D 1 D 2 D 3 D. Stock Valuation Draft: 10/24/2004. Stock Valuation

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1 Stock aluatio aft: /4/4 Stock aluatio This web documet shows how we ca use the techiques developed to hadle time value poblems to value stock. Two documets that we will efe to occasioally ae Compoudig ad iscoutig ad Auity eivatios. We will assume that the stock may be puchased ad sold at discete poits t, t, t, etc., sepaated by a costat iteval though time, ad that the fim may distibute divideds of vaious amouts to shaeholdes at these discete poits. We will futhe assume that if you puchase the stock, say at t, you will be etitled to, the divided at t but ot to, the divided at t. Of couse, if you sell you stock at ay futue poit, say at t, you would the be etitled to the divided athe tha the peso puchasig the stock at that time. This distibutio scheme mios eality if we assume that the discete poits ae all ex-divided days, o days whe puchase of the stock o loge etitles you to the ecetly aouced divided. If we assume that we puchase the stock at t ad hold the stock foeve, a time diagam of all futue cash flows fo ou cash flow will look as follows: - The puchase pice of the stock will equal. We will shotly show that we eed ot hold the stock util ifiity, ad that the value of the stock will be the same eve if we pla to sell it i the vey ea futue. I geeal, the divided amouts may vay ove time. Oe model fo the stock pice would be the peset value of all the cash flows fom t though t, discouted at a appopiate ate. This ate should be geate tha the isk- 4 Steve Feud

2 Stock aluatio aft: /4/4 fee ate because futue divideds ae ucetai. Usig the peset value discout fomula fo a steam of cash flows, the value of the stock should be: o t ( t t ( o t foget that i the above fomula, epesets a ate of etu that you thik is appopiate fo the level of isk associated with this stock. It will be much highe tha the isk-fee ate of etu, ad could be specific to the stock beig valued. I fact, this discout ate is meat to captue the isk of the stock. The highe the isk, the highe the discout ate, ad the lowe the esultig stock value. If the divideds ae ealized as expected, ad if you ca puchase the stock fo a pice equal to, you etu o the ivestmet will equal. This should be clea if you ecall the cocept of iteal ate of etu o IRR. Notice that we use symbol fo the value at t to diffeetiate fom the pice. We allow fo the possibility that the maket pice could diffe fom the theoetical value. Equatio expaded would look as follows:... ( ( ( ( ( You might wode how difficult it would be to use this equatio fo a steam of ifiite tems. I a pactical sese, divided paymets vey fa ito the futue may ot be woth much i peset value tems. A five dolla divided paid i yea 5 usig a % discout ate is woth $.46, ad afte 5 yeas is oly woth $.4. If the patte of expected divideds educes to two special cases, equatio educes to a vey compact ad simple fom. If divideds ae costat, the divided discout model educes to: ( If divideds ae gowig at a costat ate g <, the divided discout model becomes: g (4 4 Steve Feud

3 Stock aluatio aft: /4/4 Both of these esults ae deived i the documet Auity eivatios. The costat divided model, equatio, is based o the peset value of a pepetuity, ad is sometimes used to model pefeed stock sice the divideds associated with pefeed stock is moe costat tha commo stock divideds. The costat gowth model, equatio 4, is sometimes called the Godo Gowth Model. It is oly appopiate if the gowth ate g is less tha the discout ate. Othewise the value would be ifiite! Befoe we cotiue, let us demostate with seveal examples. Coside a costat divided of $, ad a discout ate of %. Because divideds ae costat, we ca use equatio : What if divideds gow at a %, the discout ate is %, ad $?. $ This meas that (.. (..4 etc. I geeal, t t- ( g o t ( g t- But we eed ot do ay of this fo the valuatio if we kow the divided at t. We use equatio 4: g... $ It should be expect that this value should be moe tha fo the costat divided example above, sice both poblems have idetical divideds at t, but i the secod example it is gowig at a % ate afte t. Now we shall tackle the questio of what happes if you do t wat to hold the stock foeve. What if we would buy the stock at t, ad sell at t? At that poit we should get ad, ad the value at t should be as follows: (5 Futhe suppose that we sell the stock at t fo a fai pice, that is,. 4 Steve Feud

4 Stock aluatio aft: /4/4 4 Steve Feud 4 We could thik of valuig the stock at t i a simila mae as above, buyig at t ad sellig at t : (6 Now usig ad substitutig (6 ito (5: ` If we cotiue i this mae we ca geealize: (... ( ( ( (7 As log as you sell the stock fo the discouted value of all futue cash flows, you will get idetical values, if you use equatio 4 (deived fom equatio, which epesets discoutig all futue divideds, o equatio 7, which epesets holdig the stock fo peiods ad the sellig it at that poit. Let us demostate this fo the same example as above. We will buy the stock at t, ad sell at t : (

5 Stock aluatio aft: /4/4 We eed to discout both the divided ad the pice of the stock that we get at t. We kow $. To get, we assume we will be able to sell it fo value, which ca be obtaied fom the Costat Gowth Model, equatio 4, which i tu equies. As you ca see, whe divideds gow at a costat ate g, you ca value the stock at ay poit i time, but you eed the amout of the divided oe peiod fowad fom the poit you wat to value the stock. I this case, it is t : Theefoe, (g (.. g..... $ If we assume that $, we ow use equatio 7 ad discout both the pice ad the divided we eceive at t, usig (oly the fial tem equied hee!: ( (. 4.. Same esult as usig the divided discout model diectly! We ae demostatig with a umeical example that it makes o diffeece if we assume that we will hold the stock foeve o hold fo a fiite peiod ad sell it. This method also allows you to value stock with divideds that do ot gow at a costat ate o have a ate of gowth which exceeds the discout ate, povided that at some poit i the futue, you divideds will evetually stat to gow at a costat ate which is below. You would eed to fid the divideds though the poit whee the costat gowth stats, the fid the value of the stock oe peiod befoe that. The discout all the divideds ad the sellig pice you would eceive up though that poit. Suppose that betwee t ad t the gowth is actually % but afte that the gowth is %. Just as above, let ad %. You ca ot use the covetioal gowth equatio 4 at t, because you gowth is ot costat, ad it is ot less tha betwee the fist ad secod time peiod. (..4 (.4 ( (.88 (..68 Afte this, divideds gow evey yea at %. You ca use the equatio 4 at t but ot befoe. At t, you ext divided is.88 at t, afte which the gowth is a costat %: $ 4 Steve Feud 5

6 Stock aluatio aft: /4/ g $8.8 Now discout the fist two divideds at t ad t ad the sellig pice at t : $. ( (. (. We would expect this stock to be woth moe tha the pevious example because thee is a geate gowth ate fo seveal peiods befoe the costat gowth ate of % kicks i. 4 Steve Feud 6