e t D u i n v D C al n e t BY KEITH R. HERRMANN

Transcription

1 ff r v D r a l g S s D u b C al u a y P m q r F h W BY KEITH R. HERRMANN

2 Lk may loals,h Cy of Durham,Norh Carola, uss db o provd log-rm fag for projs. Wh was m, rly, o alula h amou of db srv Durham would b oblgad o pay as a rsul of ssug bods, h y s faal advsor had alrady provdd h rgh aswr, bu h y wad o ofrm ha hs umbr was orr.ially,hr wr problms vrfyg hr aalyss. Usg h paym fuo Exl o alula avrag aual db srv yldd a dffr aswr ha h faal advsor had suppld. Th lvl of srss rasd as h hours wor o,ul h urka mom fally am.as a formr boss o sad, Through srss oms udrsadg. THE PAYMENT FUNCTION To sar wh h bass, oal db srv s h sum of all h rs payms ad all h prpal rpayms o a loa.avrag aual db srv s oal db srv dvdd by h umbr of yars for h loa. Th frquy of h prod payms s ypally o yar (aual), sx mohs (sm-aual), or o moh. For h sak of lary, h xampls ha follow wll b for smpl rs, whh s alulad o oly h prpal amou or h poro of h prpal ha rmas upad. If h frquy of h payms of rs ad prpal ar aual, h hr s a fuo Exl ha s dally sud for alulag h avrag aual db srv paym: h paym fuo (PMT).PMT alulas h paym for a loa basd o osa payms ad a osa rs ra.th syax o b usd s: PMT (rs ra, umbr of prods, prpal amou). Exhb 1 s a mpla ha dsplays h us of h PMT fuo. (Th umbrs hs xampl ar for llusrav purposs oly.) I our xampl, h rs ra s 8 pr, h rm of h loa s 10 yars, ad h prpal amou s $50 mllo.clls B4,B5,ad B6 Exhb 1 ar h pu lls for hs daa. Th PMT fuo s ll B8. Th o of ll B8 s -PMT(B4,B5,B6).B sur o lud a gav sg bfor h PMT fuo:th ovo Exl s o dsgush bw a amou rvd ad a paym mad by dsgag o as posv ad h ohr as gav. May popl fd mor uv o mak boh posv, Wh dffrg paym frqus, h Exl paym fuo ao b usd o auraly alula avrag aual db srv. Isad, s ssary o s up a usomzd modl. whh s aomplshd by srg a gav sg whr dad. Ths formula wll rur h avrag aual db srv of $7,451,474 pr yar.baus h rm of h loa s 10 yars, hs mas ha h oal os of h loa s $74,514,744, or 10 ms h avrag aual db srv. PMT luds addoal faurs. Thr s a paramr ha a b usd o alula loa payms f hr s a fuur valu (FV) or ash bala you wa o aa afr h las paym s mad. Thr s also a paramr, alld Typ, o da wh payms ar du.th Typ paramr a b hr omd or s a 0 f you pay a h d of a gv prod, or Typ a b s a 1 f you pay a h bgg of a gv prod. For mor ompl dsrpos of h paramrs, s h hlp provdd for PMT h faal fuos rfr so of Exl. All h xampls hs arl om valus for FV ad Typ, assumg h dfaul sgs of FV = $0 ad Typ = 0 (..,hr s o ash bala a h d of h loa ad all payms our a h d of h prod). Now l s mak hgs slghly mor omplad. Rahr ha havg aual payms, w wll say h rs ad prpal payms ar boh sm-aual. PMT a sll b usd, bu w d o mak sur w ar oss abou h us usd for spfyg h rs ra ad h umbr of prods. If w ar makg sm-aual payms o h 10- yar loa a 8 pr rs, h w us 8%/2 for ra ad 10*2 for h umbr of prods. Exhb 2 dsplays how o modfy h PMT fuo o ahv h dsrd rsul. Ths modfs h PMT fuo, dvdg h rs ra by wo, mulplyg h umbr of prods by wo, ad mulplyg wo.if h PMT aswr wr mulpld by wo,h aswr provdd would b h sm-aual db srv rahr ha h aual db srv. As show Exhb 2, h avrag aual db srv hs saro s $7,358,175.Th oal os of h loa s $73,581,750. Th avrag aual db srv hs saro s $93,299 lss pr yar ha h orrspodg aswr Exhb 1.Ths mas ha h oal os of h loa s lss wh h paym frquy s sm-aual ha wh s aual. I addo o aual ad sm-aual saros, PMT a b usd wh ohr paym frquy opos ar usd Fbruary 2010 Govrm Fa Rvw 37

3 Exhb 1:Aual Irs ad Aual Prpal Irs Ra 8.0% Toal Numbr of Prods (Trm) 10 Prpal Amou $50,000,000 Avrag Aual Db Srv $7,451,474 Formula Cll B8 No: B4, B5, ad B6 ar pu lls. =-PMT(B4,B5,B6) Exhb 2: Sm-aual Irs ad Sm-aual Prpal Irs Ra 8.0% Toal Numbr of Prods (Trm) 10 Prpal Amou $50,000,000 Avrag Aual Db Srv $7,358,175 Formula Cll B8 No: B4, B5, ad B6 ar pu lls. =-PMT((B4/2),(B5*2),B6)*2 (.g., quarrly, b-mohly, mohly, or b-wkly). Followg h xampl show Exhb 2 ad makg smlar modfaos, w a us h PMT fuo o alula avrag aual db srv udr ay of hs saros. For xampl, f w ar makg mohly payms for boh prpal ad rs, h w would us 8%/12 for ra, 10*12 for h umbr of prods, ad mulply h PMT aswr by 12. (To alula payms for h ohr paym frqus mod abov,mak h followg subsuos ra,prod, ad paym lu of h 12 w usd for mohly: for quarrly us 4, for b-mohly us 6, ad for b-wkly us 26.) May db payms ar mohly; for xampl, las or sallm fag. If h prpal ad rs payms for hs loa wr mohly, h h modfd PMT fuo would rur $7,279,656.Th oal os of h loa wh h paym frquy s mohly s $72,796,556. Th avrag aual db srv hs saro s $171,818 lss pr yar ha h orrspodg aswr Exhb 1, ad also rprss a savgs ovr h sm-aual saro.th po s ha makg mor frqu payms rsuls a lowr oal amou pad. BEYOND THE PAYMENT FUNCTION For h Durham bod ssu,h rs was o b pad smaually, ad h prpal was o b rpad aually. Wh hs dffrg paym frqus,h PMT fuo ao b usd o auraly alula avrag aual db srv.isad, s ssary o s up a usomzd modl. Exhbs 3 ad 4 dp a modl ha a b usd o alula avrag aual db srv for pad. hs saro. Th avrag aual db srv, ll C25 of Exhb 3, s $7,200,000.Th oal os of h loa s $72 mllo. Th avrag aual db srv hs saro s $251,474 lss pr yar ha h orrspodg aswr Exhb 1. Exhb 3 shows h modl,ad h lls ar populad wh umral rsuls. Exhb 4 shows h formulas bhd h umbrs Exhb 3.Wha follows s a brf dsrpo of h purpos of ah row h modl. Row sv oas h rs ra. Row gh oas h aual rpaym of h prpal o a sragh l bass. (Ths modl s for a $50 mllo loa o b pad off ovr 10 yars, ad h aual prpal rpaym s $5 mllo.) Row shows h ousadg bala of h prpal ah yar as s pad off ovr h rm of h loa. Row 10 mulpls h rs ra by h aual prpal rpaym, ad s h alulao of h rs payms. Row 11 alulas h frs of ah yar s sm-aual rs payms,ad row 12 alulas h sod sm-aual rs paym. Row 13 sums h wo smaual rs payms. (Aual rs quals h sum of wo sm-aual rs payms).row 14 alulas umulav rs, row 15 alulas oal db srv (qual o prpal plus rs), ad row16 alulas umulav db srv. I addo o h paym frquy ssu, hr ar ohr sgfa dffrs bw h mahmaal quao usd by PMT ad h assumpos Exhbs 3 ad 4. As alrady od, PMT alulas payms basd o osa (lvl) db srv ovr h rm of h loa.wha hs mas s ha, smlar o wha s of foud a ar loa or a hom morgag,pmt assums h oal db srv payms rma h sam for ah prod ad ha h amous of prpal ad rs ludd ah paym wll hag. Makg mor frqu payms rsuls a lowr oal amou 38 Govrm Fa Rvw Fbruary 2010

4 Exhb 3: 10-Yar Db Srv Calulaor Yar Irs Aual Prpal Cumulav Calulao Frs Irs Sod Irs Toal Irs Cumulav Toal Db Cumulav Ra Rpaym Prpal Paym Paym Irs Srv Db Srv 1 8.0% $5,000,000 $50,000,000 $400,000 $2,000,000 $2,000,000 $4,000,000 $22,000,000 $9,000,000 $72,000, % 5,000,000 45,000, ,000 1,800,000 1,800,000 3,600,000 18,000,000 8,600,000 63,000, % 5,000,000 40,000, ,000 1,600,000 1,600,000 3,200,000 14,400,000 8,200,000 54,400, % 5,000,000 35,000, ,000 1,400,000 1,400,000 2,800,000 11,200,000 7,800,000 46,200, % 5,000,000 30,000, ,000 1,200,000 1,200,000 2,400,000 8,400,000 7,400,000 38,400, % 5,000,000 25,000, ,000 1,000,000 1,000,000 2,000,000 6,000,000 7,000,000 31,000, % 5,000,000 20,000, , , ,000 1,600,000 4,000,000 6,600,000 24,000, % 5,000,000 15,000, , , ,000 1,200,000 2,400,000 6,200,000 17,400, % 5,000,000 10,000, , , , ,000 1,200,000 5,800,000 11,200, % 5,000,000 5,000, , , , , ,000 5,400,000 5,400,000 Assumpos Prpal Amou $50,000,000 Cos of Issua $0 Toal Fag $50,000,000 Irs Ra 8.0% Avrag Aual Db Srv $7,200,000 Th amou of prpal wll b lowr ad h amou of rs wll b hghr for h al payms. Movg forward, as ah subsqu paym s mad, h oal wll rma osa whl h amou of prpl rpad wll gradually ras ad h amou of rs pad wll gradually dras. For h Durham bod ssu, h prpal was o b rpad o a sragh l bass ovr 10 yars.th prpl for hs $50 mllo loa would b rpad 10 qual sallms of $5 mllo pr yar. Th rs s alulad basd o h ousadg loa bala for ay gv prod. Thus, for h frs yar of h loa, h rs s 8 pr of $50 mllo, or $4 mllo. I s o b pad wo sm-aual sallms of $2 mllo (wo payms ah qual o 4 pr of $50 mllo). Th db srv for h frs yar s $4 mllo rs paym plus $5 mllo prpal rpaym, for a oal of $9 mllo. For h sod yar, h ousadg prpal had b rdud o $45 mllo ($50 mllo lss h frs aual $5 mllo rpaym),ad h rs s 8 pr of $45 mllo, or $3.6 mllo (wo payms of $1.8 mllo ah).th oal db srv for h sod yar, $3.6 mllo rs plus $5 mllo prpal, s $8.6 mllo. I a smlar fasho, h rs payms dras by $400,000 vry yar ul h h ad fal yar,wh h oal db srv s $5.4 mllo (qual o $400,000 rs o h ousadg bala of $5 mllo plus $5 mllo prpal rpaym). No ha h aual prpal rpayms rma osa for all 10 yars,whl h amou of rs s rdud by $400,000 pr yar ul h loa s fally pad off.ths s o lvl db srv baus h oal payms ar o osa for all prods. Morovr, hs s dffr from h lvl db srv assumpo PMT whr h oal payms ar osa for all prods. A h ous,durham s faal aalys sad ha hs saro (8 pr aual rs ra,10-yar rm,$50 mllo prpal, sm-aual rs payms, ad aual prpal rpayms),h oal os of hs loa s $72 mllo,wh a avrag aual db srv of $7.2 mllo pr yar for 10 yars.i s amp o vrfy hs aalyss,h y frs usd h umodfd PMT fuo Exhb 1, whh rurd a Fbruary 2010 Govrm Fa Rvw 39

5 Exhb 4: 10-Yar Db Srv Calulaor Formulas Yar Irs Aual Cumulav Calulao Frs Sod Toal Cumulav Toal Cumulav Ra Prpal Prpal Irs Irs Irs Irs Db Db Rpaym Paym Paym Srv Srv 1 =+$C$23 =+C21/10 =SUM(C8:L8) =+C7*C8 =SUM($C$10:$L$10)/2 =SUM($C$10:$L$10)/2 =SUM(C11:C12) =SUM(C13:$L$13) =+C13+C8 =SUM(C15:$L$15) 2 =+$C$23 x=+c8 =C9-C8 =+D7*D8 =+C11-C10/2 =+C11-C10/2 =SUM(D11:D12) =SUM(D13:$L$13) =+D13+D8 =SUM(D15:$L$15) 3 =+$C$23 =+D8 =D9-D8 =+E7*E8 =+D11-D10/2 =+D11-D10/2 =SUM(E11:E12) =SUM(E13:$L$13) =+E13+E8 =SUM(E15:$L$15) 4 =+$C$23 =+E8 =E9-E8 =+F7*F8 =+E11-E10/2 =+E11-E10/2 =SUM(F11:F12) =SUM(F13:$L$13) =+F13+F8 =SUM(F15:$L$15) 5 =+$C$23 =+F8 =F9-F8 =+G7*G8 =+F11-F10/2 =+F11-F10/2 =SUM(G11:G12) =SUM(G13:$L$13) =+G13+G8 =SUM(G15:$L$15) 6 =+$C$23 =+G8 =G9-G8 =+H7*H8 =+G11-G10/2 =+G11-G10/2 =SUM(H11:H12) =SUM(H13:$L$13) =+H13+H8 =SUM(H15:$L$15) 7 =+$C$23 =+H8 =H9-H8 =+I7*I8 =+H11-H10/2 =+H11-H10/2 =SUM(I11:I12) =SUM(I13:$L$13) =+I13+I8 =SUM(I15:$L$15) 8 =+$C$23 =+I8 =I9-I8 =+J7*J8 =+I11-I10/2 =+I11-I10/2 =SUM(J11:J12) =SUM(J13:$L$13) =+J13+J8 =SUM(J15:$L$15) 9 =+$C$23 =+J8 =J9-J8 =+K7*K8 =+J11-J10/2 =+J11-J10/2 =SUM(K11:K12) =SUM(K13:$L$13) =+K13+K8 =SUM(K15:$L$15) 10 =+$C$23 =+K8 =K9-K8 =+L7*L8 =+K11-K10/2 =+K11-K10/2 =SUM(L11:L12) =SUM(L13:$L$13) =+L13+L8 =SUM(L15:$L$15) Assumpos Prpal Amou $50,000,000 Cos of Issua $0 Toal Fag =SUM(C19:C20) Irs Ra 8.0% Avrag Aual Db Srv =+C16/10 avrag aual db srv of $7,451,474,ad ha rsul was $251,474 (3.5 pr) oo hgh. Modfyg h PMT fuo as show Exhb 2 rurd a avrag aual db srv of $7,358,175, whh was sll $158,175 (2.2 pr) oo hgh. O h modl Exhbs 3 ad 4 was dvlopd, h y was abl o vrfy ha h faal aalys s umbr, $7.2 mllo, was orr. CONCLUSIONS Wh valuag a w db ssua, a b a good da o ru svral saros usg dffr paym frqus. As h xhbs dmosra, h paym frquy affs h oal os of h loa.wh you ar aalyzg prospv dals, mgh b bfal o s how shdulg payms of larg amous lk hs would aff h prs valu of h avalabl fuds. (A addoal faor o dsussd hs arl s ha payms wll vary dpdg o whhr h payms ar mad a h bgg or d of h prod.) PMT a b usd o alula avrag aual db srv for may yps of paym frqus, bu should o b usd wh h paym frqus for h prpal ad rs ar o h sam or wh hr s o a assumpo of lvl db srv. Somms hr s a d o alula db srv payms a saro ha dos o f o PMT; a usomzd approah s rqurd sad.th ps provdd hs arl wll hlp you valua fag opos udr a wdr vary of db srv assumpos ad paym frquy saros. KEITH R. HERRMANN s dpuy fa offr for h Cy of Durham, Norh Carola ( Cy of Durham s urrly rad rpl-a by all hr aoal rd rag ags (Moody s, Sadard ad Poor s, ad Fh). Hrrma holds a MBA from h Yal Shool of Maagm ad dgrs from h Uvrsy of Psylvaa, Kg s Collg (Uvrsy of Lodo), ad h Sorbo (Uvrsé d Pars). H a b oad a krhrrma@aol.om. 40 Govrm Fa Rvw Fbruary 2010