Download 4. Return And Risk

Chapter 4 Return and Risk
The objectives of this chapter are to enable you to:
• Understand and calculate returns as a measure of
economic efficiency
• Understand the relationships between present
value and IRR and YTM
• Understand how obtain an expected security return
from potential returns and associated probabilities
• Define and measure risk
• Understand and measure co-movement
4.A. INTRODUCTION
• The purpose of measuring investment returns
is simply to determine the economic efficiency
of an investment.
• An investment's return will express the profits
generated by an initial cash outlay relative to
the amount of that outlay.
4.B. RETURN ON INVESTMENT:
ARITHMETIC MEAN
Fund Performance
Date
June 30
July 31
Aug. 31
Sep. 30
Oct. 31
t
0
1
2
3
4
Pt
50
55
50
54
47
Pt-1 DIVt
0
50
0
55
0
50
0
54
2
rt
.100
-.091
.080
-.092
NOTES
First Month
(55/50) - 1 = .10
(50/55) - 1 = -.091
(54/50) - 1 = .08
ex-$2 dividend; [(47+2)/54)]-1= -.092
4.C. RETURN MEASUREMENT:
GEOMETRIC MEAN
Fund Performance
Date
June 30
July 31
Aug. 31
Sep. 30
Oct. 31
Nov. 30
Fund A
t Pt Pt-1 DIVt
0 50 - 0
1 55 50 0
2 50 55 0
3 54 50 0
4 47 54 2
5 51 47 0
rt
.100
-.091
.080
-.092
.081
Pt
50
80
40
60
30
45
Fund B
Pt-1 DIVt
0
50
0
80
0
40
0
60
0
30
0
rt
.60
-.50
.50
-.50
.50
Fund Returns
• Compare the prices and returns of two funds.
• Recall that the arithmetic mean return on
investment for Fund A is .0156. The arithmetic
mean return on investment on Fund B is .12.
4.D. INTERNAL RATE OF RETURN
Consider a stock whose purchase price three years ago was
$100. This stock paid a dividend of $10 in each of the three years
and was sold for $130. Its arithmetic mean annual return is 20%
10
10
10

= + 22.54

2
3
1.1 (1.1)
(1.1)
10
140
10

= 0.86

NPV  100 
2
3
1.18 (1.18)
(1.18)
NPV  100 
IRR Strengths and Weaknesses
• The primary advantage of the internal rate of return over return on
investment is that it accounts for the timeliness of all cash flows.
However, IRR does have three major weaknesses:
1.
As we have seen, IRR takes longer to calculate than does ROI.
2.
Sometimes an investment will generate multiple rates of
return; that is, more than one (r) value will equate NPV with zero.
This will occur when that investment has associated with it more
than one negative cash flow. When multiple rates are generated,
there is often no method to determine which is the true IRR.
3.
The internal rate of return is based on the assumption that
cash flows received prior to the expiration of the investment will be
re-invested at the internal rate of return.
4.E. BOND YIELDS
n
n
CFt
INT
F
NPV  0  
  P0  

t
t
(1  y) n
t 0 (1  y )
t 1 (1  y )
n
2n
CFt
INT / 2
F
NPV  0  


P



0
t
t
n
(
1

y
)
(
1

y
/
2
)
(
1

y
)
t 0
t 1
The yield to maturity (y) of the 4-year $1000 corporate bond
above is 6% if it were purchased for $1,000; the yield to
maturity would be 12.7% if the purchase price were $800.
4.F. INTRODUCTION TO RISK
•What is risk?
4.G. EXPECTED RETURN
Consider an economy with 3 potential states of nature in the next year
If the economy performs well, state 1 is realized and the stock earns 25%.
If the economy performs only satisfactorily, state 2 is realized and the stock
earns 10%.
If the economy performs poorly, state 3 is realized and the stock earns -10%.
Assume that there is a 20% for state 1 will occur, a 50% chance for state 2 and
a 30% for state 3. The expected return on the stock will be 7%:
E[RA]= (.25 ×.20) + (.10 × .50) + (-.10 × .30) = .07
4.H: VARIANCE AND STANDARD
DEVIATION
The variance of stock A returns presented in Section 4.G is .0156:
σ2 = (.25-.07)2 ×.2 + (.10-.07)2 ×.5 + (-.10-.07)2 ×.3 = .0156
n
σ=
Σ (Ri-E[R])2Pi
i=1
The standard deviation of returns on our stock is 12.49%.
4.I: HISTORICAL VARIANCE AND STANDARD
DEVIATION
• Historical stock return variances (standard deviations) can be
reasonable indicators of future variances (standard
deviations).
Table 4.4: Historical Variance and Standard Deviation of
Returns of Stock D
_
_
_
t Rt Rt - RD (Rt - RD)2
(Rt - RD)2 1/n
1 .10 -.06
.0036
.00072
2 .15 -.01
.0001
.00002
3 .20
.04
.0016
.00032
4 .10 -.06
.0036
.00072
5 .25
.09
.0081
.00162
Rd=.16
σ2 =.00340
σd =.05831
4.J. COVARIANCE
cov(A,B) = {(.25 - .07) × (.45 - .07) × .20}
+ {(.10 - .07) × (.05 - .07) × .50} + {(-.10 -.07) × (-.15 -.07) × .30}
= {.01368} + {-.0003} + {.01122} = .0246.
Table 4.5: Covariance between Returns on Stocks A and B
i Rai Rbi
1 .25 .45
2 .10 .05
3 -.10 -.15
Pi Rai-E[Ra] Rbi-E[Rb] (Rai-E[Ra])(Rbi-E[Rb])Pi
.20 .18
.38
.01368
.50 .03
-.02
-.00030
.30 -.17
-.22
.01122
COV(A,B) = .0246
Historical Covariance
Covariance measures co-movement tendency.
Table 4.6: Historical Covariance between Stocks D and E
_
_
_
_
t Rdt Ret (Rdt-Rd) (Ret-Re) (Rdt-Rd)(Ret-Re)1/n
1 .10 .15 -.06
-.05
.00060
2 .15 .18 -.01
-.02
.00004
3 .20 .25 .04
.05
.00040
4 .10 .20 -.06
0
0
5 .25 .22 .09
.02
.00036
_
_
RD=.16 .20=RE
COV(D,E) = .00140
4.K: COEFFICIENT OF CORRELATION
Correlation coefficient is a measure of co-movement
standardized to fall between -1 and 1.
COV(k,j) = σkσjρk,j
4.L: THE MARKET PORTFOLIO
• Dow Jones Industrial Average
• Standard and Poor's 500 (S&P 500)
• Shanghai SE Composite Index