13 The Time Value of Money(annuities FV )

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The Time Value of Money
Book 1. Fundamentals of Financial Management 
by J.van Horne , Jhon M. Wachowicz 
Book 2. Engineering Economy 
 by William G. Sullivan, Elin M. Wicks, C.Patrick Koelling
Types of Annuities
An Annuity represents a series of equal payments (or receipts) occurring over a specified number of equidistant periods.
Ordinary Annuity: Payments or receipts occur at the end of each period.
Annuity Due: Payments or receipts occur at the beginning of each period.
Examples of Annuities
 Student Loan Payments
 Car Loan Payments
 Insurance Premiums
 Mortgage Payments
 Retirement Savings
0 1 2 3
 $100 $100 $100
(Ordinary Annuity)
End of
Period 1
End of
Period 2
Today
Equal Cash Flows 
Each 1 Period Apart
End of
Period 3
Parts of an Annuity
3.‹#›
Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.
0 1 2 3
$100 $100 $100
(Annuity Due)
Beginning of
Period 1
Beginning of
Period 2
Today
Equal Cash Flows 
Each 1 Period Apart
Beginning of
Period 3
Parts of an Annuity
3.‹#›
Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.
FVAn = R(1 + i)n-1 + R(1 + i)n-2 + 		 ... + R(1 + i)1 + R(1 + i)0
 R R R
0 1 2 n n+1
FVAn
R = Periodic 
 Cash Flow
Cash flows occur at the end of the period
i%
. . .
Overview of an 
Ordinary Annuity – FVA
3.‹#›
Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.
	 FVA3 = $1,000(1.07)2 + 			 $1,000(1.07)1 + $1,000(1.07)0
	 = $1,145 + $1,070 + $1,000 		 = $3,215
$1,000 $1,000 $1,000
0 1 2 3 4
$3,215 = FVA3
7%
$1,070
$1,145
Cash flows occur at the end of the period
Example of an
Ordinary Annuity – FVA
3.‹#›
Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.
FVADn = R(1 + i)n + R(1 + i)n-1 + 		 ... + R(1 + i)2 + R(1 + i)1 	 = FVAn (1 + i)
 R R R R R
0 1 2 3 n–1 n
FVADn
i%
. . .
Overview View of an
Annuity Due – FVAD
Cash flows occur at the beginning of the period
3.‹#›
Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.
	FVAD3 = $1,000(1.07)3 + 			 $1,000(1.07)2 + $1,000(1.07)1
	 = $1,225 + $1,145 + $1,070 		 = $3,440
$1,000 $1,000 $1,000 $1,070
0 1 2 3 4
$3,440 = FVAD3
7%
$1,225
$1,145
Example of an
Annuity Due – FVAD
Cash flows occur at the beginning of the period
3.‹#›
Van Horne and Wachowicz, Fundamentals of Financial Management, 13th edition. © Pearson Education Limited 2009. Created by Gregory Kuhlemeyer.
Ordinary annuities 
 
FVA
n
= R(1+i) 
n-1
+ R (1+i) 
n-2
+. . . . . . . . +R(1+i) 
1
+R(1+i)
0
 
FVA
n
= Rቂ
(𝟏+𝒊))
𝒏
 −𝟏
𝒊
ቃ 
( R= periodic receipts) 
Q1. Find the amount of Rs 500 invested each of 4 years at 3% interest 
compounded annually, if invested at the end of each year. 
Sol. R= 500 
 i = 3% 
 = .03 
 n = 4 years 
 FVAn= R(1+i) 
n-1
+ R (1+i) 
n-2
+ R(1+i)
n-3
 +R(1+i)
n-4
 
 = 500(1+.03)
3
+ 500(1+.03)
2
+500(1+.03)
1
+500 
 = 2091.81 
FVAD
n
= R(1+i) 
n
+ R (1+i) 
n-1
+. . . . . . . . +R(1+i)
1
 
FVAD
n
= Rቂ
(𝟏+𝒊)
𝒏+𝟏
 −𝟏
𝒊
ቃ - R 
R= periodic receipts 
Q15. A man set aside Rs200 at the beginning of each year towards a fund for his son’s 
college education. If the money is invested at 4% per year , how much will he 
accumulated at the end of 10 ears? 
Sol. R= 200 
 n= 10 years 
 i = 4% per year 
 = .04 
 FVAD
n
= Rቂ
(𝟏+𝒊)
𝒏+𝟏
 −𝟏
𝒊
ቃ - R 
 FVAD
n
= 200ቂ
(𝟏+.𝟎𝟒)
𝟏𝟎+𝟏
 −𝟏
.𝟎𝟒
ቃ - 200 
 = 2497.27 
Q17. On his fifty first birthday and each third month thereafter Dr Habib invested 
Rs475 at 4% interest rate compounded quarterly. At the end of his fifty seven year, 
what was the amount on deposits? 
Sol. R= 475 
 i= 4% compounded quarterly 
 = 
4
4
 % 
 = .01 
 n= 6 years 
 = 24 (quarterly 6x4)