What Youll Wh ll Learn arn Today ay Ca Calculating the he PV - - PDF document

1

Aa Aaron Stevens EC1 EC171 71, 27 27 September 2011

Valuing Streams of Payments Calculating Payments

Wh What You’ll ll Learn arn Today ay

Ca Calculating the he PV PV of a a stream of paym payment ents s of

• f equal

equal am amount

• unts

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

You

• u 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

• f $1

$1.9 9 million

• n 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

• n
• n for

for a spe specifi fied pe peri riod

• d of
• f 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 Here’s s what annuity payments look like on a timeline:

today (year 0) 28 +1.9m

PV ?

. . . 30 +1.9m 1 +1.9m 2 +1.9m 29 +1.9m Years from now…

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

• r

Annui Annuity Due Due?

In an In an or

• rdinary annuity, the

e fi first paymen ment is rec ecei eived ed

• ne

e per eriod fr from m now, so it need eeds to be e discounted ed. In an In an 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.

28 +1.9m . . . 30 +1.9m 1 +1.9m 2 +1.9m 29 +1.9m Years from now… 28 +1.9m . . . +1.9m 1 +1.9m 2 +1.9m 29 +1.9m Years from now…

Va Valui uing ng Annui Annuities In n Excel

Excel Excel: PV PV = $37. $37.24 24 million

* * “ty type” i in the Excel function refers to the type of annuity

=P =PV (i, n, pmt mt, fv, type*)

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

• f 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 Yo Your Ne New w Ca Car! Do Don’t t have $22,000 in cash? It It’s s all ll good

• od…

Pay Pay only y $446/m $446/month h for 60 60 months hs!

Pa Paying ng Ov Over Ti Time me

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.

58

• 446

. . . 60

• 446

1

• 446

2

• 446

59

• 446

Months from now… +22,000 3

• 446

This This is is the he exact exact opposit

• pposite

e of

• f the

he process process of

• f

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

• n

=P =PMT (i, n, n, pv pv) ) PM PMT T = $446 $446

The The Pr Proof

• of Is

s In n The The Amor Amortization

• n

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? 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.

So Sources:

ht http://www.cre 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

• nal-d
• debt-s
• statistics

cs-1

• 1276.php

hp ht http://www.cr creditcards.c .com/press-releases/CreditCards-Weekly-Credit-Card-Rate-Report-May-19-2010.ph php

Cred edit card deb ebt is not norma mal! It It’s indicative e of f living bey eyond your mea means!

How How long

• ng 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

• 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

• f 9

f 9%, o , on w which h he ear earns ns $750/m $750/month

• nth in

in interest. Th The e Great eat Bi Big g Ins nsur uranc ance e Com Company pany of

• ffer

ers Gr Granfa an an annuity wi with payme ments of $955. $955.65 65 per per month

• nth for
• r th

the e res est t of

• f 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:

• n: 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

• v/RI

/RI/O /OFNte tebnd

Ap Application:

• n: 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

• nth?

Wh What’s s In Your Wallet?

Onc Once each year, you open n the Ma Magic Wallet an and fin d find $500 i d $500 in cash.

• cash. But

But onl

• nly onc
• nce 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