Valuing Streams of Payments Calculating Payments Aa Aaron Stevens EC1 EC171 71, 27 27 September 2011 1 What You’ll Wh ll Learn arn Today ay Ca Calculating the he PV PV of a a stream of paym payment ents s of of equal equal am amount ounts Ca Calculating lo loan payments? Am Amortizi zing an and d rev evolving loan ans an and th their p payments. nts.
Wi Winning the Lottery Y ou ou win the $5 $57 million* grand prize! * * The The gr grand and pr prize ze is ac actual tually y 30 30 annual annual paym payments ents of of $1 $1.9 9 million on per per year year. Thi This ki kind d of an an arran arrangem gemen ent is cal called ed an an an annuity: : regular payments that go on on for for a spe specifi fied pe peri riod od of of ti time. Annui Annuity Pa Payme ment nts The The winnings winnings are are $1.9million $1.9million per per year year for for 30 30 years. years. Here’s Here s what annuity payments look like on a timeline: today Years from now… (year 0) 1 2 29 28 30 . . . +1.9m +1.9m +1.9m +1.9m +1.9m PV ? Wh What’s s the present value of this annuity? Re Recall: present values are additive.
Annui Annuity Pa Payme ment nts Recal Recall that hat PV PVs ar are additive. We can fi find the PV of an an annui uity y by by addi adding g up up the he PV PV of each each year year’s s payment: Va Valui uing ng Annui Annuities In n Excel We We can ca calculate the PV of each year’s s payment: Recal Recall that hat PV PVs ar are additive. We can fi find the PV of an an annui uity y by by addi adding g up up the he PV PV of each each year year’s s payment.
Or Ordina nary Annui Annuity or or Annui Annuity Due Due? In an In an or ordinary annuity, the e fi first paymen ment is rec ecei eived ed one e per eriod fr from m now, so it need eeds to be e discounted ed. Years from now… 0 1 2 28 29 30 . . . +1.9m +1.9m +1.9m +1.9m +1.9m In an an In an annuity due, the e fi first paymen ment is rec ecei eived ed today, so it does esn’t need eed to be e discounted ed. Years from now… 0 1 2 29 28 . . . +1.9m +1.9m +1.9m +1.9m +1.9m Va Valui uing ng Annui Annuities In n Excel Excel: Excel =P =PV (i, n, pmt mt, fv, type*) PV PV = $37. $37.24 24 million * * “ty type” i in the Excel function refers to the type of annuity 0 0 = ordi dinar ary y an annuity, y, fir first paym aymen ent is rec ecei eived ed af after er 1 year year 1 = 1 = a annuity d due, fi first p payment i is r received t today
PV PV of of An An Annui Annuity fact factoring o g out t the $ he $1. 1.9M: thi this simpl plifies fies to: to: Pr Present nt Va Value ue of an n Ordi dina nary y Annui nnuity Pa Paying ng Ov Over Ti Time me Yo Your Ne New w Ca Car! Do Don’t t have $22,000 in cash? It’s It s all ll good ood… Pay Pay only y $446/m $446/month h for 60 60 months hs!
Pa Paying ng Ov Over Ti Time me The The loan loan gives gives you you $22, $22,000 000 now, now, and and you you make make regular regular lo loan an paymen payments s of f $446 $446 per per mo month h fo for 60 60 months. Months from now… 0 1 2 3 58 59 60 . . . +22,000 -446 -446 -446 -446 -446 -446 This This is is the he exact exact opposit opposite e of of the he process process of of fin findi ding g the he PV PV of an an an annuity. y. Ca Calculating Payments Co Consider the PV of annuity: wi with some me algebraic ma magic, we we can so solve fo for t r the he p payment a amount:
Excel Excel PMT T Funct Function on =P =PMT (i, n, n, pv pv) ) PM PMT T = $446 $446 The The Pr Proof oof Is s In n The The Amor Amortization on Dev Devel elop am amortizat zation tabl able e in Exc Excel el.
Exa Exampl ple: e: Cr Credit Ca Card Payments What is the e norma mal amo mount of f cred edit card deb ebt to have? e? Cred edit card deb ebt is not norma mal! It’s indicative It e of f living bey eyond your mea means! Aver erage e cred edit card deb ebt per er househ ehold with cred edit card deb ebt: $1 $15,788* *Calculated ed by dividing the e total rev evolving deb ebt in the e U.S .S. . ($8 $852.6 .6 billion as of f March 2010 data, as listed ed in the e Fed eder eral Res eser erve' e's May 2010 rep eport on consumer mer cred edit) by the e es estima mated ed numb mber er of househ eholds carrying cred edit card deb ebt (54 mi million) What is the e typical inter eres est rate e on cred edit cards? National Aver erage e 14.1 .17% per er yea ear. Typically, mi minimu mum m paymen ments are e 3-4% per er mo month. Sources: So http://www.cre ht creditca card rds.com com/cre credit-ca -card rd-n -news/cre credit-ca -card rd-i -industry ry-f -fact cts-p -pers rson onal-d -debt-s -statistics cs-1 -1276.php hp http://www.cr ht creditcards.c .com/press-releases/CreditCards-Weekly-Credit-Card-Rate-Report-May-19-2010.ph php How How long ong Will It Ta Take to Pa Pay Of Off Credi dit Card? d? Suppose e you owe e $15,788 in cred edit card deb ebt, with an annual inter eres est rate e of f 14.1 .17%. • By paying 3% per er mo month, how long does es it take e to pay it all off? ff? • How mu much do you need eed to pay per er mo month, to pay it off ff in 3 yea ears? Bui Build d Excel Excel solut ution (long g an and d sho hort).
Sp Spreadi ding ng Wealth h Ov Over Ti Time me Suppose e you have e an asset et that you want to spen end down over er time. me. What’s that called ed? Ex Examp mple: e: $1 $100,000 pres esen ent value, , 5% rate, e, spen end over er 10 yea ears (this is the e same me calculation as loan paymen ments.) .) Wh What You Learned Today An Annuities es Ca Calculating loan payments Di Distri ribut buting g pres presen ent val alues ues over er time
Announc Announceme ment nts and nd To o Do Do Re Readings: • FE, E, ch ch 4 Homew mework 3: time me value e of f mo money ey problems ems • will be e posted ed Wed ednes esday, due e nex ext Tues esday Th Thur ursday ay 9/29 9/29 - - NO LECTU LECTURE • We e will not hold a lec ecture e on Thursday due e to Rosh Rosh Ha Hashana. Bu Buying An An An Annuity Suppos Suppose e Gr Granfa Grig ha has $1 $100, 00,000 000 in a a CD bank at a ra rate o of 9 f 9%, o , on w which h he ear earns ns $750/m $750/month onth in in interest. Th The e Great eat Bi Big g Ins nsur uranc ance e Com Company pany of offer ers Gr Granfa an an annuity wi with payme ments of $955. $955.65 65 per per month onth for or th the e res est t of of his life. e. Gr Granfa is 6 is 65 y years o rs old, d, a and e d expect cts t s to l live ve unti until age age 80. 80. Th The annuity price is $100,000. Wh What do you advise Gr Granfa to to do?
Ap Application: on: Bond Bond Pr Pricing ng A bond is a contract fo for len ending/borrowing. The e bond spec ecifi fies es the e amo mount borrowed ed, the e ter erm (in yea ears), and the e inter eres est rate e and sched edule e of inter eres est paymen ments. For ex examp mple, e, the US Trea easury rec ecen ently issued ed the fo following bonds: The e price e paid by an inves estor (len ender er) buying this bond is the e Pr Pres esen ent Value e of f the bo bond’s paymen ments. http ttp://w //www.treasurydir irect ct.gov ov/RI /RI/O /OFNte tebnd Ap Application: on: Bond Bond Pr Pricing ng Bonds pay Bonds pay “co coupon” paymen ments twice e a yea ear. This 10-yea ear bond has 20 semi emi-annual inter eres est paymen ments of f $1 $1.7 .75 per er $1 $100 bond (3.5 .5% rate) e). The e fi final paymen ment (10 yea ears fr from m now) includes es a ret eturn of f the e $1 $100 principal. The e yiel eld on the e bond is 3.5 .548%. PV PV(bond) = PV PV(coupons) + PV PV(principal) PV PV(bond) = $2 $29.2 .25 + $7 $70.3 .35
Exa Exampl ple: e: Mo Mortgage Loans Su Suppose you want to buy a ho house… Be Beds: 4 B Baths: 3 S Sq. Ft.: 1,673 Lot Size: 11,805 Sq. Ft. Year Year Bui Built: 1890 890 De Description: Beautifully renovated antique colonial. 2010 updates include ro roof, f, si siding, w windows, s, d deck, k kitchen... An And it’s s priced at just $499,000. How much would the paym paymen ents cos cost you you per per mon onth? Wh What’s s In Your Wallet? Once each year, you open Onc n the Ma Magic Wallet an and fin d find $500 i d $500 in cash. cash. But But onl only onc once a year. Wh What is this Magic Wa Wallet worth?
Pe Perpetui uities Ho How w much h wo would d yo you need d to invest at at 5% 5% to repl replicat cate e the he sam ame e cas cash h fl flow? $1 $10,000 * 5% % = $5 $500 Pr Present nt Va Value ue of a Pe Perpetui uity
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