Investments 7
Download
Report
Transcript Investments 7
Return and Risk
Returns – Nominal vs. Real
Holding Period Return
Multi-period Return
Return Distribution
Historical Record
Risk and Return
Real vs. Nominal Rate
Real vs. Nominal Rate – Exact Calculation:
1 R
R i
1 R (1 r ) (1 i ) r
1
1 i
1 i
R: nominal interest rate (in monetary terms)
r: real interest rate (in purchasing powers)
i: inflation rate
Approximation (low inflation):
r R i
Example
8% nominal rate, 5% inflation, real rate?
Investments 7
R i 8% 5%
r
2.86 %
1 i
1 5%
Approximation: r R i 8% 5% 3%
Exact:
2
Single Period Return
Holding Period Return:
Percentage gain during a period
P0
P1 D1 P0
HPR
P0
t=0
HPR: holding period return
P0: beginning price
P1: ending price
D1: cash dividend
P1+D1
t=1
Example
Investments 7
You bought a stock at $20. A year later, the stock price
appreciates to $24. You also receive a cash dividend of
$1 during the year. What’s the HPR?
P D1 P0 24 1 20
HPR 1
25%
P0
20
3
Multi-period Return: APR vs. EAR
APR – arithmetic average
EAR – geometric average
HPR
APR
T
EAR (1 HPR)1/ T 1
T: length of a holding period (in years)
HPR: holding period return
APR and EAR relationship
(1 EAR)T 1
APR
T
Investments 7
4
Multi-period Return - Examples
Example 1
25-year zero-coupon Treasury Bond
HPR 329.18%
329.18
APR
0.1317 13.17%
25
EAR (1 3.2918)1/ 25 1 0.06 6%
Example 2
What’s the APR and EAR if monthly return is 1%
APR N r 121% 12%
EAR (1 r ) N 1 (1 1%)12 1 12.68%
Investments 7
5
Return (Probability) Distribution
Moments of probability distribution
Mean: measure of central tendency
Variance or Standard Deviation (SD):
measure of dispersion – measures RISK
Median: measure of half population point
Return Distribution
Describe frequency of returns falling to
different levels
Investments 7
6
Risk and Return Measures
You decide to invest in IBM, what will be
your return over next year?
Scenario Analysis vs. Historical Record
Scenario Analysis:
Economy State (s) Prob: p(s) HPR: r(s)
Boom
1
0.25
44%
Normal
2
0.50
14%
Bust
3
0.25
-16%
Investments 7
7
Risk and Return Measures
Scenario Analysis and Probability Distribution
Expected Return
E[r ] p( s)r ( s)
s
[0.25 44% 0.5 14% 0.25 (16%)] 14%
Return Variance
Var[r ] 2 p(s)(r (s) E[r ])2
s
0.25 (.44 .14) 2 0.5 (.14 .14) 2 0.25 (.16 .14) 2 0.045
Standard Deviation (“Risk”)
SD[r] Var[r] 0.045 0.2121 21.21%
Investments 7
8
Risk and Return Measures
More Numerical Analysis
Using Excel
State (s) Prob: p(s) HPR: r(s)
1
0.10
-5%
2
0.20
5%
3
0.40
15%
4
0.20
25%
5
0.10
35%
p(s)*r(s) p(s)*(r(s)-E[r])^2
-0.005
0.004
0.01
0.002
0.06
0
0.05
0.002
0.035
0.004
E[r] =
15.00%
Var[r] =
0.012
SD[r] = 10.95%
Investments 7
9
Risk and Return Measures
Example
Current stock price $23.50.
Forecast by analysts:
optimistic analysts (7): $35 target and $4.4 dividend
neutral analysts (6): $27 target and $4 dividend
pessimistic analysts (7): $15 target and $4 dividend
Expected HPR? Standard Deviation?
Economy State (s) Prob: p(s) Target P Dividend HPR: r(s)
Optimist
1
0.35
35.00
4.40 67.66%
Neutral
2
0.30
27.00
4.00 31.91%
Pessimist
3
0.35
15.00
4.00 -19.15%
E[HPR] = 26.55%
Std Dev = 36.48%
Investments 7
10
Historical Record
Annual HPR of different securities
Risk premium = asset return – risk free return
Real return = nominal return – inflation
From historical record 1926-2005
Geometric Arithmetic Standard
Risk
Real
Asset Class
Mean
Mean
Deviation Premium Return
Small Stocks
12.01%
17.95% 38.71% 14.20% 14.82%
Large Stocks
10.17%
12.15% 20.26%
8.40% 9.02%
LT Gov Bond
5.38%
5.68%
8.09%
1.93% 2.55%
T-bills
3.70%
3.75%
3.15%
0.00% 0.62%
Inflation
2.99%
3.13%
4.29%
N/A
N/A
Risk Premium and Real Return are based on APR, i.e. arithmetic average
Investments 7
11
Risk and Horizon
S&P 500 Returns 1970 – 2005
Daily
Mean
0.0341%
Std. Dev.
1.0001%
Yearly
Mean
8.9526%
Std. Dev. 15.4574%
How do they compare* ?
Mean
Std. Dev.
0.0341*260 = 8.866%
1.0001*260 = 260.026%
SURPRISED???
* There is approximately 260 working days in a year
Investments 7
12
Consecutive Returns
It is accepted that stock returns are
independent across time
Consider 260 days of returns r1,…, r260
Means:
E(ryear) = E(r1) + … + E(r260)
Variances vs. Standard Deviations:
(ryear) (r1) + … + (r260)
Var(ryear) = Var(r1) + … + Var(r260)
Investments 7
13
Consecutive Returns Volatility
Daily volatility seems to be disproportionately
huge!
S&P 500 Calculations
Daily: Var(rday) = 1.0001^2 = 1.0002001
Yearly: Var(ryear) = 1.0002001*260 = 260.052
Yearly: (ryear ) 260.052 16.126%
Bottom line:
Short-term risks are big, but they “cancel out”
in the long run!
Investments 7
14
Accounting for Risk - Sharpe Ratio
Reward-to-Variability (Sharpe) Ratio
E[r] – rf
r – rf
- Risk Premium
- Excess Return
Sharpe ratio for a portfolio:
E[rp ] rf
Risk prem ium
or SR
SR
p
of excessreturn
Investments 7
15
Normality Assumption
The normality assumption for simple returns is
reasonable if the horizon is not too short (less than a
month) or too long (decades).
Investments 7
16
Other Measures of Risk - Value at Risk
Term coined at J.P. Morgan in late 1980s
Alternative risk measurement to variance, focusing on
the potential for large losses
• VaR statements are
typically made in $ and
pertain to a particular
investment horizon, e.g.
–“Under normal market
conditions, the most the
portfolio can lose over a
month is $2.5 million at the
95% confidence level”
Investments 7
17
Wrap-up
What is the holding period return?
What are the major ways of calculating
multi-period returns?
What are the important moments of a
probability distribution?
How do we measure risk and return?
Investments 7
18