A Note on Using the Svensson procedure to estimate the risk free rate in corporate valuation

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1 A Noe on Using he Svensson procedure o esimae he risk free rae in corporae valuaion By Sven Arnold, Alexander Lahmann and Bernhard Schwezler Ocober 2011

2 1. The risk free ineres rae in corporae valuaion The risk free ineres rae is needed as an inpu facor o calculae he cos of equiy of a firm. Afer he recommendaion by he German insiue of CPA s (IDW) he so called Svensson mehod has gained significan aenion as procedure o derive an esimae for he risk free rae. This noe gives a shor overview. When esimaing he cos of equiy of a firm praciioners and researches alike face a echnical problem whenever here is a non-fla yield curve: The CAPM as a one period model asks for a single unique risk free ineres rae whereas in pracice he yields of riskless governmen bonds differ for differen imes o mauriy. 2. Yield-curve, erm srucure and he Svensson procedure Dealing wih risk free ineres raes in corporae valuaion requires knowledge of all differen ineres raes over he enire range of imes o mauriy (T). The yield curve graphically presens he yields o mauriy (YTM) of riskless governmen bonds. In conras he erm srucure displays exclusively he yield of zero-bonds as he spo raes over he differen mauriy. When looking for appropriae discoun raes he spo raes should be he firs choice; YTM are derived by looking a all bonds wih he same mauriy. The problem is ha here are very few zero-coupon bonds wihou any bankrupcy risk. Bonds issued by he German sae, or any oher bond wih an AAA raing issued by a sae, are almos free of bankrupcy risk bu hese bonds are no zero-coupon bonds. This problem can be solved by using a boosrapping procedure o derive he spo raes ou of a se of coupon bonds and heir marke prices. An addiional problem is ha even when considering all governmen bonds heir paymens will no cover he enire range of he mauriy specrum. When esimaing he erm srucure he poin is o fill up he gaps in he imes o mauriy. In an aemp o solve boh problems Svensson (1992, 1994) exended he earlier resuls by Nelson and Siegel (1987) and developed a funcional relaion o esimae he erm srucure. The funcional relaion by Svensson is given by

3 , (0.1) T T T τ 1 τ 1 τ2 T T T τ T T 1 τ 1 τ 2 1 e 1 e τ 1 e 1 τ2 f(t, β ) =β 0 +β1 + β2 e + β3 e r r(t, β) wih f being he risk free spo rae over ime o mauriy T as a funcion of he Bea-facors. β 0 o β 3 and τ 1,τ 2 are compued by a non-linear opimizaion program aiming o minimize he squared deviaion beween esimaed and rue ineres raes. The funcional form of equaion (0.1) allows for a wide range of poenial shapes of he erm srucure no covered by simple linear or log-based esimaion procedures. For German governmen bonds Deusche Bundesbank regularly esimaes and publishes he facors on is websies. hp://www.bundesbank.de/saisik/saisik_zeireihen.php?lang=de&open=&func=lis &r=www_s300_i03c In order o show poenial differences beween he esimaion procedures proposed by Nelson and Siegel (1987) and by Svensson (1994) we ve esimaed he erm srucure for Germany a he 20 h of June Graph (1) displays he resuls.

4 Yield Curves: Svensson (1994), Nelson / Siegel (1987) Svensson Bond yields by he german sae Nelson / Siegel ineres rae 5 4,95 4,9 4,85 4,8 4,75 4,7 4,65 4,6 4,55 4,5 4,45 4,4 4,35 4,3 4,25 4,2 4,15 4,1 4, ime in years

5 The Nelson-Siegel procedure overesimaes he spo raes for T higher han one year. The Svensson mehod is more precise and has he benefi o smooh ou a poenial marke mispricing due o illiquidiy as e.g. in he case of he 30 year Bundesanleihe. Graph I shows ha he assumpion of a fla erm srucure wih a unique ineres rae of all mauriies is no very realisic. In his case we observe a difference beween he spo rae wih mauriy 3 monhs and he one wih mauriy 30 years of 0.8%. Ignoring his difference would have significan influence on he value of company. Deusche Bundesbank does no direcly publish he erm srucure, bu provides daa and esimaes as inpu for he calculaion. Using he parameers published on he websie, you can esimae he German erm srucure on your own. From Q4 he erm srucure esimaes for Germany based on he Bundesbank daa will be calculaed and provided by every quarer. 3. How o proceed from here Knowing he erm srucure of riskless bond/asse does no ye compleely solve any problem in corporae valuaion. One sill has o incorporae he differen spo raes ino he discouning procedure in he case of a non-fla erm srucure. The valuaion lieraure proposes offers several ways o do his: a) Direcly applying he spo raes as discoun raes Direcly using risk free spo raes as discoun raes for presen value calculaions requires risk-adjused cash flow figures as cerainy equivalens. As he CAPM heoreically allows for calculaing marke based cerainy equivalens, his procedure is no very common. b) Deriving a single, equivalen risk free ineres rae The basic idea here is o find a unique single risk free rae ha, if applied upon he cash flows yields he same presen value han using he ime o mauriy specific spo raes. The derived unique rae is hen serving as he risk free ineres rae in he CAPM approach. The resul of his procedure depends on

6 he ime srucure of he cash flows discouned. In he valuaion lieraure you find recommendaions of a perpeual or a consan growh iming paern o be assumed. The equaion o be solved is hen = 1 [ ]( ) ( 1+ r ) [ ]( ) SCF 1+ g SCF 1+ g = r g Here r denoes he spo raes for corresponding mauriies, r he unique risk free ineres rae o be derived and g he growh rae assumed for he cash flows. As he risk free rae is direcly applied as discoun rae, he cash flows should already be risk adjused; so S(CF) denoes he cerainy equivalen of he cash flows o be valued. In pracice however his risk adjusmen is widely ignored when using his procedure. An exension of his idea is o ake he risk premium in he cos of equiy ino accoun when seing upon he equivalence equaion: = 1 [ ]( ) ( 1+ r + z) [ ]( ) ECF 1+ g ECF 1+ g = r+ z g If one uses he (unadjused) expeced value of he cash flows risk is aken ino accoun by he risk premium z. Boh approaches yield slighly differen resuls for he unique risk free rae. c) Deriving forward raes and compuing mauriy specific cos of equiy Knowing he erm srucure allows o derive he one period forward raes over he enire range of periods available. As hese forward raes can be locked in as ineres rae in he curren period wihou any furher risk, hey may well serve as risk free fuure ineres for any period. Adding a risk premium for equiy (derived via he CAPM) on op, allows hen for deriving a chain of oneperiod cos of equiy over he enire mauriy range. In his case, he valuaion of he cash flow sream should rely on a roll back approach sepwise using he one-period cos of capial. Noe ha in all he hree cases he sandard assumpion is ha he risk premium on op of he risk free rae is independen from mauriy i.e. consan for all ime o mauriies.

7 A final problem lef in a) and c) is coming from he fac ha in many counries (including Germany) he ime o mauriy specrum of bonds is limied o maximum of 20 o 30 years. Thus, for he calculaion of he erminal value here is no equivalen ineres rae of a bond wih an infinie ime o mauriy. In his case science and pracice alike recommend o use he yield of he bond wih he longes ime o mauriy available as an esimae for he reurn of a bond wih an infinie lifeime.

8 References Nelson, C. R. und A. F. Siegel (1987), Parsimonous modeling of yield curves, Journal of Business, 60, 4, pp Svensson, L. E.O. (1994), Esimaing and inerpreing forward ineres raes: Sweden , IWF Working Paper 114, Sepember. hp://www.bundesbank.de/saisik/saisik_zeireihen.php?lang=de&open=&func=lis &r=www_s300_i03c

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