INTEREST RATE RISK IN BOND INVESTMENT - UNCONVENTIONAL MEASUREMENT METHODS

Kamatni rizik obveznice najčešće se meri trajanjem i konveksnošću. Međutim, ove mere polaze od pretpostavke o ravnoj krivi prinosa i njenom paralelnom pomeranju. Za modeliranje realnijih slučajeva koriste se njihove modifikacije. Fisher-Weil-ovo trajanje služi za merenje osetljivosti na paralelno pomeranje neravne krive prinosa. Mere M-apsolutno i M-kvadrat pokazuju u kojoj meri je portfolio obveznica imunizovan na neparalelne promene krive prinosa, uzimajući u obzir dati vremenski horizont ulaganja. Neravna kriva prinosa može se aproksimirati i skupom odabranih ključnih kamatnih stopa, čija trajanja i konveksnosti mere osetljivost portfolija na promene ovih pojedinačnih kamatnih stopa.


Introduction
Interest rate risk is one of the most significant risks when it comes to investing in bonds, arising from the possibility of a bond price to shift in the opposite direction from the changes in market interest rates.Traditionally, to measure this risk we use duration and convexity.Yet, the problem with these indicators is that they do not take into account the fact that a yield curve often is not flat, and that it frequently does not record parallel shifts.
The first part of the paper explains the main indicators of interest rate risk, after which we introduce their modifications taking into consideration a non-flat yield curve.The second part will be focusing on interest rate risk indicators measuring the portfolio's cash flows dispersion in relation to the given time horizon of the concerned investment.It is demonstrated how they should be minimized in case that nonparallel shifts of the yield curve are expected.Starting from the assumption that the yield curve can be approximated by means of a finite number of the so-called key rates, the third part illustrates how to use them to model its nonparallel movements.

Classic Interest Rate Risk Measurements
Given that the bond price is determined by its maturity, coupon rate and yield to maturity, the prices of bonds with different coupon rates and different maturities will react differently to the identical change in yield to maturity.Bond duration is a measure used to compare the sensitivity of various bonds, and to calculate the expected changes in the bond portfolio's value, given that the calculation of individual changes would be inefficient.The simplest duration measure is Macaulay duration (D), calculated according to the following formula: with: y -yield to maturity, T -time to maturity, P -price, CF t -cash flow at the moment t.In other words, Macaulay duration is the weighted time until the bond's maturity, with time to maturity of each cash flow being weighted by the share of that cash flow's present value in the bond's price.This implies that the duration of a zero coupon bond equals time to maturity, given that it only has one cash flow, maturing at the end of its life cycle.For all coupon bonds duration is necessarily shorter than time to maturity, because the weight of the last cash flow will be less than 1, and the weights of earlier periods will increase.
Duration of a bond, just like its price volatility, depends on the coupon rate level.The higher the coupon rate, the higher the cash flows disbursed before maturity, and thereby also their present value (especially of earlier cash flows, due to their higher discounting factor ), hence the bond duration decreases.On the other hand, duration typically increases in parallel with maturity, only to start declining at one point in case of coupon bonds sold at a huge discount.In case of all coupon bonds (both discount and premium), as the maturity increases, duration approaches the duration of a perpetuity at the given yield to maturity (a bond with no maturity (i.e.perpetual bond) can be proven to have Macaulay duration which equals (1+y)/y).However, it turns out that the duration is also affected by the size of yield to maturity -the higher the initial yield, the lower the duration.(Increased yield leads to a reduction of all discount factors, with a relatively higher reduction of discount factors in later years, which results in bigger weights being awarded to initial cash flows.) If we divide Macaulay duration by gross yield to maturity, we get modified duration (D m ), which indicates the percentage of the price change if the yield changes by one percentage point: We can see that the first derivative is used to approximate the price change.Based on the definition of differentials, we know that it is the multiplication of the first derivative of the function and the infinitesimal change of its argument (i.e., ), which means that it represents an almost perfect approximation for very small changes of the argument (in this case, yield to maturity), and a perfect approximation only if the function is linear (when its first derivative is constant).veće promene prinosa.Budući da prvi izvod u geometrijskom smislu predstavlja tangentu na grafik funkcije, i znajući da je funkcija konveksna, zaključujemo da će aproksimacija prvim izvodom nužno dovesti do potcenjenosti cene u odnosu na njenu stvarnu vrednost.Ova potcenjenost će biti utoliko veća ukoliko kriva cena/prinos više odstupa od linearnog oblika (to jest, što je više zakrivljena / konveksna).
Trenutna forvard stopa za dospeće t -f (t)definiše se na sledeći način: odnosno: The relationship between the bond price and its yield is most frequently convex, which implies that we cannot use linear approximation for bigger changes in the yield.Given that the first derivative in geometrical terms represents a tangent on the function graph, and knowing that the function is convex, we may conclude that the first derivative approximation will necessarily lead to an underestimated price in relation to its real value.This underestimation will be all the bigger if the price/yield curve deviates more significantly from the linear form (i.e. the more curved/convex it is).
The bigger changes of yield result in more substantial deviations of projected prices from the real ones.To eliminate this drawback, an additional measurement is used, the so-called convexity, which is related to the slope of the price-yield curve.
The price change can be approximated by means of the second-degree Taylor polynomial: with C = representing the bond's convexity.Therefore, the convexity equals: The value of convexity itself does not have any useful meaning.It needs to be linked with the squared change in yield.
The point of calculating duration and convexity is in their usage to hedge the portfolio against interest rate changes.Such a hedge is referred to as immunization, because the portfolio is being immunized, i.e. made "immune" to interest rate changes.The essence of immunization is reflected in the losses based on the immunized portfolio's value being compensated by the gains in the hedged portfolio's value (and vice versa).The hedged portfolio is constructed in such a way as to make its duration and convexity equal to those of the immunized portfolio, whereas its value is the opposite (which means that a classic portfolio is immunized by means of short sales, whereas future obligations get immunized by purchasing hedge instruments).
Macaulay and modified duration imply that the yield curve is flat, because all cash flows are discounted by the same rate of return.There are other duration measurements based on more realistic assumptions.One of them is Fisher-Weil duration, defined by means of spot rates, in the following way (Urošević, Božović, 2009, p. 184): with s t being spot rates in case of continuous interest income.It measures the sensitivity of the bond price to the parallel shifts of the spot curve.If we mark the degree of spot curve movements with λ, the new price amounts to: and its sensitivity to λ is: so that: A similar measurement is the quasimodified duration, using spot rates calculated at the annual level: with:

M-Absolute and M-Square
Each term structure of zero-coupon yields responds to a unique term structure of instantaneous forward rates.We need them for the purpose of detecting more easily certain rules in respect of interest rate risk indicators examined in this chapter.
Izjednačavajući trajanje portfolija sa vremenskim horizontom ulaganja, portfolio se štiti od malih i paralelnih pomeranja krive prinosa.Za razliku od toga, minimiziranje M-apsolutnog uglavnom neće u potpunosti zaštititi portfolio od takvih promena, ali će uzeti u obzir mogućnost većih i/ili neparalelnih promena krive prinosa.Zbog toga će izbor između ove dve mere zavisiti od toga kakve se promene krive prinosa očekuju.Therefore, each change in the term structure the deviation being caused by the changes in interest rates at the moment t=0. of zero-coupon yields, ∆s , responds to a unique change in the term structure defined through instantaneous forward rates, ∆f(t).However, instantaneous forward rates are more volatile than zero-coupon yields, given that zerocoupon yields in a way represent the average of instantaneous forward rates.We observe that instantaneous forward rates are higher than Let us define the constants K 1 , K 2 the following manner: K ∆f(t) for each t K ∆f(t) for each t and K 3 in zero-coupon yields when the (zero-coupon) yield curve has an upward slope, and vice versa.
The previous section illustrates that all duration indicators (Macaulay, modified, Fisher-Weil, and quasi-modified) start from the assumption that the yield curve has parallel shifts.Immunization by means of duration will, therefore, imply that one should construct a hedged portfolio whose duration equals the duration of the immunized portfolio.In the process, it does not matter which bonds the portfolio manager uses for the purpose, i.e. because the portfolio containing, for instance, zero-coupon bonds with 2-and 10-year maturity, will be equally acceptable as the portfolio containing only zero-coupon bonds with 6-year maturity.
The interest rate risk measures that we hereby define depend on the time horizon H, at the end of which we want the portfolio to be immunized.The first such measure is called M-absolute, because it represents the weighted absolute value of the difference between time to maturity of cash flows and time horizon of the concerned investment: It can be deduced that Fisher-Weil duration is a special case of M-absolute when H=0.We can see that M-absolute equals zero only when the portfolio contains a zero-coupon bond whose time to maturity is the same as the investment's time horizon.
M-absolute is the indicator which, as opposed to duration, should be minimized.In order to understand why, we will begin from the fact that the portfolio manager wishes to minimize the deviation of the portfolio's value If we assume that CF t ≥0 for each t, we get the following (Nawalkha, Soto, Beliaeva, 2005, p. 105): K 3 depends on the changes in instantaneous forward rates, hence it is beyond the portfolio manager's control.What he can affect, though, is M A , which may be reduced by selecting securities whose cash flows are closer to the time horizon H.
By setting the portfolio's duration to be equal to the investment's time horizon, we protect the portfolio from small and parallel shifts of the yield curve.As opposed to that, the minimization of M-absolute in most cases will not completely protect the portfolio from such changes, but it will take into account the possibility of bigger and/or non-parallel shifts of the yield curve.Thus, the choice between these two measures depends on the type of the expected changes in the yield curve.
A comparison of immunization strategies based on duration, i.e.M-absolute, was conducted by Nawalkha and Chambers (2009) focusing on the data for the period 1951-1986.According to their results, the application of M-absolute more than halved the deviations from the portfolio's target value, compared with the duration-based immunization strategy.
The second measure depending on the time horizon is M-square: Given that in case of continuous interest income we apply the following formula: Vidimo da je M-kvadrat jednako nuli samo u slučaju beskuponske obveznice sa dospećem u trenutku t=H, kao i da konveksnost predstavlja specijalan slučaj M-kvadrata za H=0.
Nawalkha, Soto and Beliaeva (pp.106-107) The coefficient γ can be further broken support the following inequality: If the portfolio is immunized in terms of duration, D=H, it implies that the bottom limit of the deviation from the portfolio's target value depends on the constant K 4 (reflecting the change in term structure) and the M-square measure.In other words, for the given duration, the portfolio manager would want to minimize the portfolio's M-square.
In case that the duration is given, M-square represents a linear transformation of convexity.Convexity is a desirable characteristic of a bond, because higher convexity implies higher profit in case of declining interest rates, i.e. smaller loss in case of their growth.On the other hand, the portfolio manager would strive to minimize the portfolio's M-square.Since higher down to two segments: convexity effect (CE) and risk effect (RE): Immunization of the portfolio implies that D=H, hence we see that the immunized portfolio's yield depends on the risk-free component and the size of M-square multiplied by the ratio of the two mentioned effects.Convexity effect is always positive, regardless of whether the interest rates grow or decline.On the other hand, the risk effect depends on whether the slope of the yield curve is going up or down.If the slope increases, the first derivative of the change is positive, hence convexity inevitably entails higher M-square, the value of the coefficient γ decreases, and, this contradiction is known as convexity-Msquare paradox.
If instantaneous forward rates change by ∆f(t) at the moment t=0, the yield achieved until the time H will equal: i.e.: If the term structure remains unchanged in the period from t=0 to t=H: representing a risk-free rate of return of the in turn, the total yield drops.And vice versa, the yield increases if the slope goes down.The point of the convexity-M-square paradox is that convexity only takes into account the changes in the yield curve level, i.e. the parallel shifts in its term structure, whereas non-parallel shifts, including the changes in the slope, increase the absolute value of the risk effect.Therefore, the portfolio manager will strive to reduce the portfolio's M-square, if he expects non-parallel shifts in the term structure.
Treći problem se ogleda u tome da je promena pojedinačne ključne kamatne stope malo verovatna u praksi, iako zajednička promena svih ključnih kamatnih stopa može verno predstaviti promenu terminske strukture.Naime, izolovana promena jedne (beskuponske) kamatne stope implicira neobičnu deformaciju krive trenutnih forvard stopa.Ovaj problem se rešava posmatranjem forvard stopa umesto beskuponske krive prinosa.Taj metod se naziva pristup parcijalnih izvoda (eng.partial derivative approach).On podrazumeva da se kriva trenutnih forvard stopa podeli u više segmenata, a zatim se pretpostavlja da se sve forvard stope u jednom segmentu pomeraju paralelno.Parcijalno trajanje koje odgovara svakom segmentu se onda definiše kao relativna osetljivost portfolija na promenu forvard stope koja predstavlja taj segment.Razlika u odnosu structure changes can be approximated by means of changes in a finite number of selected, representative interest rates.Every individual researcher chooses which interest rates to use.For instance, Ho suggested 11 interest rates, with respective maturities of 3 months, 1, 2, 3, 5, 7, 10, 15, 20, 25 and 30 years.Let us assume that there were N selected interest rates, with maturities ranging from t 1 , t 2 ,..., to t N .For the sake of simplification, let us assume that the maturities of the bond's cash flows are harmonized accordingly.In that case, the change in interest rate for maturity t i will lead to the change in bond value proportionate to the key rate duration with maturity t i : with key rate duration being defined as relative sensitivity of the bond to the changes in interest rate y(t i ): Total change in the bond price equals the summation of N individual effects: We should underline that the change in interest rate with maturity t≠t i is calculated by linear interpolation of changes of its "neighboring" key rates.For instance, if ∆y(7)=0.5% and ∆y(10)=0.8%,then ∆y(8)=0.67*0.5%+0.33*0.8%=0.6%.
In case of bigger changes in term structure, it is necessary to introduce key rate convexities: A change in the bond value is then approximated in the following manner: In case of parallel shifts of the yield curve, we get the following Taylor approximation: Duration and convexity of the portfolio's key rates equal the weighted duration, i.e. convexity of all bonds in the portfolio, with the weights equaling the share of each bond in the portfolio: There are three limitations to this model observed in practice (Nawalkha, Soto, Beliaeva, 2005, p. 281).The first objection is that it does not take into account the historical movements of the term structure, which is why significant information is omitted concerning the volatility of different yield curve segments.Certain authors solved this problem by integrating covariance of interest rates changes into the model.
The second objection refers to the arbitrary selection of key rates.Given the lack of a clear criterion, the solution may be the selection of those interest rates which most substantially affect the concerned portfolio.For instance, the portfolio manager of a money market investment fund will be focusing on short-term interest rates.
The third problem is reflected in the fact that a change of individual key rates is highly unlikely in practice, even though the common change of all key rates may truthfully represent the change in the term structure.Namely, an isolated change of a single (zero-coupon) interest rate implies an atypical deformation in the instantaneous forward rates curve.This problem is solved by observing forward rates instead of the zero-coupon yield curve.This method is called partial derivative approach.It implies that the instantaneous forward rates curve is divided into several segments, and the assumption is made that all forward rates within each segment record parallel shifts.Partial duration in respect of each segment Jedno od njih je Fisher-Weil-ovo trajanje, koje se definiše pomoću spot stopa, na sledeći način (Urošević, Božović, 2009, str.184): gde su s t spot stope pri kontinualnom ukamaćenju.Ono meri osetljivost cene obveznice na paralelno pomeranje spot krive.Ako stepen pomeranja spot krive označimo sa λ, nova cena iznosi: a osetljivost na λ iznosi: tako da je Slična mera je i kvazimodifikovano trajanje, koje koristi spot stope obračunate na godišnjem nivou: gde je svakoj promeni terminske promene kamatnih stopa u trenutku t=0.strukture beskuponskih stopa, ∆s , odgovara jedinstvena promena terminske strukture Definišimo konstante K 1 , K 2 način: , obavili su Nawalkha i Chambers (2009) na podacima za period 1951-1986.Njihov rezultat je da je primena M-apsolutnog više nego dvostruko smanjila odstupanja od ciljane vrednosti portfolija u odnosu na strategiju imunizacije pomoću trajanja.Druga mera koja zavisi od vremenskog horizonta je M-kvadrat (eng.M-square): value at the moment H, i.e.V H .In other words, what is minimized is ∆V /V , H H bond with maturity H.After the approximation R(H) by means of the seconddegree Taylor polynomial we get the following equation (Nawalkha, Soto, Beliaeva, 2005, p. 109):