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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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Future (a)ue an$Compoun$ Intere"t
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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Future (a)ue an$Compoun$ Intere"t
!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 $1( - i/n
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Future (a)ue an$Compoun$ Intere"t
able 5.( shows the compounding years intothe future.
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Future (a)ue an$Compoun$ Intere"t
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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Future (a)ue an$Compoun$ Intere"t
!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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8ea) an$ Nomina) Intere"t8ate"
Bominal Interest 7ates i/ he interest rate e?pressed in current3dollar terms.
7eal Interest 7ates r/
he inflation ad
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8ea) an$ Nomina) Intere"t8ate"
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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8ea) an$ Nomina) Intere"t8ate"
#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