time value of money (annuity)

8/9/2019 time value of money (annuity)

1/23

Annuity

Future value and Present value of anordinary annuityFuture value and Present value of anannuity due

Present value of a Perpetuity


2/23

Future value of an ordinary

annuity

FV(OA)= A [{(1+i)n-1}/i]

Where:A = amount of the Periodic payment

i = discount rat

n = total no. of computation (t x m)

t = term or timem = no. of computation in a year


3/23

Example: What amount will accumulate if we deposit $5,000 atthe end of each year for the next 5 years? Assume an interest of6% compounded annually.

PV or A = 5,000

i = .06n = 5

FV(OA)= A [{(1+i)n-1}/i]

FVoa = 5,000 [ (1.3382255776 - 1) /.06 ] = 5,000 (5.637092) =28,185.46


4/23

Future Value Interest Factor for

ordinary Annuity

(FVIFA)

n

FVIFA= (1+i)t-1t=1

FVIFA = 1 x [(1+i)n-1]

i


5/23

Future value of an ordinary

annuity

FVIFA = 1 x [(1+i)n-1i

FVIFA = 1 x [(1+.06)3-1] = 3.1836 x 5000= 15918.06

Year 1 (1.00) 2 (1.06) 3 (1.1236) 4 (1.191016) 5(1.2625)

Begin 0 5000.00 10300.00 15,918.00 21,873.08

Interest(.06) 0 300 618.00 955.08 1312.38

Deposit 5000.00 5000.00 5000.00 5000.00 5000.00End 5000.00 10300.00 15,918.00 21,873.08 28,185.46


6/23

Present value of an ordinary

annuity

PV(OA)= A [{1-(1+i)-n}/i]

Where:A = amount of the Periodic payment

i = discount rate

n = total no. of computation (t x m)

t = term or time

m = no. of computation in a year


7/23

Example: What amount must you invest today at6% compounded annually so that you can withdraw$5,000 at the end of each year for the next 5 years?

PMT = 5,000i = .06n = 5

PV(OA)= A [{1-(1+i)-n}/i] PVoa = 5,000 [(1 - (1/(1 + .06)5)) / .06] = 5,000

(4.212364) = 21,061.82


8/23

ordinary Annuity

(PVIFA)

n

PVIFA= 1t=1 (1+i)1

PVIFA = 1 x [1-{1/(1+i)n}]i


9/23

Present value of an ordinary

annuity

PVIFA = 1 x [1-{1/(1+i)n}]

iPVIFA = 1 x [1-{1/(1+.06)3}] = 2.6730 x 5000=13,365

.06

Year 1 (.9434) 2 (.89) 3 (.8396) 4 (.7921) 5(.7473)

Begin 21,061.82 17,325.53 13,365.06 9,166.96 4,716.98

Interest(.06) 1,263.71 1,039.53 801.90 550.02 283.02

Withdraw -5000.00 -5000.00 -5000.00 -5000.00 -5000.00

End 17,325.53 13,365.06 9,166.96 4,716.98 .00


10/23

Future value of an annuity due

FV(AD)= FV(OA) x (1+i)

Where: FV(OA) = A [{(1+i)n-1}/i]i = discount raten = total no. of computation (t x m)


11/23

Example: What amount will accumulate if wedeposit $5,000 at the beginning of each year forthe next 5 years? Assume an interest of 6%

compounded annually. PV or A = 5,000

i = .06n = 5

FV(AD)= FV(OA) x (1+i) FVoa = 28,185.46 (1.06) = 29,876.59


12/23

Future value of an annuity due

FVIFA(annuity due)= FVIFA x (1+i)

FVIFA(ad) = {1 x [(1+.06)3-1]} x (1.06).06

= 3.374616 x 5000=16,873.08

Year 1 (1.00x1.06) 2 (1.06x1.06) 3(1.1236x1.06)

4

(1.191016x1.06)

5

(1.2625x1.06)

Begin 0 5000.00 10300.00 15,918.00 21,873.08

Deposit 5000.00 5000.00 5000.00 5000.00 5000.00

Interest(.06) 300 618.00 955.08 1312.38 1,691.13

End 5,300.00 10,918.00 16,873.08 23,185.46 29,876.59


13/23

Present value of an annuity due

PV(AD)= PV(OA) x (1+i)

Where: PV(OA)= A [{1-(1+i)-n}/i]


14/23

Example: What amount must you invest today a 6%interest rate compounded annually so that you canwithdraw $5,000 at the beginning of each year for

the next 5 years? PMT = 5,000

i = .06n = 5

PV(AD)= PV(OA) x (1+i) PVoa = 21,061.82 (1.06) = 22,325.53


15/23

Present value of an annuity due

PVIFA(ad) = PVIFA x (1+i)PVIFA = ( 1 x [1-{1/(1+.06)3}]) x (1.06)

.06=2.8334 x 5000= 14,166.96

Year 1(.9434x1.06)

2

(.89x1.06)

3

(.8396x1.06)4

(.7921x1.06)5

(.7473x1.06)

Begin 22,325.53 18,365.06 14,166.96 9,716.98 5,000.00

Interest(.06)

1,039.53 801.90 550.02 283.02Withdraw -5000.00 -5000.00 -5000.00 -5000.00 -5000.00

End 18,365.06 14,166.96 9,716.98 5,000.00 .00


16/23

Perpetuity

A constant stream of identical cash flows withno end.

PV of a perpetuity= A

iWhere:A = amount of the Periodic payment

i = discount rate


17/23

EX. Assume that a perpetual bond has an $80-per-year interest payment & the discount rate

is 10%. The PV of this is $800.

PV of a perpetuity= $80

10%


18/23

Mixed Streams

Future value and Present valueof a mixed stream


19/23

Mixed stream

Stream of unequal periodic cash flows thatreflect no pattern.


20/23

Future value of a mixed stream

nFVIFA = (1+i)t-1t=1

Where: i = 8%

Year Cash flow No. of yearsearning interest

FVIFA Future value

1 11,500 2 1.166 13,409

2 14,000 1 1.080 15,120

3 12,900 0 1.000 12,900

FV of mixed stream 41,429


21/23

Present value of a mixed

stream

n

PVIFA= 1t=1 (1+i)1

Where: i = 9%

Year Cash flow PVIFA Future value

1 400 .917 366.8

2 800 .842 673.6

3 500 .772 386FV of mixed stream 41,429


22/23

FV(OA)= A [{(1+i)n

-1}/i]FVIFA = 1 x [(1+i)n-1]

IPV(OA)= A [{1-(1+i)-n}/i]

PVIFA = 1 x [1-{1/(1+i)n}]

i


23/23

FV(AD)= FV(OA) x (1+i)FVIFA(annuity due)= FVIFA x (1+i)

PV(AD)= PV(OA) x (1+i)

PVIFA(ad) = PVIFA x (1+i)

PV of a perpetuity= Ai

time value of money (annuity)

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