Chapter 3, Pinches
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“Time Value of Money” (from Pinches)
Chapter 3
Future Value (FV) + Present Value (PV)
t = # of periods in which interest is computed
n
t
I = interest paid per period = k (PV0) and k = .10
If PV0; n = 1 
FV1 = PV0 + I = PV0 + k (PV0) = PV0 (1+k) = 500 (1+ 0.10) = 550
If n = 3  FV3 = PV0 1  k  = 500 (1.331) = 665.5
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In general FVn = PV0 (1 + k) n or using FV table of factors: FVn = PV0 (FVk,n)
FVk,n = FV factor for n periods @ k
PV0= value today of future series of discounted CF’s
FVN
= FVn ( PVk,n ) PV factor from Tab B.1 (= PVk,n)
(1  k)n
If k = .10 and FV3 = 665.5  PV0 = 665.5 (.751) = 500
(5a)
PV0 =
FVk,n + PVk,n are reciprocals or inverses of each other  PVk,n =
e.g.: PV10%,3 
1
(for all k,n)
FVk,n
1
1

 .751
FV10,3 1.331
FVk,n
PVk,n
10%
3.0
5%
2.0
1.0
0%
1.0
0.5
n
5%
10%
n
Annuity - a series of equal CF’s for a specified n
Ordinary - payment @ end of period
E.g.: $600 for 4 years, what is final amount when each payment is reinvested @ 10%.
Chapter 3, Pinches
n 1
FVn  PMT (1 k)t  PMT(FVAk,n )
(6a)
t 0
FVAk ,n 
(1  k )  1
k
n
*
PMT = annuity payment

= sigma or sum of
FVA k,n = FV factors from annuity table (Tab. B.4)
If PMT = $600; k = .10; n = 4
 FV4  $600(FVA10%,4 )  $600(4.641)  $2784.6
PV of Ordinary Annuity - how much you are willing to pay for series of payments
which begin in 1 year.
(6b) 

n
1
t =1 (1  k)t
of discounted payments = PV0  PMT
(6b) = PMT (PVA k,n )  PV0
[1 
( PVAk ,n ) 
1
(1  k ) n
k
]
**
(PVA k,n ) = PV factors from an annuity table (Tab. B.2)
* and **  No need for Tables → use financial calculator
If PMT = $600; k=.10; n=4
 PV0 = $600 (3.170) = $1902
PV of an uneven series
k = .12
Year
CF
1
100
2
150
3-8
325
n
(6c)
n
FVt

FVt (PVk,t )

t
t 1 (1  k)
t 1
PV0  
Step 1:
PV of 325 for t = 3 - 8  6 years  PVA12,6 = 4.111
 4.111 (325) = 1336.08 = value of annuity @ end of 2nd year
Step 2:
1336.08 must be discounted back to t = 0  PV12, 2 = .797
 .797 x 1336.08 = $1064.86 (A)
2
Chapter 3, Pinches
Step 3:
discount years 1 + 2 payments
year 1: 100 x .893 = 89.33 (B)
year 2: 150 x .797 = 119.55 (C)
Step 4:
A+B+C = 1273.71 = PV of this uneven series
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Perpetuities – an infinite annuity
PMT1
Value of a Perpetuity =
 (Perp.)0
k
PMT1 = $140
(Perp.)0 
140
 $2000
.07
k = .07
Determining Interest Rates: determination of discount rate when CF’s + T are known.
Individual CF – borrow $1000 today and agree to repay $2,012.07 in lump-sum in 5
years
1000
 PV0  1000  PVk,n FVn  PVk,n (2,012.07)  PVk,n 5 
 .497
2,012.07
PV Table  k = 15%
Annuities – borrow 2,124.9 today and repay @ end of each of next 3 years an amount
equal to 900
 PV0  PMT(PVAk,n )  n = 3
2124.90 = 900( PVA k,3 )  PVAk,3 
2124.90
 2.361
900
PVA Table  k = 13%
May be necessary to approximate k if PV factor does not appear on table.
Uneven Series - requires “guessing” to approximate k
E.g.: Invest 352.31 today and receive payments of 80 @ t = 1; 125 @ t = 2; 225 @ t =
3
“guess” k = 12% 
t
1
2
3

CF x PV12,n
80 x .893
125 x .797
225 x .712
= PV
= 71.44
= 99.62
= 160.20
 33126
.
Since 331.26 < 352.31  k = 12 is too high  try 9% 
Chapter 3, Pinches
t
1
2
3
PV9,n
CF
80
125
225
 k = 9% equates
 PV
.917
.842
.772
4
PV
= 73.36
= 105.26
= 173.70
  352.31
to value of investment  9% is interest rate.
Future Series - A promise to pay $10,000 in 7 years
If k = 8%, how much must be invested @ end of each period to meet such a promise?
FVn
10,000
 FVn  PMT(FVA8,7 )  PMT 

 $1120.70
FVA8,7
8.923
 each year this amount must be allocated to meet future promise.
Loan AMZ (optional) - borrow 48,040 for 3 years @ 12%. What is size of payments?
48,040
PV0  PMT(PVA12,3 )  48,040  PMT(2.402) PMT 
 20, 000
2.402
AMZ Schedule
t
PMT
1
2
3
20,000
20,000
20,004.75
Int. = I
5,764.8
4,056.58
2,143.37
Principal Repayment =
PR = PMT - I
14,235.20
15,943.42
17,861.38
Balance = (bal)-1 - PR
33,804.80
17,861.38
0
Interest in year 1 = .12 (amount borrowed) = .12 (48,040)
Interest in year 2 = .12 (33,804.80)
Interest in year 3 = .12 (17,861.38) *
Last payment = * + 17,861.38 = 20,004.75
= int. + nominal balance
Effective Interest Rates - actual return after adjusting of nominal rate to compounding
period.
k
 General formula FVn  PV0 (1 ) mn
(8a)
m
m = # of times per year interest is compounded
m = 1  annual
m = 2  semi-annual
m = 4  quarterly
m = 12  monthly
m = 365  daily
(8a)  m  FVn
Chapter 3, Pinches
(8a)  PV0 
FVn
mn
 k 
1

 m 

300
8  162.08
 .16 
1
 2 
m
 k 
Effective interest rate = ke  1 
 m   1
If FV4  300; k = .16; m = 2  PV0 
(8b)
(8b)  if m = 1  ke= k and as m‘s  ke ‘s
E.g.:
Situation A: n = 1; k = 12.5%; m = 4
4
 .125 
 ke = 1 
 1  .13098  13.098%

4 
Situation B: n = 1; k = 12.2%; m = 365
365
 .122 
 ke = 1 
 365   1  .12973  12.973%
 Situation B provides cheapest source for financing.
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