1. Annuities
Due
Chapter 13
McGraw-Hill Ryerson©

2. Learning Objectives Annuities Due Annuities Due Calculate
After completing this chapter, you will be able to:
Calculate
LO 1.
… the Future Value and Present Value of
Annuities Due
LO 2.
…the payment size, number of payments, and interest rate in
Annuities Due

3. Membership dues are usually paid in advance
Annuities Due
1st.
If you lease equipment, a vehicle, or rent property, the typical lease contract requires payments at the beginning of each period of coverage
15th.
Membership dues are usually paid in advance
Only a small modification to the math of Ordinary Annuities is needed to handle
Annuities Due

4. are made at the beginning of the period
Classification of
Annuities
Ordinary Annuity
Annuitiy Due
Payments / Deposits
are made at the
end of the period
Payments / Deposits
are made at the beginning of the period

5. Compare the payment interval to the compounding interval13
Annuities
Due
Classification of
Annuities
13 - 6
Is the payment at the end or at the beginning of each payment interval?
Compare the payment interval to the compounding interval
Annuity
Category
End
Equal
Ordinary Simple
End
Not Equal
Ordinary General
Simple Annuity Due
Beginning
Equal
General Annuity Due
Beginning
Not Equal
McGraw-Hill Ryerson©

6. [ ] [ ] * (1+ i) = PMT * (1+ i) = PMT Annuity Due i i Formulae FVdue
LO 1.
FVdue
= PMT
(1+ i)n
- 1 [ i ]
Formula
* (1+ i)
PVdue
= PMT
1-(1+ i)-n [ i ]
Formula
* (1+ i)
If the payments form a general annuity due, use i2 = (1+i)c – 1 in the formula for i

7. Clues to help identify annuities due
Tips
“Payments at the beginning of each…..” 1. 2.
“Payments…in advance” 3.
“First payment … made today”
“Payments ….. starting now” 4.

8. How much will Elyse accumulate in her RRSP by age 60 if she makes semi-annual contributions of $1, starting on her 30th birthday?
Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made on her 60th birthday.

9. Financial Q 2nd BGN 2nd SET 2nd QUIT BGN BGN END Step 2…
Step 1 – Set your calculator to the “BGN mode”
How much will Elyse accumulate in her RRSP by age 60 if she makes semi-annual contributions of $1,000 starting on her 30th birthday? Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made on her 60th birthday. Q
Your calculator is now set for annuity due
Notice the tiny BGN above the 0
BGN
BGN
END
2nd
BGN
SET
2nd
QUIT
2nd
Step 2…

10. Elyse’s RRSP value on her 60th birthday
Financial
Step 2 – Solve for the FV of the annuity due
How much will Elyse accumulate in her RRSP by age 60 if she makes semi-annual contributions of $1,000 starting on her 30th birthday? Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made on her 60th birthday. Q
BGN
Elyse’s RRSP value on her 60th birthday
FV= 247,510.31 2
2nd
ON/OFF
P/Y
1000 60
PMT N 8 PV
ENTER
CPT FV
QUIT
2nd
Algebraic
Calculation

11. Elyse’s RRSP value on her 60th birthday[ FV
= PMT
(1+ i)n
- 1 i ]
* (1+ i)
Formula
c = 1
i =
0.08/2
PMT =
1000
n = 60
How much will Elyse accumulate in her RRSP by age 60 if she makes semi-annual contributions of $1,000 starting on her 30th birthday? Assume that the RRSP earns 8% compounded semi-annually and that no contribution is made on her 60th birthday. Q
Elyse’s RRSP value on her 60th birthday
247,510.31
9.5196 60
1.04 y x - 1 .
0.04 X
1000 X
1.04

12. A lottery offers the winner a choice between a $300,000 cash prize, or quarterly payments of $7,000 beginning immediately and continuing for 20 years Which alternative should the winner pick if money is worth % compounded quarterly?
What do we have to find?
Need to find the PV today of the payment alternative and compare with the $300,000 cash.

13. Your calculator is now set for annuity due
Financial
Step 1 – Set your calculator to the “BGN mode”
A lottery offers the winner a choice between a $300,000 cash prize, or quarterly payments of $7,000 beginning immediately and continuing for 20 years Which alternative should the winner pick if money is worth % compounded quarterly? Q
Your calculator is now set for annuity due
BGN
END
BGN
2nd
BGN
SET
2nd
QUIT
2nd
Step 2…

14. Cash Value of the payment option
Financial
Step 2 – Solve for the PV of the annuity due
A lottery offers the winner a choice between a $300,000 cash prize, or quarterly payments of $7,000 beginning immediately and continuing for 20 years Which alternative should the winner pick if money is worth % compounded quarterly? Q
BGN
Cash Value of the payment option
PV= 283,775.83 4
2nd
ON/OFF
P/Y
7000 80
PMT N 8 FV
CPT PV
ENTER
The $300,000 cash is a better offer …$16,224 more in current day dollars…($300,000 - $283,775.83)
QUIT
2nd
Algebraic
Calculation

15. [ ] [ ] * (1+ i) = PMT Q i 1-(1+ i)-n PVdue *(1.02) = $283,775.83
A lottery offers the winner a choice between a $300,000 cash prize, or quarterly payments of $7,000 beginning immediately and continuing for 20 years. Which alternative should the winner pick if money is worth 8% compounded quarterly?
PVdue
= PMT
1-(1+ i)-n [ i ]
* (1+ i)
(1 -
( /4)-80)
.02
*(1.02)
= 7000 [ ]
= $283,775.83
The $300,000 cash is a better offer… $16,224 more in current day dollars…($300,000 - $283,775.83)

16. Step 1 – Set your calculator to the “BGN mode”
Financial
LO 2.
Step 1 – Set your calculator to the “BGN mode” Q
Step 2 – Solve for the PMT of the annuity due
You have accumulated $104,000 in your RRSP. Your goal is to build it to $250,000 with equal contributions every months for the next 7 years. If you earn 8.5% compounded semi-annually, and start today, find the size of your contributions.
Step 1
Step 2
PMT= -3,286.10
BGN
BGN
2nd
BGN 2
2nd
QUIT
ENTER
P/Y 14
8.5
SET
2nd N
250000 FV
QUIT
2nd
104000 PV
Payment needed is $
CPT
PMT
Algebraic
Calculation

17. [ ] Step Q *(1+ i) = PMT - 1 i Extract necessary data... c = 1 i =
Formula
FVdue
= PMT [
(1+ i)n
- 1 i ]
*(1+ i)
Step
Extract necessary data...
c = 1
i =
0.085/2
PMT = ?
n = 14
You have accumulated $104,000 in your RRSP. Your goal is to build it to $250,000 with equal contributions every months for the next 7 years. If you earn 8.5% compounded semi-annually, and start today, find the size of your contributions. Q
FV =
PV =
Find the FV 7 years from now of the $104,000 already saved. a.
FV = (1+.085/2)14
= $186,250.84
Subtract this value from the $250,000 target to get the FV of the annuity. b.
$250,000 - $186,250.84
= $63,749.16

18. [ ] ] [ Step Choose appropriate formula and Solve FV = PMT (1+ i)n - 1?
0.085/2
PMT = 14 1
FV =
$63,749.16
PV = 0
(1.0425)14 – 1
0.0425
* (1.0425)
= PMT ] [
= PMT( )
PMT = $

19. Q
A car sells for $27,900. The manufacturer offers an interest rate of 1.8% compounded monthly on a three year lease If the residual value is $14,500, find the lease payments assuming a $2,500 down payment.
Purchase Price
Down Payment
Present Value of Lease Payments
Present Value of Residual Value = +
$27,900
$2,500
Present Value of Lease Payments
Present Value of $14,500 = +
$25,400
Present Value of Lease Payments = +
Present Value of $14,500

20. Step 1 – Set your calculator to the “BGN mode”
Financial
Step 1 – Set your calculator to the “BGN mode”
Step 2 – Solve for the PMT of the annuity due
Step 1
Step 2
A car sells for $27, The manufacturer offers an interest rate of 1.8% compounded monthly on a three year lease If the residual value is $14,500, find the lease payments assuming a $2,500 down payment. Q
BGN
PMT=
BGN
2nd
BGN 12
2nd
QUIT
ENTER
P/Y 36
1.8
SET
2nd N
14500 FV
QUIT
2nd
25400 PV
Payment needed is $322.50
CPT
PMT
Algebraic
Calculation

21. Find the PV of the $14,500 residual value
Step
Extract necessary data...
c = 1
i =
0.018/12
PMT = ?
n = 36
A car sells for $27, The manufacturer offers an interest rate of 1.8% compounded monthly on a three year lease If the residual value is $14,500, find the lease payments assuming a $2,500 down payment. Q a.
Find the PV of the $14,500 residual value
PV = (1+.018/12)-36
= $13,738.32 b.
PV annuity =
SP – DP – PV of residual value
= $27,900 - $ $13,738.32
= $11,661.68
See Next

22. [ ] [ ] * (1+ i) = PMT Step i 1-(1+ i)-n PVdue
Choose appropriate formula and Solve
Step
PVdue
= PMT
1-(1+ i)-n [ i ]
* (1+ i)
c = 1
i =
0.018/12
PMT = ?
n = 36
FV =
PV = $11,661.68 [ ] =
1 - (1.0015)-36
0.0015
* (1.0015)
PMT
= PMT ( )
PMT = $ Lease Payment

23. Q i = .06/4 QUIT 2nd BGN SET 2nd P/Y QUIT ENTER I/Y PV PMT FV CPT N
How long will it take to accumulate $1 million in an RRSP if the first quarterly payment of $2000 is made today? Assume the RRSP earns 6% compounded quarterly.
i = .06/4
PV = $0
n =?
FV = $
Step 1
Step 2
143 Quarterly payments
N =
BGN
BGN
QUIT
2nd
BGN
SET
2nd
P/Y 4
QUIT
ENTER
35.75 years 6
I/Y PV
2000
PMT FV
CPT N
Algebraic
Calculation

24. …using the algebraic method to determine Number of Payments
Calculation
…using the algebraic method to determine Number of Payments
PMT
FVdue i * ( ) + 1 ln [ ] n
(1+ i)
Formulae
PMT i * ( ) + 1 ln [ ] n
PVdue
(1+ i)

25. How long will it take to accumulate $1 million in an RRSP if the first quarterly payment of $2000 is made today? Assume the RRSP earns 6% compounded quarterly. Q
n =?
PV = $0
FV = $
i = .06/4
PMT
FVdue i * ( ) + 1 ln [ ]
n =
(1+ i)
Formula ] [
1,000,000 * .015
ln (1.015)
2000(1.015) 1+
n = ln =
0.0149
= i.e. 143 quarterly payments
or years

26. Effective Interest rate on the monthly payment plan = 13.04%Q
A $100,000 life insurance policy requires an annual premium of $420 or a monthly premium of $37. In either case, the premium is due at the beginning of the period of coverage. What is the effective rate of interest charged to those who pay monthly?
2nd
P/Y
BGN
mode
Set to
P/Y =
BGN
C/Y =
I/Y = 12
ENTER 12 N
420 PV 37
PMT 1
ENTER
QUIT
2nd FV
CPT
I/Y
Effective Interest rate on the monthly payment plan = 13.04%

27. This completes Chapter 13