Principles of Annuities - CIFPs · Simple Annuity Due - a series of constant, future payments occurring at the beginning of each payment period, and each payment period has the same

Principles of AnnuitiesThe Mathematical Foundation of Retirement Planning

Knut LarsenBrigus Learning Inc.Friday, June 5, 2015

Learning Objectives for this Presentation

1. Know the structure of the course2. Know the difference between a simple annuity and a

general annuity, and the difference between an ordinary annuity and an annuity due

3. Apply formulas and calculate the Future Value and the Present Value of an ordinary simple annuity

4. Apply the formula and calculate the equivalent periodic rate

5. Describe a deferred annuity and the procedure for calculating the Present Value of a deferred annuity

6. Describe a perpetuity and a constant-growth annuity

June 5, 2015 Knut Larsen, Brigus Learning Inc. 2

Structure of the Course4 units, 10 chapters, 2 appendices, 32 short scenarios5 review questions after each unitAppendix A: some mathematical detailsAppendix B: the use of HP10B II+ financial calculator20 multiple choice question quiz at the end of the course


Structure of the CourseNo calculus, only basic mathAn understanding of Time Value of Money conceptsUnit 1: Simple AnnuitiesUnit 2: General Annuities Unit 3: Finding the periodic payment, rate, term Unit 4: Special annuities Deferred annuities Perpetuities Constant-growth annuities


Time Value of MoneyMoney has a different value at different points in time,

since money can earn a returnFundamental concepts: Future Value (FV) Present Value (PV) Payment (PMT) Periodic Rate of Return/Discount Term (number of periods, n)


Future Value (FV)


nrPMTFV )1( +×=

Present Value (PV)


nrPMTPV

)1( +=

Fundamental Annuities


“Annuity” ►- a series of constant payments occurring at intervals of the same length

1. Ordinary Annuities- payments occur at the END of each payment period

2. Annuities Due- payments occur at the BEGINNING of each payment period

Simple Annuities


“Simple” ►- the length of the payment period and the length of the compounding (discounting) period are the same

1. Ordinary Simple Annuity- a series of constant, future payments occurring at the end of each payment period, and each payment period has the same length as the compounding (discounting) period

2. Simple Annuity Due- a series of constant, future payments occurring at the beginning of each payment period, and each payment period has the same length as the compounding (discounting) period

FV of an Ordinary Simple Annuity


Definition ►- The Future Value (FV) of annuity is the sum of the future value of each constant periodic payment

)(...)()( 21 nPMTFVPMTFVPMTFVFV +++=

FV of any Periodic Payment


PMTFV kt =)( ,

the future value of the payment made at the end of period t, valued at the end of period k(t , k = 1, 2, ... ,n; k ≥ t)

FV of Payment 1 in Period 1


PMT PMTFV =)( 1,1



r)PMT(1rPMTPMT

PMTFV

+=×+

=)( 2,1



2r)PMT(1r)r)(1PMT(1

rr)PMT(1r)PMT(1 PMTFV

+

=++=×+++

=)( 3,1

FV of Payment 1 in Period n


1

,1

)1()1(

)1)()(

−+

=++

=+

=

n

2-n

1-n1,

n

rPMTrr)PMT(1

rFV(PMT PMTFV



2

,2

)1(

)(−+

=n

n

rPMT

PMTFV



3

,3

)1(

)(−+

=n

n

rPMT

PMTFV

FV of Payment n in Period n


PMTrPMTrPMT

PMTFVnn

nn

=+

=+

=−

0

,

)1()1(

)(

FV of all Payments in Period n


)(...)()( 21 nPMTFVPMTFVPMTFVFV +++=

1)1( −+ nrPMT2)1( −+ nrPMT

PMT

PMTrPMTrPMTFV nn +++++= −− ...)1()1( 21

FV of all Payments in Period n


[ ]1...)1()1(

)(...)()(21

21

+++++×

=+++=−− nn

n

rrPMTPMTFVPMTFVPMTFVFV

−+×=

rr)(1PMTFV

n 1

Application of the FV-formula


−+×=

rr)(1PMTFV

n 1

Savings plan:1. $1,000 at the end of each month2. Term = 10 years3. Annual Rate of Return = 6% compounded monthly

−+×=

×

1206.0

11206.0

000,1$12)(10)(1

FV



−+×=

rr)(1PMTFV

n 1


−+×=

005.01005.0000,1$

(120))(1FV



−+×=

rr)(1PMTFV

n 1


−

×=005.0

1000,1$ ... 1.81939673FV



−+×=

rr)(1PMTFV

n 1


×=

005.0000,1$ ... 0.81939673FV



−+×=

rr)(1PMTFV

n 1


[ ]... 163.87934FV ×= 000,1$



−+×=

rr)(1PMTFV

n 1


34.879,163$≅FV

Summary so far:


−+×=

rr)(1PMTFV

n 1

1. Annuity- a series of constant payments occurring at intervals of the same length

2. Ordinary Annuities- payments occur at the END of each payment period

3. Annuities Due- payments occur at the BEGINNING of each payment period

4. Simple - the length of the payment period and the length of the compounding (discounting) period are the same

5. The Future Value (FV) of annuity - the sum of the future value of each constant periodic payment

PV of an Ordinary Simple Annuity


Definition ►- The Present Value (PV) of an annuity is the sum of the present (discounted) value of each future constant periodic payment

)(...)()( 21 nPMTPVPMTPVPMTPVPV +++=



Definition ►- The Present Value (PV) of an ordinary, simple annuity is the sum of the present (discounted) value of each future constant periodic payment

nrPMT

rPMT

rPMTPV

)1(...

)1(1 2 +++

++

+=




+

+++

++

×= nrrrPMTPV

)1(1...

)1(1

11

2




+

−×=

rrPMTPV

n)1(11




+−×=

−

rrPMTPV

n)1(1

Application of the PV-formula


Retirement plan:1. $2,000 received at the end of each month2. Term = (retirement period) = 25 years3. Annual Discount Rate = 4% compounded

monthly

+−×=

−

rrPMTPV

n)1(1




monthly

+−×=

×−

1204.0

)1204.01(1

000,2$)1225(

PV




monthly

+−×=

−

...003333.0...)003333.01(1000,2$

300

PV




monthly

−×=

−

...003333.0...)003333.1(1000,2$

300

PV




monthly

−

×=...003333.0

1000,2$ ..0.3684917.PV




monthly

×=

...003333.0000,2$ ..0.6315082.PV




monthly

[ ]9...189.452482PV ×= 000,2$




monthly

378,904.96PV $≅

Simple Annuity Due - FV


dueannuity simplea of value future FVannuity impleordinary s an of value future FV

D

O

==

)1( rFVFV OD +×=

)1(1)1( rrrPMTFV

n

D +×

−+×=

+−+×=

+

rrrPMTFV

n

D)1()1( 1

Simple Annuity Due - PV


dueannuity simplea of value present PVannuity impleordinary s an of value present PV

D

O

==

)1( rPVPV OD +×=

)1()1(1 rr

rPMTPVn

D +×

+−×=

−

+−+×=

+−

rrrPMTPV

n

D

1)1()1(

General Annuities


1. A series of constant payments occurring at intervals of the same length, but the length of the payment period is different from the length of the compounding period

2. In principle, the kind of reasoning used for simple annuities also applies to general annuities

3. Instead of using the given periodic rate of return or discount, use the equivalent periodic rate

Equivalent Periodic Rate


Definition ►

the equivalent periodic rate is the rate per payment period that is equivalent to the given rate per compounding period



Example:• a Canadian mortgage with 4% annual interest

compounded semi-annually and monthly payments• the given rate per compounding period is 2% and the

equivalent rate is the value of i that satisfies the following equation:

1)1(02.0 6 −+= i6)1(02.1 i+=

)1(02.1 61

i+=

i+=102.16



Example:• a Canadian mortgage with 4% annual interest

compounded semi-annually and monthly payments• the given rate per compounding period is 2% and the

equivalent rate is the value of i that satisfies the following equation:

1021i 61 −= )/(.1- ..1.0033058.i =

..0.0033058.i =

0.33%i ≅

FV of an Ordinary General Annuity


−+×=

i)i(1PMTFV

n 1

FV of an Ordinary General AnnuityAn Application


−+×=

i)i(1PMTFV

n 1

Investment plan:1. invest $1,000 at the end of each month for the next 20

years2. Expected rate of return = 5% annually, compounded

annually





annually

1)1(1−+= m

cri

r = the given annual rate = 5% = 0.05c = the number of compounding periods per year = 1

m = the number of payment periods per compounding period = 12





annually

1)105.01( 12

1−+=i

105.1 ...083333.0 −=i1−= ...1.00407412i ...0.00407412=



−+×=

i)i(1PMTFV

n 1



annually



−+×=

...0.00407412...)0.00407412(1FV

240 1000,1$



annually



−

×=...0.00407412

..2.6532977.FV 1000,1$



annually



×=

...0.00407412

..1.6532977.FV 000,1$



annually



[ ]3...405.804485FV ×= 000,1$



annually



9$405,804.4FV ≅



annually

PV of an Ordinary General Annuity


+×=

i)i(1-1PMTPV

-n

Determining the Periodic PaymentOrdinary Simple Annuity


−+×=

r)r(1PMTFV

n 1

−+

×=1n)r(1

rFVPMT


−+×=

i)i(1PMTFV

n 1

−+

×=1n)i(1

iFVPMT

Determining the Periodic PaymentOrdinary General Annuity

Determining the Periodic Rate


−+×=

r)r(1PMTFV

n 1

r = ?

Use a handheld calculator or a spreadsheet

.....no simple algebraic solution





PMT = $2,000 - payable at the end of each month; simple ordinary annuity

n = 25 years

PV = $342,120

r = ?

An illustration:





1. Clear the calculator for old data2. toggle the “BEG/END” so that “BEG” does NOT show3. enter P/YR = 12 (monthly compounding)4. enter PMT = 20005. enter N = 300 (= 25 years x 12 months)6. enter PV = - 342,120 (don’t forget the minus sign!)7. ...and solve for the given annual rate by pressing I/YR8. The result is I/YR = r = 0.05 = 5%

Procedure:

Determining the Term


−+×=

r)r(1PMTFV

n 1

r)ln(1

r)PMTFV(1 ln

n+

×+=



Illustration:Martha can set aside and invest $800 at the end of each month. She needs $100,000 at the end of her saving period. She expects a 6% annual rate of return, compounded monthly. How long will it take her to reach her financial target?

PMT = $800FV = $100,000; ordinary, simple annuityGiven annual rate = 6%r = 0.06/12 = 0.005 (=0.5% monthly)n = ?



PMT = $800FV = $100,000Given annual rate = 6%R = 0.06/12 = 0.005 (=0.5% monthly)

r)ln(1

r)PMTFV(1 ln

n+

×+=






)0050ln(1

0.005)$800

$100,000(1 lnn

.+

×+=




)005ln(10.005)125(1 lnn

.×+

=




)005ln(10.625)(1 lnn

.+

=




)005ln(1(1.625) lnn

.=




…1510.00498754…60.48550781n =




months 1.3 years, 8…97.344115n ≈=

Special AnnuitiesDeferred Annuity definition: A deferred annuity is a series of constant periodic

payments that will commence sometime in the future Most practical application: PV calculations Procedure:

1. Calculate the capital required to support the series of specified periodic payments for a specified number of periods; this means –determine the present value of an annuity at the time when the payments are to begin

2. Discount that capital amount to its equivalent capital value (present value) now


Special AnnuitiesPerpetuities definition: A perpetuity is an annuity with periodic payments that

continue indefinitely Most practical application: PV calculations, since the FV of any

perpetuity is infinitely large


∞→

+−×=

−

n forr

r11PMTPVn)(lim

rPMT

rPMTPV =×=

1

Special AnnuitiesConstant-Growth Annuities definition: A constant-growth annuity is an annuity with periodic

payments that grow at a constant rate from one period to the next the initial periodic payment and the rate of periodic payment

growth are known


grgrPMTFV

nn

−+−+

×=)1()1(

grrg

PMTPVn

n

−++

−×= )1(

)1(1

Learning Objectives for this Presentation

1. Know the structure of the course2. Know the difference between a simple annuity and a

general annuity, and the difference between an ordinary annuity and an annuity due

3. Apply formulas and calculate the Future Value and the Present Value of an ordinary simple annuity

4. Apply the formula and calculate the equivalent periodic rate

5. Describe a deferred annuity and the procedure for calculating the Present Value of a deferred annuity

6. Describe a perpetuity and a constant-growth annuity


SummaryCharacteristics of Annuities: simple or general ordinary or due immediate or deferred constant-payment or constant-growth finite or infinite (perpetuities)

Understand the logic behind annuities: Mathematical logic supports common sense Provides insight to the impact of changes


That’s it!

Thank you for listening!

Contact info:[email protected](416) 532-0999


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Principles of Annuities - CIFPs · Simple Annuity Due - a series of constant, future payments occurring at the beginning of each payment period, and each payment period has the same