price, which is being generally set by supply and demand. Factors that determine bond prices will be reviewed in detail in the next article. Now, let us focus on one of the very significant parameters, which are being closely observed by bond investors: the so-called yield to maturity (often abbreviated to YTM ). The yield to maturity is a number presented in percentage points, which says to the bond investor what is going to be the real annual profit (yield) when purchasing a given bond. The key parameter, or say a variable, for the calculation of the yield to maturity is the actual price of the bond, for which the investor can buy it. The investor knows that the particular bond generates a fixed annual coupon, and that in the maturity of the bond, he gets back the nominal value of the bond. Let us show the concept of the yield to maturity in an exem

The Aspects of Trading Fixed Income (Aspekty obchodování s dluhopisy)

Bonds are traded in markets in percentage points of their nominal values. Whether in financial media, banks, or on stock exchanges, we can often see bond prices quoted as, for example, 101 %, 103.2 %, etc. Practically it means that one piece of a bond, which has, say, a nominal value of USD 1 000, can be really bought for USD 1 010 in case of a price of 101 %, or for USD 1 032 in case of a price of 103.2 %. Bond traders do speak their specific language. It can be heard, for example, something like: "I sell the nominal of a million dollars of bond XYZ for 101". It means he wants to sell, say, 1000 pieces of a bond with a nominal value of USD 1000 a piece for a price of 101 %, i.e. he expects to receive for the whole nominal a sum of USD 1 010 000. Primary versus Secondary Market A new bond issue is usually first offered by the issuer to big institutional investors within a so-called primary auction . By the primary auction the bond gets to the market. After the pri