rates of return pdf
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Interest, Discount & Return Rates
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
Present Value: No Intermediate Cash Flow 3
N
N
k)(1PV FV
k)(1FV 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
Nk)(11+
Nk)(1+
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
Present Value w/ periodic compounding and no intermediate cash flow
Nm
mk1PVFV
⋅
⎟⎠⎞
⎜⎝⎛ +⋅=
Nm
mk1
FVPV ⋅
⎟⎠⎞
⎜⎝⎛ +
=
¨ 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
%116.5112%51k
12
=−⎟⎠⎞
⎜⎝⎛ +=
%417.1%)116.51(m%5 m
1
=−+=
( )5
521
%116.51FVPV
125%1
FVPV
+=
⎟⎠⎞
⎜⎝⎛ +
= ⋅
5
Using annual nominal rate Using annual effec-ve rate
ki is effec-ve annual rate ki is nominal annual rate
Present Value w/ periodic compounding and intermediate cash flow
6
∑= +
=N
1ii
i
i0 )k1(
CFV
i 0 1 2 m·∙N
PV
CFi
∑⋅
=⎟⎠⎞
⎜⎝⎛ +
=Nm
1ii
i
i0
mk1
CFVm: number of periods per year e.g., m=12 N: number of years m�N: total number of periods over N years
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
1i)(1n)(1r
i)(1r)(1n)(1
−≈
−+
+=
+⋅+=+
7
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
Interest Rates
(Simple annual rates) Yield Curve
5.000%
5.020%
5.040%
5.060%
5.080%
5.100%
5.120%
5.140%
0 5 10 15 20
Effective An
nual Rate
Annual Compounding Periods (m)
Con-nuous Compounding 10
?mk1iml
m gcompoundin continous For
mk1PVFV
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%
kkw
w
m
w
1w
w
kwm
ew1
1imlmk
1iml
,ew11iml
w11
mk1
)w ,m as 1,k :(Note
kwm and mk
w1 therefore
kmw Define
=⎟⎠⎞
⎜⎝⎛ +=⎟
⎠⎞
⎜⎝⎛ +
≡⎟⎠⎞
⎜⎝⎛ +
⎟⎠⎞
⎜⎝⎛ +=⎟
⎠⎞
⎜⎝⎛ +
∞→∞→<
⋅==≡
⋅
∞→∞→
∞→
⋅
Con-nuous Compounding 11
1ii
1i
ii
v
1i
i
v1ii
v
SlnSln SSlnv
eSS
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
Mean Rate: Simple Return Rates 12
SS
SSSr
1i1i
1iii
−−
− Δ=
−=
What’s the average or mean quarterly simple rate of return?
%6691.4
3.4483%5.4545%3.7736%6.0000%
41a
=
⎟⎟⎠
⎞⎜⎜⎝
⎛+
++=
t i Si ri0.00 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= 1m
rii=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)
Mean Rate: Simple Return Rates 13
03.120$)046691.01(100$ )a1(SS 4404 =+⋅=+⋅=
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
1SS 1)r1(g
m1
0
mm1
m
1ii −⎥
⎦
⎤⎢⎣
⎡=−⎥
⎦
⎤⎢⎣
⎡+= ∏
=
( ) 4.6635%11.0344831.0545451.0377361.060000g 41
=−⋅⋅⋅=
00.120$)046635.01(100$)g1(SS 4404 =+⋅=+⋅=
Periodic Rate
MeanPeriodic Mean Rate
Arithmetic aGeometric g
v Arithmetic u
r
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
Mean Rate: Log Return Rates 15
1ii
1i
ii
SlnSln
SSlnv
−
−
−=
=
u= 1m
vii=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%41u
=
+++=
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 vi0.00 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%
Mean Rate of Return 16
$120.00 e$100.00eS
$120.00 e$100.00eSS
.182322μ0
.045580*4u404
=
⋅=⋅=
=
⋅=⋅= ⋅
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
−
Another Example 17
( ) %0000.06.7659%-‐2.7652%-‐14.6603%5.1293%-‐41u =+=
( ) %3800.06.5421%-‐2.7273%-‐15.7895%5.0000%-‐41a =+=
( )
%0000.01100$100$
%0000.010.03460.97271579.10.9500g
41
41
=−⎟⎠
⎞⎜⎝
⎛=
=−⋅⋅⋅=
00.100$eSeSS 000.0*40
u404 =⋅=⋅= ⋅
00.100$)0000.01(100$)g1(SS 4404 =+⋅=+⋅=
53.101$)3800.01(100$)a1(SS 4404 =+⋅=+⋅=
t i Si ri vi
0.00 0 100.00$ 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
Stock Prices Over 100 days
si ##=si%1 ⋅ 1+a+εi( )
si ##=si%1 ⋅ 1+a( )si ##=si%1 ⋅ 1+g( )si ##=si%1 ⋅e
u
a is the mean of a random variable – the simple rate of return ε is a varia-on from the mean – an ‘error’ term
Sta-s-cs For Daily Simple Return Rates 19
Histogram For Daily Simple Return Rates 20
SN=S0
0100200300400500600700800900
1000110012001300140015001600
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
-‐22.5%
-‐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
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
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
vi
i
i
==+
==+
=Simple rate of return
Future value factor
Natural log rate of return
SPX Monthly Mean Rates: 1950 -‐ 2011 25
%65779.
r7391r
n1 a
739
1ii
n
1ii
=
== ∑∑==
%%56784.
1)]r(11)]r(1g7391
739
1ii
n1
n
1ii
=
−⎥⎦
⎤⎢⎣
⎡+=−⎥
⎦
⎤⎢⎣
⎡+= ∏∏
==
%56623.
v7391 )rln(1
n1 u
739
1ii
n
1ii
=
=+= ∑∑==
r 1+r ln(1+r) v ev
E[r]=a E[1+r] E[ln(1+r)] E[v]=u E[ev]
0.65779% 100.65779% 0.56623% 0.56623% 100.65779%
Arithmetic Mean
1+r
g
0.56784%
Geometric Mean
$-‐
$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 sa1s s
⋅=
+⋅=
+⋅=
−
−
−
SPX Monthly Variance Rates: 1950 -‐ 2011 27 27
( )[ ] [ ]
( )
( )
%1783918.
uv1397
1
uv1n
1s
svarvr1lnarv
739
1i
2i
n
1i
2i
2
2
=
−−
=
−−
=
==+
∑
∑
=
=
( )
( )
%1761733.
%6561736.r7391
ar1n
1
d]e[arv]r1[arv]r[arv
739
1i
2i
n
1i
2i
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[ev]=d
0.17835% 0.17835% 0.18077% 0.18077% 0.17835%Var[r]=d2 Var[1+r]=d2 Var[ln(1+r)]=s2 Var[v]=s2 Var[ev]=d2
0.0017835 0.0017835 0.0018077 0.0018077 0.0017835
Standard Deviation
Variance
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