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13-3
MH Ryerson
Learning ObjectivesLearning Objectives
After completing this chapter, you will be able to:
> Calculate the future value and presentvalue of annuities due.
> Calculate the payment size, number ofpayments, and interest rate in annuitiesdue.
13-4
MH Ryerson
Classification of AnnuitiesClassification of Annuities
Ordinary annuity -regular
deposits/paymentsmade at the end of
the period
Annuity due -regular
deposits/paymentsmade at the
beginning of theperiod
Jan. 31 Monthly Jan. 1
Jun. 30 Quarterly Apr. 1
Dec. 31 Semiannually Jul. 1
Dec. 31 Annually Jan. 1
13-5
MH Ryerson
TABLE 13.1Distinguishing Characteristics of Annuity Categories
Annuity Category
Is the payment at theend or at the beginning
of each payment interval?
Compare thepayment interval to thecompounding interval.
Ordinary simple annuity End EqualOrdinary general annuity End Not equalSimple annuity due Beginning EqualGeneral annuity due Beginning Not equal
Distinguishing Characteristics of Annuity CategoriesDistinguishing Characteristics of Annuity Categories
13-6
MH Ryerson
PV
Time Diagram for an n-Payment Annuity DueTime Diagram for an n-Payment Annuity Due
PMT PMTPMT PMT PMT PMT PMT PMT
0 1 2 3 4 5 n-2 n-1 n
Interval Number
FV
13-7
MH Ryerson
Future Value of a Simple Annuity DueFuture Value of a Simple Annuity Due
FV(due) = PMT[(1+ i)n - 1]
i* ( 1 + i )
= FV * (1 + i)
13-8
MH Ryerson
How much will Elyse accumulate in herHow much will Elyse accumulate in herRRSP by age 60 if she makes semi-annualRRSP by age 60 if she makes semi-annualcontributions of $1000 starting on her 30thcontributions of $1000 starting on her 30thbirthday? Assume that the RRSP earns 8%birthday? Assume that the RRSP earns 8%compounded semi-annually and that nocompounded semi-annually and that nocontribution is made on her 60th birthday.contribution is made on her 60th birthday.
Set your financial calculator to the BGN mode
8/2 =
n
I/Y
FV
PMT
comp
PV0
30*2=
1000 +/- FV(due) = $66 519.97
Financial Calculator sol�n:
13-9
MH Ryerson
How much will Elyse accumulate in herHow much will Elyse accumulate in herRRSP by age 60 if she makes semi-annualRRSP by age 60 if she makes semi-annualcontributions of $1000 starting on her 30thcontributions of $1000 starting on her 30thbirthday? Assume that the RRSP earns 8%birthday? Assume that the RRSP earns 8%compounded semi-annually and that nocompounded semi-annually and that nocontribution is made on her 60th birthday.contribution is made on her 60th birthday.
Algebraic Solution:
FV (due) =$66 519.97
.08 / 2 = sto1
+ 1 = sto 2
yx 60 =
- 1 =
/ rcl 1 * 500 * rcl 2 =
Now let�s use the formulato answer the same question
FV(due) = PMT[(1+ i)n - 1]
i* ( 1 + i )
13-10
MH Ryerson
Present Value of an Annuity DuePresent Value of an Annuity Due
= PV * (1 + i)
PV(due) = PMT[1 - (1+ i)-n ]
i* ( 1 + i )
13-11
MH Ryerson
A lottery offers the winner a choice betweenA lottery offers the winner a choice betweena $300 000 cash prize, or quarterlya $300 000 cash prize, or quarterlypayments of $7000 beginning immediatelypayments of $7000 beginning immediatelyand continuing for 20 years. Whichand continuing for 20 years. Whichalternative should the winner pick if moneyalternative should the winner pick if moneyis worth 8% compounded quarterly?is worth 8% compounded quarterly?
Set your financial calculator to the BGN mode
8/4 =
n
I/Y
PV
PMT
comp
FV0
20*4=
7000 +/-
PV(due) = $283 776
The cash prize should betaken - it is worth $16224more in current dollars
Financial Calculator sol�n:
13-12
MH Ryerson
A lottery offers the winner a choice betweenA lottery offers the winner a choice betweena $300 000 cash prize, or quarterlya $300 000 cash prize, or quarterlypayments of $7000 beginning immediatelypayments of $7000 beginning immediatelyand continuing for 20 years. Whichand continuing for 20 years. Whichalternative should the winner pick if moneyalternative should the winner pick if moneyis worth 8% compounded quarterly?is worth 8% compounded quarterly?
( ) ( )ii
iPMT
iPVPVn
+111
=
)+(1(due)
×
+−
×=−
Now let�s use the formulato answer the same question
.08 / 4 = sto1
+ 1 = sto 2
yx 80 +/- =
- 1 = +/-
/ rcl 1 * 7000 * rcl 2 =
PV(due) = $283 776The cash prize should be taken
13-13
MH Ryerson
Mark has already accumulated $104 000 inMark has already accumulated $104 000 inhis RRSP. His goal is to build it to $250 000his RRSP. His goal is to build it to $250 000with equal contributions every 6 months forwith equal contributions every 6 months forthe next 7 years. If he earns 8.5% cmpd sa,the next 7 years. If he earns 8.5% cmpd sa,and starts today, find the size of hisand starts today, find the size of hiscontributions?contributions?
Set your financial calculator to the BGN mode
8.5/2 =
n
I/Y
PMT
FV
comp
PV104000
7*2=
250000
Payment needed = $3286.10
Financial Calculator sol�n:
13-14
MH Ryerson
A car sells for $27 900. The manufacturerA car sells for $27 900. The manufactureroffers an interest rate of 1.8% compoundedoffers an interest rate of 1.8% compoundedmonthly on a three year lease. If themonthly on a three year lease. If theresidual value is $14 500, find the leaseresidual value is $14 500, find the leasepayments assuming $2500 down payment.payments assuming $2500 down payment.
−
valueresidual the
of luePresent va +
payments lease the
of luePresent va =
payment
Down
price
Purchase
−
500 $14 the
of luePresent va +
payments lease the
of luePresent va =
$2500
900 $27
13-15
MH Ryerson
A car sells for $27 900. The manufacturerA car sells for $27 900. The manufactureroffers an interest rate of 1.8% compoundedoffers an interest rate of 1.8% compoundedmonthly on a three year lease. If themonthly on a three year lease. If theresidual value is $14 500, find the leaseresidual value is $14 500, find the leasepayments assuming $2500 down payment.payments assuming $2500 down payment.
Set your financial calculator to the BGN mode
1.8/12 =
n
I/Y
PMT
FV
comp
PV25400 +/-
3*12 =
14500 Lease Payment = $332.50
13-16
MH Ryerson
Calculating the Number of PaymentsCalculating the Number of Payments
using the algebraic methodusing the algebraic method
( )( )i
iPMT
FVi
n+
+
×+
1ln
1
(due)1ln
=
( )( )i
iPMT
PVi
n+
+
×−−
1ln
1
(due)1ln
=
13-17
MH Ryerson
How long will it take to accumulate $1 millionHow long will it take to accumulate $1 millionin an RRSP if the first quarterly payment ofin an RRSP if the first quarterly payment of$2000 is made today? Assume the RRSP earns$2000 is made today? Assume the RRSP earns8% cmpd quarterly.8% cmpd quarterly.
Set your financial calculator to the BGN mode
8/4 =
n
I/Y
PMT
FV
comp
PV0
2000 +/-
1000000 120.18 payments = 120*3 =360 months = 30 years + �a bit�
13-18
MH Ryerson
How long will it take to accumulate $1 millionHow long will it take to accumulate $1 millionin an RRSP if the first quarterly payment ofin an RRSP if the first quarterly payment of$2000 is made today? Assume the RRSP earns$2000 is made today? Assume the RRSP earns8% cmpd quarterly.8% cmpd quarterly.
( )( )i
iPMT
FVi
n+
+
×+
1ln
1
(due)1ln
= ( )
( )iiPMT
PVi
n+
+
×−−
1ln
1
(due)1ln
=
Which do we use?Does this involve FV or PV?
Algebraic Sol�n
13-19
MH Ryerson
How long will it take to accumulate $1 millionHow long will it take to accumulate $1 millionin an RRSP if the first quarterly payment ofin an RRSP if the first quarterly payment of$2000 is made today? Assume the RRSP earns$2000 is made today? Assume the RRSP earns8% cmpd quarterly.8% cmpd quarterly.
120.18 payments = 120*3 =360 months = 30 years + �a bit�
( )( )i
iPMT
FVi
n+
+
×+
1ln
1
(due)1ln
=
Algebraic Sol�n
.08 / 4 = sto1
+ 1 = sto2
lnx sto3
rcl1 * 1000000
/ 2000 / rcl2 =
+ 1 = lnx / rcl3 =
13-20
MH Ryerson
A $100 000 life insurance policy requires anA $100 000 life insurance policy requires anannual premium of $420 or a monthlyannual premium of $420 or a monthlypremium of $37. In either case, the premiumpremium of $37. In either case, the premiumis due at the beginning of the period ofis due at the beginning of the period ofcoverage. What is the effective rate ofcoverage. What is the effective rate ofinterest charged to those who pay monthly?interest charged to those who pay monthly?
Set your financial calculator to the BGN mode12
I/Y
n
PMT
FV
comp
PV420
37 +/-
0
1.0269
f = (1+i)m - 1
= (1 + 0.010269)12 - 1
= 0.1304 = 13.04%