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Duration and Convexity of Bonds SJAS 2014, 11 (1): 53-66 ISSN 2217-8090 UDK: 336.763.3 DOI: 10.5937/sjas11-4766 Original paper/Originalni naučni rad DURATION AND CONVEXITY OF BONDS Slobodan Čerović1,*, Marina Pepić2, Stanislav Čerović3, Nevena Čerović4 1Singidunum University, Department of Tourism and Hospitality, 32 Danijelova Street, Belgrade, Serbia 2Nati onal Bank of Serbia 12 Kralja Petra Street, Belgrade, Serbia 3Singidunum University, Master’s graduate student 32 Danijelova Street, Belgrade, Serbia 4University of Belgrade, Faculty of Economics, Master’s student 6 Kamenička Street, Belgrade, Serbia Abstract: The wide impact that interest rate changes have on business performance, the fact that all market participants are, more or less, exposed to interest rate risk, as well as high volatility in interest rates in recent years, make interest rate risk one of the most significant risks. It is impossible to neutralize interest rate risk completely, but it is desirable to reduce it to a minimum. In order to effectively manage it, interest rate risk must first be identified and measured. This paper aims to show the two methods of measuring the interest rate risk - duration and convexity. Key words: The concept of duration is a good indicator of changes in the price of bonds Macaulay, but only for small changes in the interest rates. In case of major changes, modified, the duration gives overestimated/underestimated approximation of the effective, bond price, because bond price-yield relationship is not linear. Therefore, empirical and dollar duration, when measuring interest rate risk, convexity of bonds must be taken into duration of a portfolio, account. Modified duration and convexity taken together provide the best modified and effective convexity, approximation of the sensitivity of bond prices to changes in interest rates. convexity of a portfolio. INTRODUCTION terest rate changes have on business performance, it is necessary to reduce it to a minimum. In order to e wide impact that interest rate changes have manage it e ectively, exposure to interest rate risk on business performance, the fact that all market must rst be identi ed and measured. participants are, to a higher or lesser degree, ex- is paper represents an attempt to clarify the two posed to interest rate risk (regarding the value and basic methods of measuring interest rate risk – du- structure of balance and o -balance positions sen- ration and convexity, and to show their advantages, sitive to interest rate changes, volatility in interest as well as drawbacks. e third method of measur- rate, and duration of exposure to risk), as well as ing interest rate risk – value at risk (VaR) is not the high volatility in interest rates during the recent subject of this paper. years, make interest rate risk one of the most sig- e paper will rst explain the notion and charac- ni cant risks. erefore, it is of the utmost impor- teristics of the concept of duration, then the types of tance to manage this kind of risk adequately. duration, and nally the advantages and limitations It is di cult to completely neutralize interest rate of this concept. A er that the concept of convexity risk, however, in regard to the great impact that in- will be discussed, i.e. characteristics, types, advantages * E-mail: [email protected] 53 et al. SJAS 2014 11 (1) 53-66 andČerović limitations S. Duration of this and concept. convexity ofApplication bonds of the e required yield is a yield generated by market for concepts of duration and convexity to portfolio will bonds with xed maturity date and xed level of risk. be clari ed as well. In case of multiple annual coupon payments, bond price will be (Šoškić and Živković, 2006, p. 259): DURATION Ct M n A bond is an instrument of indebtedness that mm P = t mn (2) obliges the bond issuer (borrower) to repay the lender t1= ii 11 (creditor) the borrowed assets plus the interest within mm a certain period of time (Fabozzi, 2000, p. 1). ere are coupon and zero-coupon bonds. With coupon bonds where m is – number of coupon payments in a a periodic interest (coupon) payment is present. With year. zero-coupon bonds yield is the di erence between the purchase price of a bond and its face value, i.e. its With changes in market interest rates the required selling price in case it is sold before maturity. yield for the bond changes as well. When the coupon rate is equal to the required yield, the bond price will e nominal yield is bond yield based on coupons be equal to the principal. When the coupon rate is (Šoškić and Živković, 2006, p. 236). Since coupon greater than the required yield, the bond price will rate is xed and not sensitive to market changes, it be greater than the nominal value and this bond will is not of importance to the investors when making a be selling at a premium. When the coupon rate is less decision whether a bond investment is worthwhile. than the required yield, the bond price will be less erefore, current yield and yield to maturity are than the nominal value and this bond will be selling most o en used as yield indicators. at a discount. Current yield is a quotient of the coupon yield for Regarding zero-coupon bonds, as we said before, a certain period and the purchase price (Mishkin and yield is the di erence between purchase bond price Eakins, 2005, p. 258). Current yield is greater than and its nominal value, i.e. selling price if sold prior nominal (coupon) yield when the price of a bond to maturity. e zero-coupon bond price is equal to is less than nominal, and vice versa, current yield is the present value of the sum received on the maturity less than nominal when the price of a bond is greater date and it is calculated according to the following than nominal. Current yield is sensitive to changes in formula (Šoškić and Živković, 2006, p. 260): bond price and therefore a good indicator of whether a bond investment is worthwhile. M When reaching long-term decisions yield to ma- P = (3) turity of a bond is used for yield calculation. e ( 1 i ) n yield to maturity is the discount rate at which the value of current cash ows from the bond is equal Not only is the overall cash ow that bond rejects to the price of the bond (Šoškić and Živković, 2006, important for investment decision, but also the pe- p. 237). At the moment of the otation of a bond riod in which it happens. Maturity of the bond gives yield to maturity is equal to the market interest rate. information only on the date of the nal payment, If the market interest rate changes during a period but not on the size and date of coupon payments of time, the bond price will change as well. e price prior to the last payment of coupon and principal. of a coupon bond is equal to (Šoškić and Živković, e longer the period to maturity and the higher the 2006, p. 259): coupon rate and yield to maturity, the more impor- tant coupon payments become when compared to n payments upon the maturity of the bond. Maturity Ct M P = t n (1) of the bond, therefore, is not an adequate indicator t =1 ( 1 i) ( 1 i) of the time required to make investment in bonds worthwhile. us, the concept of bond duration has where P is – price of a bond, C – coupon value, been developed. M – nominal bond value, n – number of years to Duration is a measure of interest rate risk of bonds maturity, i – required yield or market interest rate. and it is used to determine the average period of ma- 54 et al. SJAS 2014 11 (1) 53-66 turity of the bond. e concept of duration measures market rates.Čerović Nevertheless, S. Duration for greater and convexity changes of bonds in price sensitivity of bonds or bond portfolios to the interest rates, the results are not that good. Namely, changes in interest rates (Choudhry, 2005, p. 32). because of the convexity of the price/yield curve (to Bonds with longer duration have higher changes be elaborated upon further on), estimated change in price than bonds with shorter duration, and that in bond value for greater rate changes will result in represents a greater risk. underestimated/overestimated bond value. Bond duration is an approximation of the per- centage change in bond price regarding the change Duration and yield to maturity in interest rate of 100 basis points. It is calculated according to the formula (Fabozzi, 1996, p. 49): Relation between yield to maturity and duration is shown by the following formula (La Grandwille, P P 2001, p. 81): D = − (4) 2 P( ∆i ) d D −1 =−1 i S (6) where D is – duration of bond, P – initial bond d i price, Δi – change in yield, P+ - estimated bond value in case of yield rise for Δi, P - estimated bond value - where S is dispersion measured by a variance of in case of yield drop for Δi. Values for P+ and P- are obtained using the model for bond valuation. It is the bond payment number. Because of ‘-‘ symbol dD/ important to stress that the parallel shi of the yield di will always be negative, which means that there is curve is expected, that is, the yield change Δi is the an inverse relation between duration and yield, i.e. same for all maturities. duration drops with the rise of yield to maturity and vice versa.
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