This slide set is a work in progress and is embedded in my Principles of Finance course, which is also a work in progress, that I teach to computer scientists and engineers http://awesomefinance.weebly.com/

1.
Rates:
Interest,
Discount
&
Return

2. Learning
Objec-ves
¨ Present
and
future
value
¨
Discount
rates
¨ Rate
compounding
¨ Nominal
and
real
rates
¨ Interest
rates
¨ Mean
return
rates
¤ Arithme-c
¤ Geometric
¨ We’ll
skip
the
probability
distribu-ons
for
rates
of
return
2

3. Present
Value:
No
Intermediate
Cash
Flow
3
N
N
k)(1PV
FV
k)(1
FV
PV
+⋅=
+
=
0
1
2
N
PV
FV
FV:
Future
value
PV:
Present
value
k:
effec-ve
periodic
discount
or
future
value
rate
N:
number
of
periods
:
Discount
factor
:
Future
value
factor
N
k)(1
1
+
N
k)(1+

4. Present
Value
w/
No
Intermediate
Cash
Flow
¨ Example
¤ k
=
annual
effec-ve
discount
rate
=
5.116%
¤ N
=
5
years
¤ PV
=$100.00
¤ FV
=
PV·∙(1+.05116)5
=
$128.33
i=0
1
2
3
4
5
PV
FV
4

5. Present
Value
w/
periodic
compounding
and
no
intermediate
cash
flow
Nm
m
k
1PVFV
⋅
⎟
⎠
⎞
⎜
⎝
⎛
+⋅=
Nm
m
k
1
FV
PV ⋅
⎟
⎠
⎞
⎜
⎝
⎛
+
=
¨ Annual
effec+ve
rate
includes
effect
of
periodic
compounding
¨ Annual
nominal
rate
does
not
include
effect
of
periodic
compounding
¨ Example
¤ 5%
annual
compounded
monthly
n k
=
5%,
annual
nominal
rate
n m
=
12,
compounding
frequency
¤ Annual
effec-ve
rate
is
¤ N
is
number
of
years
¤ Effec-ve
and
nominal
monthly
rate
%116.51
12
%5
1k
12
=−⎟
⎠
⎞
⎜
⎝
⎛
+=
%417.1%)116.51(
m
%5 m
1
=−+=
( )5
521
%116.51
FV
PV
12
5%
1
FV
PV
+
=
⎟
⎠
⎞
⎜
⎝
⎛
+
= ⋅
5
Using
annual
nominal
rate
Using
annual
effec-ve
rate

6. ki
is
effec-ve
annual
rate
ki
is
nominal
annual
rate
Present
Value
w/
periodic
compounding
and
intermediate
cash
flow
6
∑= +
=
N
1i
i
i
i
0
)k1(
CF
V
i
0
1
2
m·∙N
PV
CFi
∑
⋅
=
⎟
⎠
⎞
⎜
⎝
⎛
+
=
Nm
1i
i
i
i
0
m
k
1
CF
V
m:
number
of
periods
per
year
e.g.,
m=12
N:
number
of
years
mŸN:
total
number
of
periods
over
N
years

7. Real
and
Nominal
Rates
¨ n
=
nominal
rate
¨ r
=
real
rate
¨ i
=
infla-on
rate
¨ Example
¤ n=3%
¤ i=2%
¤ r
=0.98%
≈1%
¨ Cash
flows
and
discount
rates
must
be
congruent
¤ Nominal
is
typical
inr
1
i)(1
n)(1
r
i)(1r)(1n)(1
−≈
−
+
+
=
+⋅+=+
7

8. Interest
Rates
¨ Rate
of
return
on
debt
securi-es
¤ Bonds
n Fixed
‘coupon’
rate
¤ Cer-ficates
of
deposit
¤ Notes
n Floa-ng
rate
¤ Mortgages
¤ Commercial
paper
8
Govt
Rates
BLS
CPI
BLS
CPI
Chart
BLS
FAQs
CD
Rates

9. Interest
Rates
(Simple
annual
rates)
Yield Curve

10. 5.000%
5.020%
5.040%
5.060%
5.080%
5.100%
5.120%
5.140%
0 5 10 15 20
Effective
Annual
Rate
Annual
Compounding
Periods
(m)
Con-nuous
Compounding
10
?
m
k
1iml
m
gcompoundin
continous
For
m
k
1PVFV
m
w
m
=⎟
⎠
⎞
⎜
⎝
⎛
+
∞→
⎟
⎠
⎞
⎜
⎝
⎛
+⋅=
∞→
k
is
annual
nominal
rate,
m
is
number
of
compounding
periods
per
year
5%
annual
nominal
rate
is
e.05
–
1
con-nuously
compounded
annual
effec-ve
rate:
5.1271%
k
kw
w
m
w
1
w
w
kwm
e
w
1
1iml
m
k
1iml
,e
w
1
1iml
w
1
1
m
k
1
)w
,m
as
1,k
:(Note
kwm
and
m
k
w
1
therefore
k
m
w
Define
=⎟
⎠
⎞
⎜
⎝
⎛
+=⎟
⎠
⎞
⎜
⎝
⎛
+
≡⎟
⎠
⎞
⎜
⎝
⎛
+
⎟
⎠
⎞
⎜
⎝
⎛
+=⎟
⎠
⎞
⎜
⎝
⎛
+
∞→∞→<
⋅==≡
⋅
∞→∞→
∞→
⋅

11. Con-nuous
Compounding
11
1ii
1i
i
i
v
1i
i
v
1ii
v
SlnSln
S
S
lnv
e
S
S
eSS
ePVFV
i
i
−
−
−
−
−=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
=
=
⋅=
⋅=FV
=
PV·ek
k
=
5%
k
is
nominal
rate
over
some
period
ek
is
the
future
value
factor
e.05
=
1.051271
e-­‐k
is
the
discount
factor
e-­‐.05
=
0.951229
ek-­‐1
is
the
con-nuously
compounded
rate
e.05-­‐1
=
0.051271
Si
are
sequen-al
stock
prices
Con-nuously
compounded
future
value
factor
Natural
log
rate
of
return

12. Mean
Rate:
Simple
Return
Rates
12
S
S
S
SS
r
1i1i
1ii
i
−−
− Δ
=
−
=
What’s
the
average
or
mean
quarterly
simple
rate
of
return?
%6691.4
3.4483%
5.4545%3.7736%6.0000%
4
1
a
=
⎟⎟
⎠
⎞
⎜⎜
⎝
⎛
+
++
=
t i Si ri
0.00 0 100.00$
0.25 1 106.00$
6.0000%
0.50 2 110.00$
3.7736%
0.75 3 116.00$
5.4545%
1.00 4 120.00$
3.4483%
Example:
Quarterly
historical
price
record
for
1
year
Compute
the
sequence
of
simple
rates
of
return
from
security
price,
S
a=
1
m
ri
i=1
m
∑ ''''
n
=
number
of
periods
in
a
historical
return
record,
associated
with
n+1
prices
m
=
number
of
periods
in
a
year
(in
this
example
m=n
as
a
special
case)

13. Mean
Rate:
Simple
Return
Rates
13
03.120$)046691.01(100$
)a1(SS 44
04 =+⋅=+⋅=
⌢
No,
it
over
es-mates
the
price
What’s
the
mean
rate
of
return
that
results
in
the
actual
price,
S4
?
Does
this
mean
rate
over
4
quarters
reproduce
the
stock
price
at
the
end
of
1
year
?
That’s
the
geometric
mean
rate
of
return,
g
1
S
S
1)r1(g
m
1
0
m
m
1
m
1i
i −⎥
⎦
⎤
⎢
⎣
⎡
=−⎥
⎦
⎤
⎢
⎣
⎡
+= ∏=
( ) 4.6635%11.0344831.0545451.0377361.060000g 4
1
=−⋅⋅⋅=
00.120$)046635.01(100$)g1(SS 44
04 =+⋅=+⋅=
⌢
Periodic
Rate
Mean
Periodic
Mean
Rate
Arithmetic a
Geometric g
v Arithmetic u
r

14. Mean
Rate:
Simple
Return
Rates
a
is
the
periodic
(e.g.,
quarterly)
arithme-c
mean
rate
of
return
g
is
the
periodic
(e.g.,
quarterly)
geometric
mean
rate
of
return
‘Periodic’
herein
means
daily,
weekly,
monthly,
quarterly,
but
not
annual
So
how
do
we
-me-­‐scale
these
periodic
mean
return
rates?
For
example:
Scale
the
quarterly
mean
rates
to
an
annual
mean
return
Via
mul-plica-on
?
Via
compounding
NO
( ) ( )
026%
20.
1-­‐
4.6691%1
1-­‐a1
18.6541%
4.6635%
·∙
4
g
·∙
m
18.6764%
4.6691%
·∙
4
a
·∙
m
4m
=
+=+
==
==
( ) ( ) %000.021-­‐
4.6635%1
1-­‐g1 4m
=+=+
But
compounding
the
geometric
mean
rate
does
produce
the
annual
rate
–
by
defini-on
-­‐
but
ignores
the
intermediate
rate
fluctua-ons
but
compounding
is
s-ll
an
annoying
mathema-cal
opera-on
Sn> S0 1+a+e( )
m

15. Mean
Rate:
Log
Return
Rates
15
1ii
1i
i
i
SlnSln
S
S
lnv
−
−
−=
=
u=
1
m
vi
i=1
m
∑
The
periodic
arithme-c
mean
natural
log
return
rate
is
Now
the
natural
log
rate
of
return
( )
%5580.4
3.3902%5.3110%3.7041%5.8269%
4
1
u
=
+++=
18.2322%4.5580%4u4μ =⋅=⋅=
Mul-ply
the
quarterly
natural
log
mean
return
rate
by
4
to
get
the
annual
log
mean
return
rate?
t i Si ri vi
0.00 0 100.00$
0.25 1 106.00$
6.0000% 5.8269%
0.50 2 110.00$
3.7736% 3.7041%
0.75 3 116.00$
5.4545% 5.3110%
1.00 4 120.00$
3.4483% 3.3902%
Average
4.6691% 4.5580%

16. Mean
Rate
of
Return
16
$120.00
e$100.00eS
$120.00
e$100.00eSS
.182322μ
0
.045580*4u4
04
=
⋅=⋅=
=
⋅=⋅= ⋅
⌢
Now
check
whether
the
natural
log
mean
return
rate
reproduces
the
year
end
stock
price
Annual
and
other
accumulated
rates
of
return
can
be
determined
by
mul-plying
the
log
mean
periodic
rate
of
return
factor
discount
annual
e
factor
value
future
annual
e
returnof
rate
annualμ
μ
μ
−

17. Another
Example
17
( ) %0000.06.7659%-­‐2.7652%-­‐14.6603%5.1293%-­‐
4
1
u =+=
( ) %3800.06.5421%-­‐2.7273%-­‐15.7895%5.0000%-­‐
4
1
a =+=
( )
%0000.01
100$
100$
%0000.010.03460.97271579.10.9500g
4
1
4
1
=−⎟
⎠
⎞
⎜
⎝
⎛
=
=−⋅⋅⋅=
00.100$eSeSS 000.0*4
0
u4
04 =⋅=⋅= ⋅
⌢
00.100$)0000.01(100$)g1(SS 44
04 =+⋅=+⋅=
⌢
53.101$)3800.01(100$)a1(SS 44
04 =+⋅=+⋅=
⌢
t i Si ri vi
0.00 0 100.00$
0.25 1 95.00$
-­‐5.0000% -­‐5.1293%
0.50 2 110.00$
15.7895% 14.6603%
0.75 3 107.00$
-­‐2.7273% -­‐2.7652%
1.00 4 100.00$
-­‐6.5421% -­‐6.7659%
Average
0.3800% 0.0000%

18. 18
Stock
Prices
Over
100
days
si
##=si%1
⋅ 1+a+εi( )
si
##=si%1
⋅ 1+a( )
si
##=si%1
⋅ 1+g( )
si
##=si%1
⋅eu
a
is
the
mean
of
a
random
variable
–
the
simple
rate
of
return
ε
is
a
varia-on
from
the
mean
–
an
‘error’
term

19. Sta-s-cs
For
Daily
Simple
Return
Rates
19

20. Histogram
For
Daily
Simple
Return
Rates
20

21. 0
100
200
300
400
500
600
700
800
900
1000
1100
1200
1300
1400
1500
1600
Jan-­‐50 Jun-­‐55 Dec-­‐60 Jun-­‐66 Nov-­‐71 May-­‐77 Nov-­‐82 May-­‐88 Oct-­‐93 Apr-­‐99 Oct-­‐04 Mar-­‐10
SPX
(^GSPX)
Daily
Prices:
1950
-­‐
2012
21
15,722
Daily
Prices
January
1950
to
September
2012

22. -­‐22.5%
-­‐20.0%
-­‐17.5%
-­‐15.0%
-­‐12.5%
-­‐10.0%
-­‐7.5%
-­‐5.0%
-­‐2.5%
0.0%
2.5%
5.0%
7.5%
10.0%
12.5%
Jan-­‐50 Jun-­‐55 Dec-­‐60 Jun-­‐66 Nov-­‐71 May-­‐77 Nov-­‐82 May-­‐88 Oct-­‐93 Apr-­‐99 Oct-­‐04 Mar-­‐10
SPX
Daily
Simple
Return
Rates:
1950
-­‐
2012
22
15,721
simple
daily
return
rates
January
1950
to
September
2012

23. SPX
Monthly
Ln
Return
Rates:
1950
-­‐
2012
23
-­‐40% -­‐35% -­‐30% -­‐25% -­‐20% -­‐15% -­‐10% -­‐5% 0% 5% 10% 15% 20%
Monthly
Natural
Log
Return
Rates

24. End
Date Adj
Close
S r 1+r ln(1+r) v ev
8/1/11 1,119.46$
-­‐13.373% 86.627% -­‐14.356% -­‐14.356% 86.627%
7/1/11 1,292.28$
-­‐2.147% 97.853% -­‐2.171% -­‐2.171% 97.853%
6/1/11 1,320.64$
-­‐1.826% 98.174% -­‐1.843% -­‐1.843% 98.174%
5/2/11 1,345.20$
-­‐1.350% 98.650% -­‐1.359% -­‐1.359% 98.650%
4/1/11 1,363.61$
2.850% 102.850% 2.810% 2.810% 102.850%
3/1/11 1,325.83$
-­‐0.105% 99.895% -­‐0.105% -­‐0.105% 99.895%
2/1/11 1,327.22$
3.196% 103.196% 3.146% 3.146% 103.196%
1/3/11 1,286.12$
2.2646% 102.2646% 2.2393% 2.2393% 102.2646%
12/1/10 1,257.64$
6.530% 106.530% 6.326% 6.326% 106.5300%
11/1/10 1,180.55$
-­‐0.229% 99.771% -­‐0.229% -­‐0.229% 99.7710%
10/1/10 1,183.26$
3.686% 103.686% 3.619% 3.619% 103.6856%
9/1/10 1,141.20$
8.755% 108.755% 8.393% 8.393% 108.7551%
SPX
Monthly
Ln
Return
Rates:
1950
-­‐
2011
24
( )
( ) %2393.2vr1ln
%2646.102er1
%2646.2r
ii
v
i
i
i
==+
==+
=Simple
rate
of
return
Future
value
factor
Natural
log
rate
of
return

25. SPX
Monthly
Mean
Rates:
1950
-­‐
2011
25
%65779.
r
739
1
r
n
1
a
739
1i
i
n
1i
i
=
== ∑∑ ==
%%56784.
1)]r(11)]r(1g
739
1
739
1i
i
n
1
n
1i
i
=
−⎥
⎦
⎤
⎢
⎣
⎡
+=−⎥
⎦
⎤
⎢
⎣
⎡
+= ∏∏ ==
%56623.
v
739
1
)rln(1
n
1
u
739
1i
i
n
1i
i
=
=+= ∑∑ ==
r 1+r ln(1+r) v
e
v
E[r]=a E[1+r] E[ln(1+r)] E[v]=u E[e
v
]
0.65779% 100.65779% 0.56623% 0.56623% 100.65779%
Arithmetic
Mean
1+r
g
0.56784%
Geometric
Mean

26. $-­‐
$250
$500
$750
$1,000
$1,250
$1,500
$1,750
$2,000
12/18/4910/22/56 8/27/63 7/1/70 5/5/77 3/9/84 1/12/91 11/16/97 9/20/04 7/26/11
Actual
Arithmetic
Mean
Geometric
Mean
Natural
Log
Mean
SPX
Monthly
Prices:
1950
-­‐
2011
26
( )
( )
u
1ii
1ii
1ii
es
s
g1s
s
a1s
s
⋅=
+⋅=
+⋅=
−
−
−

27. SPX
Monthly
Variance
Rates:
1950
-­‐
2011
27
27
( )[ ] [ ]
( )
( )
%1783918.
uv
1397
1
uv
1n
1
s
svarvr1lnarv
739
1i
2
i
n
1i
2
i
2
2
=
−
−
=
−
−
=
==+
∑
∑
=
=
( )
( )
%1761733.
%6561736.r
739
1
ar
1n
1
d]e[arv]r1[arv]r[arv
739
1i
2
i
n
1i
2
i
2v
=
−=
−
−
=
==+=
∑
∑
=
=
r 1+r ln(1+r) v
ev
SD[r]=d SD[1+r]=d SD[ln(1+r)]=s SD[v]=s SD[e
v
]=d
0.17835% 0.17835% 0.18077% 0.18077% 0.17835%
Var[r]=d
2
Var[1+r]=d
2
Var[ln(1+r)]=s
2
Var[v]=s
2
Var[e
v
]=d
2
0.0017835
0.0017835
0.0018077
0.0018077
0.0017835
Standard
Deviation
Variance