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Stephen G.CECCHETTI Kermit L.SCHOENHOLTZ

Future Value, Present Value

and Interest RatesCopyright 2011 by The McGraw-Hill Companies, Inc. ll rights reser!e".McGraw-Hill#Irwin

Chapter Four

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A rie! Hi"tor# o! Len$in%

Lenders have been despised throughout history.

Credit is so basic that we find evidence of

loans going back five thousand years.

It is hard to imagine an economy without it. Yet, people still take a dim view of lenders

because they charge interest.

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Intro$u'tion

Credit is one of the critical mechanisms wehave for allocating resources.

Although interest has historically beenunpopular, this comes from the failure toappreciate the opportunity cost of lending.

Interest rates Link the present to the future.

ell the future reward for lending today.

ell the cost of borrowing now and repaying later.

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(a)uin% *onetar# +a#ment"No, an$ in the Future

!e must learn how to calculate and comparerates on different financial instruments.

!e need a set of tools"

#uture value $resent value

%ow and why is the promise to make a

payment on one date more or less valuable than

the promise to make it on a different date&

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Future (a)ue an$Compoun$ Intere"t

#uture valueis the value on some future date ofan investment made today.

'()) invested today at *+ interest gives '()* in a

year. o the future value of '()) today at *+

interest is '()* one year from now.

he '()) yields '*, which is why interest rates are

sometimes called a yield.

his is the same as a simple loan of '()) for a year

at *+ interest.

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he higher the interest rate or the higher theamount invested, the higher the future value.

2ost financial instruments are not this simple,so what happens when time to repaymentvaries.

!hen using one3year interest rates to computethe value repaid more than one year from now,we must consider compound interest. Compound interest is the interest on the interest.

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!hat if you leave your '()) in the bank fortwo years at *+ yearly interest rate&

he future value is"

'()) - '())).)*/ - '())).)*/ - '*).)*/ 0 '(().4*

'())(.)*/(.)*/ 0 '())(.)*/4

In general

#1n0 $1( - i/n

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able 5.( shows the compounding years intothe future.

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Converting nfrom years to months is easy, butconverting the interest rate is harder.

If the annual interest rate is *+, what is the monthly

rate&

Assume imis the one3month interest rate and n

is the number of months, then a deposit made

for one year will have a future value of

'())( - im/(4.

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!e know that in one year the future value is'())(.)*/ so we can solve for im"

( - im/(40 (.)*/

( - im/ 0 (.)*/(6(4 0 (.))5(

hese fractions of percentage points are calledbasis points.

Abasis pointis one one3hundredth of a percentage

point, ).)( percent.

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Invest '()) at *+ annual interest

%ow long until you have '4))&

he 7ule of 84"

9ivide the annual interest rate into 84 o 846*0(5.5 years.

(.)*(5.5 0 4.)4

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+re"ent (a)ue

#inancial instruments promise future cashpayments so we need to know how to valuethose payments.

$resent valueis the value today in the present/of a payment that is promised to be made in thefuture.

:r, present value is the amount that must beinvested today in order to reali;e a specificamount on a given future date.

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+re"ent (a)ue

olve the #uture 1alue #ormula for $1"#1 0 $1 x(-i/

so

his is

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+re"ent (a)ue

!e can generali;e the process as we did for futurevalue.

$resent 1alue of payment received nyears in the

future"

ni

FVPV

/( +=

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+re"ent (a)ue

#rom the previous e=uation, we can see thatpresent value is higher"

(. he higher future value of the payment, #1n.

4. he shorter time period until payment, n.>. he lower the interest rate, i.

$resent value is the single most important

relationship in our study of financial

instruments.

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Ho, +re"ent (a)ueChan%e"

(. 9oubling the future value of the payment,without changing the time of the payment or

the interest rate, doubles the present value.

his is true for any percentage.

4. he sooner a payment is to be made, the more

it is worth.

ee figure 5.(

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Fi%ure 4. +re"ent (a)ue o!533 at 6 Intere"t

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Ta7)e 4.2 +re"ent (a)ue o!533 +a#ment

Higher interest rates

are associated with

lower present values,

no matter what the

size or timing of the

payment.At any fixed interest

rate, an increase in

the time reduces its

present value.

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7isk re=uires compensation, but securingproper compensation means understanding the

risks of what is purchased.

If interest rates rise, losses on a long3term bondare greater than losses on a short3term bond.

Long term bonds are more sensitive to the risk that

interest rates will change.

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Investors might mis

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he search for yield can bid up prices of risky

securities and depress the market compensation

for risk below a sustainable level.

!hen risk comes to fruition, like when defaultsincrease, the prices of riskier securities fall

disproportionately, triggering financial losses.

9uring the 4))834))@ crisis, the plunge of

corporate and mortgage security prices show

how markets reprice risk when the search for

yield has gone too far.

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!e can turn a monthly growth rate into acompound3annual rate using what we havelearned in this chapter. Investment grows ).*+ per month

!hat is the compound annual rate&

#1n0$1(-i/n0 ())?(.))*/(40().(8

Compound annual rate 0 .(8+

Bote" .(8 (4?).)*0.)/

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!e can also use this to compute the percentagechange per year when we know how much an

investment has grown over a number of years.

An investment has increased 4) percent over five

years" from ()) to (4).

#1n0 $1( - i/n

(4) 0 ())( - i/*

i 0 ).)>8(

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!hat is the difference between waiting a yearto buy a car or buying it now&

oday"

If you take '5))) in savings, 5 year loan at

.8*+ interest, your payments are '4>8 per

every '(),))) borrowed.

You can afford '>))6month so you can get a

loan up to '(4,D* so with your '5))), you can

get a car that costs '(,*D.

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!ait a year& $ut the '>)) per month payment into savings each

month at 5+ interest.

You will have '8D>D at the end of the year" the

future value of the '5))) plus (4 monthly

contributions of '>)).

If you now took out a >3year loan at .8*+ you can

now afford to borrow '@8D(.

Added to savings, you can buy a car worth '(8,(D.

%ave to compare the e?tra '())) you have in a

year to current costs of old car.

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Interna) 8ate o! 8eturn

Imagine that you run a tennis racket company andthat you are considering purchasing a new

machine.

2achine costs '( million and can produce >))) rackets

per year. You sell the rackets for '*), generating '(*),))) in

revenue per year.

Assume the machine is only input, have certainty about

the revenue, no maintenance and a () year lifespan.

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Interna) 8ate o! 8eturn

Ealance the cost of the machine against therevenue.

'( million today vs. '(*),))) a year for ten years.

o find the internal rate of return, we take thecost of the machine and e=uate it to the sum of

the present value of each of the yearly

revenues.

olve for i3 the internal rate of return.

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Interna) 8ate o! 8eturnE9amp)e

$1,000,000 =

$10,000

!1+ i"1 +$10,000

!1+ i"# +$10,000

!1+ i" + ......+$10,000

!1+ i"10

olving for i, i0.)D(5 or D.(5+

o long as your interest rate at which you borrow

the money is less than D.(5+, then you should buythe machine.

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$100,000 = PV!1+ i"

$100,000

!1.0%"1+$100,000

!1.0%"#+$100,000

!1.0%"+L +

$100,000

!1.0%"%%+$100,000

!1.0%"%= $#,0,00%

Can you retire when youFre 5)&

Assume

Live to D*

Interest rate 0 5+ !ant to have '()),))) per year

You will need

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here is a common problem faced by thosewishing to retire" hould you take a single lump3sum payment or a

series of annual payments&

You must consider the present value of both toanswer the =uestions.

$eople are impatient, with an e?traordinarily highpersonal discount rate.

he personal discount rate is the value placed ona dollar today versus a dollar a year from now.

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on$ a"i'"

he most common type of bond is a couponbond.

Issuer is re=uired to make annual payments, calledcoupon payments.

he annual interest the borrower pays ic/, is thecoupon rate.

he date on which the payments stop and the loan isrepaid n/, is the maturity dateor term to maturity.

he final payment is theprincipal, face value, orparvalueof the bond.

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(a)uin% the +rin'ipa)

Assume a bond has a principle payment of '()) and its maturity date is nyears in the future. he present value of the bond principal is"

he higher the n, the lower the value of the payment.

PBP =

F

!1+ i"n =$100

!1+ i"n

( ) i th C

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nCP

i

C

i

C

i

C

i

CP

/(......

/(/(/( >4( +++

++

++

+=

(a)uin% the Coupon+a#ment"

hese resemble loan payments. he longer the payments go, the higher their total

value.

he higher the interest rate, the lower the present

value. he present value e?pression gives us a general

formula for the string of yearly coupon payments madeover n years.

( ) i th C

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(a)uin% the Coupon+a#ment" p)u" +rin'ipa)

nnBPCPCB

i

F

i

C

i

C

i

C

i

CPPP

/(/(

......

/(/(/(

>4( ++

+++

++

++

+=+=

!e can

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on$ +ri'in%

he relationship between the bond price andinterest rates is very important.

Eonds promise fi?ed payments on future dates, so

the higher the interest rate, the lower their present

value.

The value of a bond varies inversely with the

interest rate used to calculate the present value

of the promised payment.

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Credit cards are veryuseful, but sometimes too

easy.

!e can use the present

value e=uation to calculate

how long it will take topay off a card given fi?ed

payments.

2onthly payment more

important than interest

rate.

8 ) $ N i ) I t t

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8ea) an$ Nomina) Intere"t8ate"

Eorrowers care about the resources re=uired torepay.

Lenders care about the purchasing power of the

payments they received. Neither cares solely about the number of

dollars, they care about what the dollars buy.

8ea) an$ Nomina) Intere"t

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Bominal Interest 7ates i/ he interest rate e?pressed in current3dollar terms.

7eal Interest 7ates r/

he inflation ad

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he nominal interest rate you agree on i/ must bebased on expected inflation e/ over the term of the

loan plus the real interest rate you agree on r/.

i ! r " e his is called theFisher #$uation.

he higher e?pected inflation, the higher the nominal

interest rate.

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his figure shows thenominal interest rate

and the inflation rate

in >* countries and

the euro area in early4)().

8ea) an$ Nomina) Intere"t

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#inancial markets =uote nominal interest rates. !hen people use the term interest rate, they are

referring to the nominal rate.

!e cannot directly observe the real interestrateG we have to estimate it.

r ! i % e

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Stephen G.CECCHETTI Kermit L.SCHOENHOLTZ

Future Value, Present Value

and Interest Rates

En$ o!

Chapter Four

chap004 (3)

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