defferal annuities
TRANSCRIPT
-
8/21/2019 Defferal Annuities
1/54
11
McGraw-Hill Ryerson
11-1
Chapter 11
Ordinary
Annuities
McGraw-Hill Ryerson
-
8/21/2019 Defferal Annuities
2/54
11
McGraw-Hill Ryerson
11-2
Calculate the
LearningObjectives
After completing this chapter, you will be able to:
number of paymentsin ordinary
and deferred annuities
payment sizein ordinary and
deferred annuities
interest ratein ordinary annuities
LO-1
LO-2
LO-3
-
8/21/2019 Defferal Annuities
3/54
11
McGraw-Hill Ryerson
11-3
Using your financial calculator
we need to reorganize the formulae to solve
algebraically
solve for payment numberor
size or interest rateusing the
same steps as before
-
8/21/2019 Defferal Annuities
4/54
11
McGraw-Hill Ryerson
11-4
Finding the Payment Size.
PMT
-
8/21/2019 Defferal Annuities
5/54
11
McGraw-Hill Ryerson
11-5
Your life partner somehow convinced you that you cantafford the car of your dreams, priced at $28800. You are
advised to Save up for 4 yearsand then buy the car forcash. How much would you have to save each month, if
you could invest with a return of10%compounded monthly?
You need to decide if this situation involves a PVor a FV and then use the appropriate formula...
PMT
As you have to save up the$28,800,i.e. in the future,
FV= $28,800
Assume you have no savings PV= 0
FindingPaymentSizeof an
Ordinary Simple Annuity
-
8/21/2019 Defferal Annuities
6/54
11
McGraw-Hill Ryerson
11-6
Your life partnersomehow convinced youthat you cant afford the
car of your dreams,priced at $28800. (At leastnot right now).You are
advised toSave up for 4 yearsand
then buy the car for cash.
How much would youhave to save each month, if
you could invest with areturn of 10%
compounded monthly?
FindingPaymentSizeof an
Ordinary Simple Annuity
48
12
PMT = - 490.44
10 0
28800
Formula solution
-
8/21/2019 Defferal Annuities
7/54
11
McGraw-Hill Ryerson
11-7
[FV =PMT (1+ i)n - 1i
]i
PV =PMT 1-(1+ i)-n[ ]
Which Formula?
Algebraic Method of Solving for PMT
(a) If the paymentsform a Simple Annuity go directly to 2.1.
If the annuitys
PV is known,substitute values
of PV, n,andi
into PVformula.
If the annuitys
FVis known,
substitute values
of FV, n,and i
into FVformula.
3. & 4.
(b) If the paymentsform a General Annuity, find candi2
2.
-
8/21/2019 Defferal Annuities
8/54
11
McGraw-Hill Ryerson
11-8
Calculate the quantity within the square brackets.
Rearrange the equationto solve for PMT.
i[FV =PMT (1+ i)n- 1]
iPV =PMT 1-(1+ i)
-n[ ]
Which Formula?
Algebraic Method of Solving for PMT
3.
4.
Applying Method
-
8/21/2019 Defferal Annuities
9/54
11
McGraw-Hill Ryerson
11-9FindingPaymentSizeof an
Ordinary Simple Annuity
Which Formula?Your life partner
somehow convinced youthat you cant afford the
car of your dreams,priced at $28800. (At leastnot right now).You are
advised toSave up for 4 yearsand
then buy the car for
cash.How much would you
have to save each month,if you could invest with a
return of 10%
compounded monthly?
[FV =PMT (1+ i)n - 1i
]
As the annuitys FVis known,
therefore, the FVformula is used
2.
Extract necessarydata...
FV =28800 n = 4*12 = 48
i = .10/12 c= 1 PMT = ?PV =0
-
8/21/2019 Defferal Annuities
10/54
11
McGraw-Hill Ryerson
11-10
12.10
28800
481
1
Your life partnersomehow convinced youthat you cant afford the
car of your dreams,priced at $28800. (At leastnot right now).You are
advised toSave up for 4 yearsand
then buy the car for
cash.How much would you
have to save each month,if you could invest with a
return of 10%
compounded monthly?
0.00831.00831.48940.48940.008358.7225490.44
[FV =PMT (1+ i)n - 1i
]FormulaFV =28800 n = 4*12 = 48
i = .10/12 c= 1 PMT = ?
PV =0
another example
-
8/21/2019 Defferal Annuities
11/54
11
McGraw-Hill Ryerson
11-11
The
-
8/21/2019 Defferal Annuities
12/54
11
McGraw-Hill Ryerson
11-12
Your parents are discussing the terms of the$100 000 mortgagethat they have offered to
hold in the purchase of your first home.
They are considering aninterest rateof 5%compounded monthly.
If you were to take 20 yearsto repay the mortgage,
find the sizeof the monthly payment.
2405
0
100 000
PMT = -659.9612
n =12*20 = 240PV =$100000 FV =0
Formula solution
-
8/21/2019 Defferal Annuities
13/54
11
McGraw-Hill Ryerson
11-13
Your parents arediscussing the terms ofthe 100 000 mortgage
that they have offered tohold in the purchase ofyour first home. They
are considering aninterest rate of 5%
compounded monthly.If you were to take 20
yearsto repay themortgage, find the size
of the monthly payment.
i=.05/12
Extract necessary data...
n =12*20 = 240
PV =$100000
FV =0C=1
-
8/21/2019 Defferal Annuities
14/54
11
McGraw-Hill Ryerson
11-14
Choose appropriate formula and Solve
As the annuitys PVis known, the PVformula is used2.
i=.05/12n =12*20 =240 PV =$100000i
PV =PMT 1-(1+ i)-n[ ]Formula
12.05 1
100 000
240 1
0.00421.00420.3686-0.6314151.530.0015659.96
Size of monthlymortgagepayment
-
8/21/2019 Defferal Annuities
15/54
11
McGraw-Hill Ryerson
11-15
Amount
$
How much interest will you pay your
parents over the 20 year period?
Monthly Payment Number of Payments
659.96 240 158,390.40
Amount Borrowed 100,000.00
Total Interest Paid 58,390.40
x
-
8/21/2019 Defferal Annuities
16/54
11
McGraw-Hill Ryerson
11-16
PMT = -700
As this amountof interest
shocks you,you discuss thepossibility of
making paymentsof $700/month,
to save some timeand interest costs.
Determinethe timeit wil l take you torepay yourmortgage
at this new rate.
700
N = 217.52
Formula solution
218 payments = 18 yrs 2months
-
8/21/2019 Defferal Annuities
17/54
11
McGraw-Hill Ryerson
11-17
Formula)i
PMT
PV i
+
*
1ln
1ln[ ]n
0.0042
i = .05/12 PMT=$700PV= $100,000 C= 1n 0
12.05 1
700
100 000
1
1.00420.00420.5952-0.4048-0.9045-217.52217.52
218payments= 18 yrs 2months
-
8/21/2019 Defferal Annuities
18/54
11
McGraw-Hill Ryerson
11-18
1. Base formula+
=
i
-ni)PMTPV
(11
[ ]2. To isolaten, divide bothsides by PMT
PMTPMT
Continue
Developing the Formula
PMT=PV +
i
-ni)(11[ ]
+=
i
-ni)PMTPV
(11 ][
Formula)i
PMT
+
PV i*
1ln
1ln[ ]n
-
8/21/2019 Defferal Annuities
19/54
11
McGraw-Hill Ryerson
11-19
(a) Multiply both sides by i
3. Continue to isolate n.
PMT=
PV+
i
-ni)(11[ ]
PMT=PV
+ -ni)(11
i[ ]*i *i
(b) Reorganize equation
(c) Now Take the natural logarithm
(ln or lnx) of both sides-n* ln
PMT
*i
=
PV- -n1 +i)(1
-n+i)(1
+i)(1 = ln
(d) Solving forn
divide both sides by ln(1+i)ln(1+i) ln(1+i)
from 2.
=PMT
*iPV1 -
[ ]PMT*iPV1 -
-n* ln +i)(1 = ln[ ]PMT*iPV1 -PMTPV*i)i1ln
1ln
n[ ]
-
8/21/2019 Defferal Annuities
20/54
11
McGraw-Hill Ryerson
11-20
700.00 217.52 152,264.00
Total Interest Saved 6,126.40
Approximately how much money do you save
in interest charges by paying $700/month,
rather than $659.91/month?
Amount
$
Monthly Payment Number of Paymentsx
158,390.40659.96 240
-
8/21/2019 Defferal Annuities
21/54
11
McGraw-Hill Ryerson
11-21
If you could
see your way to a
further increase
of $25/month,
(a) how muchfaster would you
pay off the
mortgage, and
(b) approximatelyhow much less
interest would be
involved?
725
PMT = -725N = 205.62
Paying $725206 payments = 17 yrs 2months
Formula solution
-
8/21/2019 Defferal Annuities
22/54
11
McGraw-Hill Ryerson
11-22
i = .05/12 PMT=$725PV= $100,000 C= 1n 0
0.0042
12.05 1
725
100 000
1
1.00420.00420.5747-0.4253-0.8550-205.52
206payments= 17 yrs 2months
205.52
Formula)i
PMT
+
PV i*
1ln
1ln[ ]n
-
8/21/2019 Defferal Annuities
23/54
11
McGraw-Hill Ryerson
11-23
(b)Total Interest Saved 3,189.50
700.00 217.52 or 218 152,264.00
Amount
$
Monthly Payment Number of Paymentsx
725.00 205.62 or 206 149,074.50
(a) Payments Saved 12
-
8/21/2019 Defferal Annuities
24/54
11
McGraw-Hill Ryerson
11-24
-
8/21/2019 Defferal Annuities
25/54
11
McGraw-Hill Ryerson
11-25
York Furniture has a promotion on a bedroom setselling for $2250. Buyers will pay
no money down and no payments for 12 months.
The first of 24 equal monthly paymentsis due
12 monthsf rom the purchase date.What should the monthlypaymentsbeif York Furnitureearns 10%compounded monthly on its accountreceivableduring boththe deferral period and
the repayment period?
Since you want the furni ture now, this involves a PV
PMT
PV= $2250 Once you repay the loan, FV= 0
Payments are deferredfor 11months.
DEFERRAL
FindingPaymentSizein a
Deferred Annuity
-
8/21/2019 Defferal Annuities
26/54
11
McGraw-Hill Ryerson
11-26York Furniture has a promotion on a
bedroom set selling for $2250. Buyers will payno money down and no payments for 12 months.
The first of 24 equal monthly paymentsisdue 12 monthsf rom the purchase date.
What should the monthlypaymentsbeif YorkFurniture earns 10%compounded monthly
on its accountreceivableduring boththe deferralperiodand the repayment period?
In effect, York furniture has
given a loan to a buyerof $2,250on the day of the sale!
When the payments begin, the buyer owes $2,250
plus accrued interest!
-
8/21/2019 Defferal Annuities
27/54
11
McGraw-Hill Ryerson
11-27
d= 11i= 0.10/12
n= 24
$2250 PMT PMT PMT Payments
$2250
PVAnnuity
FV
PVof the payments at
the end of month 11
FVof the $2,250 loan
at the
end of month 11
=
Months0 11 12 13 35 36
-
8/21/2019 Defferal Annuities
28/54
11
McGraw-Hill Ryerson
11-28
11
12
FV = 2,465.06
10 0
Find the amount owedafter 11months:
2250
$2,465.06is the PVof the annuity
York Furniturehas a promotion
on a bedroom set sellingfor $2250. Buyers will pay
no money down and no
payments for 12 months.The first of 24 equalmonthly paymentsis due
12 monthsfrom thepurchase date. Whatshould the monthlypaymentsbeif York
Furniture earns 10%compounded monthly on
its accountreceivableduring boththe deferral
periodandthe repayment per iod?
FindingPaymentSizein a
Deferred Annuity
-
8/21/2019 Defferal Annuities
29/54
11
McGraw-Hill Ryerson
11-29
2465.06
24
FV = 2,465.06
Now find the PMT of the annuity
0
24 monthly paymentsof $113.75wil l repay the loan.
PV = - 2,465.06PMT = 113.75
York Furniturehas a promotion
on a bedroom set sellingfor $2250. Buyers will pay
no money down and no
payments for 12 months.The first of 24 equalmonthly paymentsis due
12 monthsfrom thepurchase date. Whatshould the monthlypaymentsbeif York
Furniture earns 10%compounded monthly on
its accountreceivableduring boththe deferral
periodandthe repayment per iod?
Formula solution
FindingPaymentSizein a
Deferred Annuity
-
8/21/2019 Defferal Annuities
30/54
11
McGraw-Hill Ryerson
11-30
FV = PV(1 + i)nFormula
FV = 2250(1 + 0.10/12)11= $2,465.06
2465.06
+=
i
-ni)PMTPV
(11[ ]= PMT[1-(1+.10/12)-24]
.10/12
= $113.75PMT
York Furniturehas a promotion
on a bedroom set sellingfor $2250. Buyers will pay
no money down and no
payments for 12 months.The first of 24 equalmonthly paymentsis due
12 monthsfrom thepurchase date. Whatshould the monthlypaymentsbeif York
Furniture earns 10%compounded monthly on
its accountreceivableduring boththe deferral
periodandthe repayment per iod?
24 monthly paymentsof $113.75
will repay the loan.
Find the amount owedafter 11months:
FindingPaymentSizein a
Deferred Annuity
-
8/21/2019 Defferal Annuities
31/54
11
McGraw-Hill Ryerson
11-31
i .e....Number Of Payments
-
8/21/2019 Defferal Annuities
32/54
11
McGraw-Hill Ryerson
11-32
$20,000is invested in a fundearning 8%compounded quarterly.
The first quarterly withdrawal
of $1,000 will be taken from the fundfive years from now.How many withdrawalswill it take to
deplete the fund?N
Payments are deferredfor 19quarters
DEFERRAL
FindingNumber Of Paymentsin a
Deferred Annuity
The FVof $20,000after the deferral, becomes thePV of the annuity...
-
8/21/2019 Defferal Annuities
33/54
11
McGraw-Hill Ryerson
11-33
Years
0 4.75 5 6 7 8
d= 19
$20,000
PV1
n= ?
This FV1 then becomes the PVof theannuity of $1000/quarter
The $20000
earnsinterest for 4years 9months
Payments of $1000/quarter
FV1
i= 0. 08/4 = .02PMT = $1000
-
8/21/2019 Defferal Annuities
34/54
11
McGraw-Hill Ryerson
11-34
$20,000isinvestedin a fund
earning 8%compounded
quarterly.The first quarterlywithdrawal
of $1000 will betaken from thefundf ive years
from now.How many
withdrawalswill it take to
deplete the fund?
19
48 0
Find the FVof$20,000in 4.75 years
20000
$29,136.22is the PVof the annuity
FV = 29,136.22
FindingNumber Of Paymentsin a
Deferred Annuity
-
8/21/2019 Defferal Annuities
35/54
11
McGraw-Hill Ryerson
11-35
1000
Now find the PMT of the annuity
044.1 quarterly paymentswill deplete the
fund(44 full paymentsand 1partial)
29136.22
FV = 29,136.22PV = - 29136.22N = 44.1
Formula solution
$20,000isinvestedin a fund
earning 8%compounded
quarterly.The first quarterlywithdrawal
of $1000 will betaken from thefundf ive years
from now.How many
withdrawalswill it take to
deplete the fund?
FindingNumber Of Paymentsin a
Deferred Annuity
-
8/21/2019 Defferal Annuities
36/54
11
McGraw-Hill Ryerson
11-36
FV = PV(1 + i)nFormula
FV= 20000(1 + 0.08/4)19
= $29,136.22
Find the FVof$20,000in 4.75 years
PMT
PV *i
)i1ln
1ln
n
[ ]
ln(1.02)
=44.1 paymentsor 11 years
$20,000isinvestedin a fund
earning 8%compounded
quarterly.The first quarterlywithdrawal
of $1000 will betaken from thefundf ive years
from now.How many
withdrawalswill it take to
deplete the fund?
FindingNumber Of Paymentsin a
Deferred Annuity
[ln 1 - ]29136.22 *.021000
-
8/21/2019 Defferal Annuities
37/54
11
McGraw-Hill Ryerson
11-37
Whennumber of compoundings per year
number of payments per year
-
8/21/2019 Defferal Annuities
38/54
11
McGraw-Hill Ryerson
11-38
Since you get paid
every second
Thursday you
decide to pay $350
everytwo weeksto make your
budgeting easier.
F ind the new term
of your mortgage if
the interest charges
remain at 5%
compounded
monthly.
12
26350
P/Y = 26
415bi-weekly
payments or15 yrs 11.4 months
C/Y= 12PMT = -350N = 414.74
Formula solution
-
8/21/2019 Defferal Annuities
39/54
11
McGraw-Hill Ryerson
11-39
Since you get paid
every second
Thursday you
decide to pay $350
everytwo weeksto make your
budgeting easier.
F ind the new term
of your mortgage ifthe interest charges
remain at 5%
compounded
monthly.
Determine cStep 1
C= 12 / 26= .4615
i2= (1+i)c- 1
i2= (1+ .05/12) .4615-1
i2= 0.0019
Use cto determinei2Step 2
C=number of compoundings per yearnumber of payments per year
Step 3
-
8/21/2019 Defferal Annuities
40/54
11
McGraw-Hill Ryerson
11-40
as the valueforiin the appropriate annuity formula
Step 3 Use this rate i2= 0.0019
Formula)i
PMT
+
PV i*
1ln
1ln[ ]n
1.00190.00190.5428-0.4571-0.7828
1
350100 000
1
0.0019-414.74
415 paymentsor 15 yrs 11.4 months
$100 000 T
-
8/21/2019 Defferal Annuities
41/54
11
McGraw-Hill Ryerson
11-41
# ofPayments
PaymentAmount
Total CostScenario
1.
2.
$100,000 Twenty-year
Mortgage
Interest 5%per annum
$659.96 $158,390.40
$152,264.00
$149,074.50
$700.00
3.$725.00
$145,250.004. $350.00
Terms
Per
month
Permonth
Per
month
Everytwo
weeks
240
218
206
415 $145,250.004.Every
twoweeks
415
Best Scenario
11 42
-
8/21/2019 Defferal Annuities
42/54
11
McGraw-Hill Ryerson
11-42
350
You are nowconsidering delaying
the purchaseof your first house to
allow for a largerdown payment. If
you save $350per pay,how longwould it taketo have anadditional
$15000, if you canearn 8%compounded
monthly on yoursavings?
12
26
150000
N = 40.32
8
= FV
New requiredFormula
11 43
-
8/21/2019 Defferal Annuities
43/54
11
McGraw-Hill Ryerson
11-43
Determine cStep 1
C= 12 / 26= .4615
i2= (1+i)c- 1
i2= (1+ .08/12).4615
-1
i2= 0.0031
Use cto determinei2Step 2
C=number of compoundings per year
number of payments per year
Step 3
You are nowconsidering delaying
the purchase
of your first house toallow for a largerdown payment. Ifyou save $350per
pay, how longwould
it take to haveanadditional$15000,if you can earn 8%compounded monthly
on your savings?
11 44
i[ ]
-
8/21/2019 Defferal Annuities
44/54
11
McGraw-Hill Ryerson
11-44
You are nowconsidering delaying
the purchase
of your first house toallow for a largerdown payment. Ifyou save $350per
pay, how longwould
it take to haveanadditional$15000,if you can earn 8%compounded monthly
on your savings?
15000
1
1.00310.00310.13161.1316
Formula n)i
PMT
+
FV i*
1ln
1ln[ ]+
0.12370.003140.3
40.3 bi-weekly payments
= approx 1yr 7months
1
350
11 45
i[ ]
-
8/21/2019 Defferal Annuities
45/54
11
McGraw-Hill Ryerson
11-45
1. Base formula-
=
i
PMTFV+
ni)(1 1
[ ]2. To isolaten, divide bothsides by PMT PMTPMT
continued
Developing the Formula
PMT=FV
Formula)i
PMT
+
FV i*
1ln
1ln[ ]n
+
-=
iPMTFV
+ni)(1 1[ ]
-
i+
ni)(1 1[ ]
11 46
-
8/21/2019 Defferal Annuities
46/54
11
McGraw-Hill Ryerson
11-46
from2.PMT
=FV
-
i+
ni)(1 1[ ]
(a) Multiply both sides by i3. Continue to isolate n
PMT
FV= -
i+
ni)(1 1[ ]*i*i
[ ]PMTFV
-+ni)(1 1
*i
(b) Reorganize equation =PMT
FV+
ni)(1 1 *i
(c) Now Take the natural logarithm
(ln or lnx) of both sides nln (1+ i) = ln[ ]PMT*iFV
1 +
(d) Solving forn
divide both sides by ln(1+i)ln(1+i) ln(1+i)
nln (1+ i) = ln[ ]PMT*iFV1 +PMT
FV * i
)
i1ln
1 +ln
n[ ]
11 47
-
8/21/2019 Defferal Annuities
47/54
11
McGraw-Hill Ryerson
11-47
You already have
$10000saved
for your down
payment. If you
save $350per pay,
how longwould it take
to have an additional
$15000? Assumeyou can earn 8%
compounded monthly
on all of your savings.
12
26
Alreadyentered
N = 37.25
350
10000
25000
8
37.5 bi-weekly payments
= approx 1 yr 5months
11 48
-
8/21/2019 Defferal Annuities
48/54
11
McGraw-Hill Ryerson
11-48
You already have
$10000saved
for your down
payment. If yousave $350per pay,
for the next 2 years,
find the size ofyour
available downpayment. Assume
you can earn 8%
compounded monthly
on all of your savings.
Alreadyentered
12
26
FV = 31430.12
10000
52
8
Formula solution
350
11 49
-
8/21/2019 Defferal Annuities
49/54
11
McGraw-Hill Ryerson
11-49
3 Steps
Formula Solution
This is more complicatedto solve when usingalgebraic equations!
1.
2.
3.
Find the FVof the $10 000in 2 years
Find the FVof the $350 per pay
Add totals together
The $10 000continues to earn
interestdur ing the new savings per iod!
You already have
$10000saved
for your down
payment. If yousave $350per pay,
for the next 2 years,
find the size ofyour
available downpayment. Assume
you can earn 8%
compounded monthly
on all of your savings.
11 50
-
8/21/2019 Defferal Annuities
50/54
11
McGraw-Hill Ryerson
11-50
Formula Solution
FV = PV(1 + i)nFormula
= 10000(1 + 0.08/12) 24
= $11,728.88
1.
2.
i2= (1+i)c- 1
= (1+ .08/12).4615-1
= 0.0031
3.
$
11,728.88
= 350[(1+.0031)521]
.0031
= $19701.24 19,701.24
31,430.12Total
=
PMTFV
-
i
+ni)(1 1
[ ]
You already have
$10000saved
for your down
payment. If yousave $350per pay,
for the next 2 years,
find the size ofyour
available downpayment. Assume
you can earn 8%
compounded monthly
on all of your savings.
11 51
-
8/21/2019 Defferal Annuities
51/54
11
McGraw-Hill Ryerson
11-51
11 52
-
8/21/2019 Defferal Annuities
52/54
11
McGraw-Hill Ryerson
11-52
A life insurance company advertises that$50,000will purchase a 20-year annuitypaying $341.13at the end of each month.
What nominal rate of returndoes the annuityinvestment earn?
1
12
341.13240
050000
C/Y = 1I/Y = 5.54
The annuity earns 5.54% pa
11-53
-
8/21/2019 Defferal Annuities
53/54
11
McGraw-Hill Ryerson
11-53
to solve for i without
a financial calculator
11-54
-
8/21/2019 Defferal Annuities
54/54
1111-54
This completes Chapter 11