business mathematics jerome chapter 11
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McGraw-Hill Ryerson©
Chapter 11
OrdinaryAnnuities
McGraw-Hill Ryerson©
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McGraw-Hill Ryerson©
Calculate the…
Learning ObjectivesAfter completing this chapter, you will be able to:
… number of payments in ordinary and deferred annuities
… payment size in ordinary and deferred annuities
… interest rate in ordinary annuities
LO-1
LO-2
LO-3
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Using your financial calculator
we need to reorganize the formulae to solve algebraically
… solve for payment number or size or interest rate using the
same steps as before …
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Finding the Payment Size….
PMT
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Your life partner somehow convinced you that you can’t afford the car of your dreams, priced at $28800. You are advised to… “Save up for 4 years and 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?
You need to decide if this situation involves… a PV or 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
Finding Payment Size of an
Ordinary Simple Annuity
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Your life partner somehow convinced you that you can’t afford the
car of your dreams, priced at $28800. (At least
not right now). You are advised to… “Save up for 4 years and 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?
Finding Payment Size of an
Ordinary Simple Annuity
48
12PMT = - 490.44
10 028800
Formula solution
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[FV = PMT (1+ i)n - 1i ]i
PV = PMT 1-(1+ i)-n[ ]
Which Formula? Algebraic Method of Solving for PMT
(a) If the payments form a Simple Annuity go directly to 2. 1.
If the annuity’s PV is
known, substitute values of PV, n,
and i into PV formula.
If the annuity’s FV is known,
substitute values of FV, n, and i
into FV formula.
3. & 4.
(b) If the payments form a General Annuity, find c and i2
2.
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Calculate the quantity within the square brackets.
Rearrange the equation to 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…
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Finding Payment Size of an
Ordinary Simple Annuity
Which Formula? Your life partner somehow convinced you that you can’t afford the
car of your dreams, priced at $28800. (At least
not right now). You are advised to… “Save up for 4
years and 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 annuity’s FV is known, therefore, the FV formula is used
2.
Extract necessary data...FV = 28800 n = 4*12 = 48
i = .10/12 c = 1 PMT = ?PV = 0
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12.10
28800
481
1
Your life partner somehow convinced you that you can’t afford the
car of your dreams, priced at $28800. (At least
not right now). You are advised to… “Save up for 4
years and 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?
0.00831.0083 1.48940.48940.008358.7225490.44
[FV = PMT (1+ i)n - 1i ]Formula
FV = 28800 n = 4*12 = 48i = .10/12 c = 1 PMT = ?
PV = 0
…another example
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The
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Your parents are discussing the terms of the $100 000 mortgage that they have offered to
hold in the purchase of your first home. They are considering an interest rate of 5% compounded monthly. If you were to take 20 years to
repay the mortgage, find the size of the
monthly payment.
24050
100 000
PMT = -659.9612
n =12*20 = 240PV = $100000
FV = 0
Formula solution
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Your parents are
discussing the terms of the 100 000 mortgage
that they have offered to hold in the purchase of your first home. They
are considering an interest rate of 5%
compounded monthly. If you were to take 20
years to repay the mortgage, find the size
of the monthly payment..
i = .05/12
Extract necessary data...
n =12*20 = 240
PV = $100000
FV = 0C =1
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Choose appropriate formula and Solve
As the annuity’s PV is known, the PV formula is used2.
i = .05/12n =12*20 =240 PV = $100000i
PV = PMT 1-(1+ i)-n[ ]Formula
12.05 1
100 000
240 1
0.00421.0042 0.3686-0.6314151.530.0015659.96
Size of monthly mortgage
payment
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Amount$
How much interest will you pay your parents over the 20 year period?
Monthly Payment Number of Payments659.96 240 158,390.40
Amount Borrowed 100,000.00
Total Interest Paid 58,390.40
x
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PMT = -700
As this amount of interest
shocks you, you discuss the
possibility of making payments
of $700/month, to save some time
and interest costs.
Determine the time
it will take you to repay your
mortgage at this new
rate.
700
N = 217.52
Formula solution
218 payments = 18 yrs 2months
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Formula iPMTPV i
1ln
1ln[ ]n
0.0042
i = .05/12 PMT = $700PV = $100,000 C = 1 n 0
12.05 1
700
100 000
1
1.0042 0.00420.5952-0.4048-0.9045-217.52 217.52
218 payments = 18 yrs 2months
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1. Base formulai
-ni)PMTPV
(11[ ]2. To isolate n, divide both
sides by PMT PMTPMT
…Continue…
Developing the Formula
PMTPV
i-ni)(11[ ]
i
-ni)PMTPV
(11 ][
Formula iPMT
PV i*
1ln
1ln[ ]n
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(a) Multiply both sides by i
3. Continue to isolate n.PMT
PV i
-ni)(11[ ]
PMTPV -ni)(11
i[ ] *i *i
(b) Reorganize equation
(c) Now Take the natural logarithm (ln or lnx) of both sides
-n* ln
PMT *iPV -n1 i)(1
-ni)(1
i) (1 ln
(d) Solving for n… 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
1lnn
[ ]
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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 Paymentsx158,390.4
0659.96 240
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If you could see your way to a further increase
of $25/month, (a) how much
faster would you pay off the
mortgage, and (b) approximately
how much less interest would be
involved?
725
PMT = -725N = 205.62
Paying $725 206 payments = 17 yrs
2months
Formula solution
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i = .05/12 PMT = $725PV = $100,000 C = 1 n 0
0.0042
12.05 1
725
100 000
1
1.0042 0.00420.5747-0.4253-0.8550-205.52
206 payments = 17 yrs 2months
205.52
Formula iPMT
PV i*
1ln
1ln[ ]n
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(b)Total Interest Saved 3,189.50
700.00 217.52 or 218 152,264.00
Amount$
Monthly Payment Number of Paymentsx
725.00 205.62 or 206 149,074.50(a) Payments Saved 12
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York Furniture has a promotion on a bedroom set selling for $2250. Buyers will pay “no money down and no payments for 12 months.”
The first of 24 equal monthly payments is due 12 months from the purchase date. What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both the deferral period
and the repayment period?
Since you want the furniture now, this involves a PV
PMT
PV = $2250 Once you repay the loan, FV = 0 Payments are deferred for 11 months.
DEFERRAL
Finding Payment Size in a
Deferred Annuity
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York Furniture has a promotion on a bedroom set selling for $2250. Buyers will pay “no money down and no payments for 12 months.” The first of 24 equal monthly payments is due 12 months from the purchase date.
What should the monthly payments be if York Furniture earns 10% compounded monthly on its account receivable during both
the deferral period and the repayment period?In effect, York furniture has given a loan to a buyer of $2,250
on the day of the sale!
When the payments begin, the buyer owes $2,250
plus accrued interest!
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d = 11i = 0.10/12
n = 24
$2250 PMT PMT PMT Payments
$2250
PVAnnuity
FV
PV of the payments at the end of month 11
FV of the $2,250 loan at the end of month 11
=
Months0 11 12 13 35 36
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11
12
FV = 2,465.06
10 0
Find the amount owed after 11 months:
2250
$2,465.06 is the PV of the annuity
York Furniture
has a promotion on a bedroom set selling
for $2250. Buyers will pay “no money down and no
payments for 12 months.” The first of 24
equal monthly payments is due 12 months from the
purchase date. What should the monthly payments be if York
Furniture earns 10% compounded monthly on
its account receivable during both the deferral
period and the repayment period?
Finding Payment Size in a
Deferred Annuity
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2465.0624
FV = 2,465.06
Now find the PMT of the annuity …
0
24 monthly payments of $113.75 will repay the loan.
PV = - 2,465.06PMT = 113.75
York Furniture
has a promotion on a bedroom set selling
for $2250. Buyers will pay “no money down and no
payments for 12 months.” The first of 24
equal monthly payments is due 12 months from the
purchase date. What should the monthly payments be if York
Furniture earns 10% compounded monthly on
its account receivable during both the deferral
period and the repayment period? Formula solution
Finding Payment Size in a
Deferred Annuity
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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 Furniture
has a promotion on a bedroom set selling
for $2250. Buyers will pay “no money down and no
payments for 12 months.” The first of 24
equal monthly payments is due 12 months from the
purchase date. What should the monthly payments be if York
Furniture earns 10% compounded monthly on
its account receivable during both the deferral
period and the repayment period?
24 monthly payments of $113.75 will repay the loan.
Find the amount owed after 11 months:
Finding Payment Size in a
Deferred Annuity
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i.e....Number Of Payments
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$20,000 is invested in a fund earning 8% compounded quarterly. The first quarterly withdrawal of $1,000 will be taken from the fund five years from now. How many
withdrawals will it take to deplete the fund?
N
Payments are deferred for 19 quarters
DEFERRAL
Finding Number Of Payments in a
Deferred Annuity
The FV of $20,000 after the deferral, becomes the PV of the annuity ...
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Years0 4.75 5 6 7 8
d = 19$20,000
PV1
n = ?
This FV1 then becomes the PV of the annuity of $1000/quarter
The $20000 earns
interest for 4 years 9 months
Payments of $1000/quarter
FV1
i = 0. 08/4 = .02PMT = $1000
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$20,000 is invested in a fund earning 8% compounded
quarterly. The first
quarterly withdrawal
of $1000 will be taken from the fund five years
from now. How
many withdrawals will it
take to deplete the fund?
19
48 0
Find the FV of $20,000 in 4.75 years
20000
$29,136.22 is the PV of the annuity
FV = 29,136.22
Finding Number Of Payments in a
Deferred Annuity
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1000
Now find the PMT of the annuity …
044.1 quarterly payments will deplete the
fund(44 full payments and 1 partial)
29136.22
FV = 29,136.22PV = - 29136.22N = 44.1
Formula solution
$20,000 is invested in a fund earning 8% compounded
quarterly. The first
quarterly withdrawal
of $1000 will be taken from the fund five years
from now. How
many withdrawals will it
take to deplete the fund?
Finding Number Of Payments in a
Deferred Annuity
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FV = PV(1 + i)nFormula
FV = 20000(1 + 0.08/4)19
= $29,136.22
Find the FV of $20,000 in 4.75 years
PMTPV *i
i1ln
1lnn
[ ]
ln(1.02)
= 44.1 payments or 11 years
$20,000 is invested in a fund earning 8% compounded
quarterly. The first
quarterly withdrawal
of $1000 will be taken from the fund five years
from now. How
many withdrawals will it
take to deplete the fund?
Finding Number Of Payments in a
Deferred Annuity
[ln 1 - ]29136.22 *.021000
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When…number of compoundings per year
number of payments per year
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Since you get paid every second Thursday you
decide to pay $350 every two weeks
to make your budgeting easier.
Find the new term of your mortgage if the interest charges
remain at 5% compounded
monthly.
12
26350
P/Y = 26
415 bi-weekly payments or 15 yrs 11.4 months
C/Y= 12PMT = -350N = 414.74
Formula solution
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Since you get paid every second
Thursday you decide to pay $350
every two weeks to make your
budgeting easier. Find the new term of your mortgage
if the interest charges remain at 5% compounded
monthly.
Determine c Step 1
C= 12 / 26 = .4615
i2 = (1+i)c - 1
i2 = (1+ .05/12) .4615-1
i2 = 0.0019
Use c to determine i2 Step 2
C =number of compoundings per year
number of payments per year
Step 3
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as the value for “i” in the appropriate annuity formula
Step 3 Use this rate i2 = 0.0019
Formula iPMT
PV i*
1ln
1ln[ ]n
1.0019 0.00190.5428-0.4571-0.7828
1
350100 000
1
0.0019-414.74
415 payments or 15 yrs 11.4 months
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# of Payments
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
Per month
Per month
Every two
weeks
240
218
206
415 $145,250.004. Every two
weeks415
Best Scenario
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350
You are now considering delaying the purchase of your first house to
allow for a larger down payment. If
you save $350 per pay, how long would it take to have an additional
$15000, if you can earn 8% compounded
monthly on your savings?
12
26
150000
N = 40.32
8
= FV
New requiredFormula
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Determine c Step 1
C= 12 / 26 = .4615
i2 = (1+i)c - 1
i2 = (1+ .08/12) .4615-1
i2 = 0.0031
Use c to determine i2 Step 2
C =number of compoundings per year
number of payments per year
Step 3
You are now considering delaying the purchase of your first house to
allow for a larger down payment. If you save $350 per
pay, how long would it take to have
an additional $15000, if you can earn 8%
compounded monthly on your
savings?
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You are now considering delaying the purchase of your first house to
allow for a larger down payment. If you save $350 per
pay, how long would it take to have
an additional $15000, if you can earn 8%
compounded monthly on your
savings?
15000
1
1.00310.00310.13161.1316
Formula n iPMT
FV i*
1ln
1ln[ ]+
0.1237 0.003140.3
40.3 bi-weekly payments = approx 1yr 7months
1
350
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1. Base formula
iPMTFV ni)(1 1[ ]
2. To isolate n, divide both sides by PMT PMTPMT
…continued…
Developing the Formula
PMTFV
Formula iPMT
FV i*
1ln
1ln[ ]n
+
iPMTFV
ni)(1 1[ ]
i ni)(1 1[ ]
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…from 2. PMTFV
i ni)(1 1[ ]
(a) Multiply both sides by i 3. Continue to isolate n …
PMTFV
i ni)(1 1[ ] *i *i
[ ]PMT FV
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
n ln(1+ i) ln[ ]PMT * iFV1 +
(d) Solving for n… divide both sides by
ln(1+i) ln(1+i) ln(1+i)
n ln(1+ i) ln[ ]PMT * iFV1 +
PMTFV * i
i1ln
1 +lnn
[ ]
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You already have
$10000 saved for your down
payment. If you save $350 per pay,
how long would it take to have an additional $15000?
Assume you can earn 8%
compounded monthly on all of your savings.
12
26
Already entered
N = 37.25
350
10000
25000
8
37.5 bi-weekly payments = approx 1 yr 5months
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You already have
$10000 saved for your down
payment. If you save $350 per pay, for the next 2 years, find the size of your available down
payment. Assume you can earn 8%
compounded monthly on all of your savings.
Already entered
12
26
FV = 31430.12
10000
52
8
Formula solution
350
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3 Steps
Formula Solution
This is more complicated to solve when
using algebraic equations!
1.
2.
3.
Find the FV of the $10 000 in 2 years
Find the FV of the $350 per pay
Add totals together
The $10 000 continues to earn interest during the new savings period!
You already have
$10000 saved for your down
payment. If you save $350 per pay, for the next 2 years, find the size of your available down
payment. Assume you can earn 8%
compounded monthly on all of your savings.
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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)52 –1].0031
= $19701.24 19,701.2431,430.12Total
PMTFV
i ni)(1 1[ ]
You already have
$10000 saved for your down
payment. If you save $350 per pay, for the next 2 years, find the size of your available down
payment. Assume you can earn 8%
compounded monthly on all of your savings.
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A life insurance company advertises that $50,000 will purchase a 20-year annuity
paying $341.13 at the end of each month.
What nominal rate of return does the annuity investment earn?
1
12
341.13240
050000
C/Y = 1I/Y = 5.54
The annuity earns 5.54% pa
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…to solve for i without
a financial calculator
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This completes Chapter 11