15.401 Recitation
1: Present Value
Learning Objectives

Review of Concepts
o Compounding/discounting
o PV/FV
o Real
R l vs. nominal
i l rate
t
o Annuities and perpetuities

Examp
ples
o CD
o Auto loan
o Scholarship fund
o Project planning
2010 / Yichuan Liu
2
Review: Compounding / Discounting

We can…
o move money forward in time by compounding.
o move money backward in time by discounting.
x (1+r)m‐n
t=

x (1+r)s‐n
(1+r)m‐n∙Y
Y
(1+r)s‐n∙Y
m
n
s
Note:
o Only relative time matters
o Multiplying by (1+r)m‐n = dividing by (1+r)n‐m.
2010 / Yichuan Liu
3
Review: APR vs. EAR

Annual percentage rate (APR) vs. equivalent
annual return (EAR):
N
 APR 
EAR  1
 1
N 


(N = comp. freq.)
N t
Note:
o always use the EAR when compounding and discounting
o Due to interest comppounding,
g, the EAR is hig
gher than the
APR whenever the compounding frequency is higher than
once a year.
2010 / Yichuan Liu
4
Continuous Compounding (optional)

Given a fixed APR, higher compounding frequency
leads to higher EAR. Suppose we take compounding
frequency to infinity, then
N
 APR 
EAR   lim1
1  e APR 1.

n
N 

e  2.71828183...

The continuously compounded EAR is the highest
possible EAR for a given APR.
2010 / Yichuan Liu
5
Review: PV / FV

Cash flow:
C0
C1
C2
……
CT‐1
CT
0
1
2
……
T‐1
T
PV0

CT … …
T+1 … … periods
FVT
Present value (PV):
C1
C2
PV0  C 0 


1
2
1 r  1 r 
 Future value (FV) :
FVT  C 0 1 r   C1 1 r 
T 1
T

 CT 1 r   CT 1 1 r   
0
2010 / Yichuan Liu
1
1
6
Review: Nominal vs. Real Interest Rate
Nominal‐real
Nominal
real interest rate conversion:
1 rnominal
1 rreal 
1 i
 Nominal‐real cash flow conversion:
C nominal
C real 
1 i
 When you discount or compound,

o Either use the nominal cash flow and the nominal interest
rate
o Or use the real cash flow and the real interest rate
o Do not mix and match
2010 / Yichuan Liu
7
Review: Annuity/Perpetuity

Annuity:
0
C
C
……
C
1
2
……
T‐1
C
1 
PV0  1

r  1 r T 

C
T periods
Perpetuity:
0
C
C
C ……
1
2
3 ……
periods
C
PV0 
r
2010 / Yichuan Liu
8
Review: Growing Annuity/Perpetuity

Growing Annuity:
C
0
C(1+g) … … C(1+g)T‐2 C(1+g)T‐1
1
2
……
T‐1
T



C
1 g 
PV0 
.
1
T 
r  g  1 r  

T periods
Growing Perpetuity (r > g):
0
C
C(1+g)
1
2
C(1+g)2 … …
3
……
periods
C
PV0 
.
rg
2010 / Yichuan Liu
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Example 1: CD

You can invest $10,000 in a CD offered by your bank.
The CD matures in 5 years and the bank quotes you
a rate of 4.5%. How much will you have in 5 years, if
the 4.5% is
a)
b)
c)
2010 / Yichuan Liu
EAR
Quarterly APR
Monthly APR
10
Example 1: CD

Answer:
a) 10,000 1.0455  $12,461.82
b) r
EAR
4
045 
 0.045
 1
  1.04576
4 

10 000 1.04576
10,000
04576  $12,507.51
$12 507 51
5
12
c) r  1  0.045   1.04594
04594


EAR

12 
10,000
10,000
 1.04594  $12
12,,517
517..96
5
2010 / Yichuan Liu
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Example 2: Auto Loan

You would like to buy a new car for $22,000. The
dealer requires a down payment of $10,000 and
offers you 6% APR financing (compounded monthly)
for 5 years for the remaining balance. What is your
monthly payment?
2010 / Yichuan Liu
12
Example 2: Auto Loan

Answer: let C be the monthly payment, then


C
1
22000 
10000.
1
125 
00.06
06 /12  1 00.06
06 /12  
C  $231.99.
2010 / Yichuan Liu
13
Example 3: Scholarship Fund

You would like to establish a scholarship fund that
will help outstanding students with financial
difficulties pay their college tuition.
o Starting today, you hope to give 50 students $20,000 each
in today’s money (i.e., adjusted for inflation) every year.
o The effective nominal interest rate is 5%/yr.
o Inflation is 2%/yr.

How much money do you need now if you want the
f
fund
to last fforever?
2010 / Yichuan Liu
14
Example 3: Scholarship Fund

Answer:
o Method 1: nominal amount + nominal interest rate
1m 1.02
1m 
 35
35m
m.
1.05 1.02
o Method 2: real amount + real interest rate
rreal
1m
1.05

1  2.9412%  1m 
 35m.
0.029412
1.02
o Note: same answer!

You need $35 million today.
2010 / Yichuan Liu
15
Example 4: Project Planning


GeneriCorp is considering whether or not to expand
into a new market. The company faces the following
cash flow (in $million) if it decides to expand:
‐200
‐400
‐300
+100
+500
+600
t=0
1
2
3
4
5
years
A committee appointed by the CEO determined that
the appropriate discount rate is 9%
9%. Should the
company take on the expansion project?
2010 / Yichuan Liu
16
Example 4: Project Planning

Answer:
400 300
100
500
600




NPV  200 
2
3
4
1 09 11.09
1.09
09 11.09
09 11.09
09 1
1.09
095
 $1.91m.

Positive
P
iti NPV = take
t k th
the project;
j t th
though
h NPV is
i
dangerously close to zero.
2010 / Yichuan Liu
17
MIT OpenCourseWare
http://ocw.mit.edu
15.401 Finance Theory I
Fall 2008
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