Chapter 10: LESSONS FROM MARKET HISTORY

10.1 Returns Rt +1=

Dt +1 (P t+1 −P t )
+ =.05+.09=.14
10.1.1 Dollar Returns Pt Pt
- Holder có:
+ Dividend: income component of holder’s Ex10.1 Calculating Returns
return Pt = $25
+ Capital gain / capital loss (negative Dt+1: $2
capital gain) Pt+1 = $35
- Total dollar return = Dividend income +  Dividend yield?
 Capital gains yield? = % chênh lệch
Captital gain/loss
giá lúc bán so với giá lúc mua
- Total cash if stock is sold = Initial  Total return?
invstment + Total dollar return =
Proceeds from stock sale + Dividends
- If u hold your stock and don’t sell it at
end-year.
+ Should u still consider capital gain as
part of your return?  yes
+ Does this violate our previous PV rle
that only cash matters?  no
10.1.2 Percentage Returns
t: year Dt (P1−P 0) $ 2 ($ 35−25)
Pt: price of stock at the beginning of the R 1= + = + =.08+ .40=.48
P0 P0 $ 25 $ 25
year (giá cổ phần ở đầu năm)
10.2 Holding Period Returns
Dt+1: D paid on stock during the year (cổ
Roger Ibbotson, Rex Sinquefield: studies
tức chi trả trong năm)
present year-by-year historial rates of
Rt+1: total return on the investment (tổng
return on 5 important types of financial
TSSL)
instruments in US.
Returns: what you would have earned if
you had help portfolios of the following:
1. Large-company common stocks: is
based on Standard & Poor’s (S&P) 500
(contain 500 largest companies in US)
2. Small-company common stocks:
composed of stock corresponding to
smallest 20% of companies listed in NYSE,
again as measured by market value of
outstanding stock (20% cổ phiếu có giá trị
Pt = $37 nhỏ nhất trên NYSE)
Dt+1: $1.85 3. Long-term corporate bonds: a portfolio
 Dividend yield? = Dt+1/Pt = $1.85/$37 of high-quality corporate bonds with 20y
= .05 maturities
 Capital gain yield? = (Pt+1 − Pt ) /Pt = 4. Long-term US gov bonds: is based on
($40.33 − 37) / $37 = .09 gov bonds with maturities of 20y
 Total return? = 5. US Treasury bills: is based on Treasury
bills (T-bills for short) with 1m maturity
 If returns were 11%, -5%, 9% in a 3y Sharpe ratio:
period, an investment of $1 at the + = risk premium of asset / standard
beginning of the period would be worth: deviation
(1 + R1) × (1 + R2) × (1 + R3) = $1 × (1 + is a measure of return to level of risk
+ .11) × (1 − .05) × (1 + .09) = $1.15 taken
15%: a 3y holding period return + sometimes referred to as reward-to-
10.3 Return Statistics risk ratio (tỷ số phần thưởng cho rủi ro)
Có 2 number mô tả distribution of returns (reward: average excess return, risk:
là: standard deviation)
+ Arithmetic average return (10.3) Ex10.4 Sharpe Ration
+ Risk in return (10.5) - average risk premium = .087%
- Arithmetic average return - standard deviation = .198
- Frequency distribution  sharpe ratio? = .087/.198 = .439
- Average/mean 10.5.2 Normal Distribution and Its
Ex10.2 Calculating Average Returns Implications for Standard Deviation
T = 4: .1370, .3580, .4514, and −.0888 - Normal distribution:
.1370+.3580+.4514−.0888 + bell-shaped curve
R= =2.144∨21.44 %
4 + symmetric, not skewed
10.4 Average Stock Returns and - Classical statistics:
Risk-Free Returns + standard deviation: spread of a normal
- Excess return on the risky asset: distribution (standard deviation =
difference between risky returns and risk- variance2)
free returns + normal distribution: probability of
- Equity risk premium: average excess having return is above/below mean by a
return on common stocks, because it is the certain amount depends only on standard
additional return from bearing risk deviation
10.5 Risk Statistics
- Spread/dispersion of a distribtion:
measure how much return can deviate
from mean return
+ distribution is very spread out 
return occur are cery uncertain
+ distribution whose returns are all
within a few percentage points of each
other is tight  returns are less uncertain 10.6 More on Average Returns
Ex10.3 Volatility 10.6.1 Arithmetic versus Geometric
- Returns: 1370, .3580, .4514, and
Averages
−.0888
- Average return: .2144 Buy a stock for $100
 variance? First year: fall to $50
1 2 2 2 2
Second year: rise back to $100
Var= [ ( R 1−R ) + ( R 2−R ) + ( R3 −R ) + ( R4 −R ) ]  Average return = (-50+100)/2 = .25
T −1
1  which is correct, 0 or .25? both true
¿ [ ( .1370−.2144 ) + ( .3580−.2144 ) + ( .4514−.2144 ) + ( .0888−.2144 ) ]
2 2 2 2
3 + 0: geometric average return (TSSL TB
¿ .0582 nhân)  what was your average
SD=√ .0582=.2413 compound return per year over a
10.5.1 Variance particular period?
 + .25: arithmetic average return (TSSL Table 10.5 Annualized Equity Risk
TB cộng)  what was your return in an Premiums and Sharpe Ratios for 17
average year over a particular period? Countries, 1900–2010
10.6.2 Calculating Geometric Average
Returns

+ R2) × ⋅ ⋅ ⋅ × (1 + RT)]1/T – 1
Geometric average return = [(1 + R1) × (1

+ Step 1: take each of T annual returns

R1, R2..., RT; add 1 to each
+ Step 2: multiply all the numbers from
step 1
+ Step 3: raise the result from step 2 to
the power of 1/T
+ Step 4: subtract 1 from the result of
step 3 Table 10.3: Geometric versus Arithmetic
Anual return: .1, .12, .03, -.09 Average Returns: 1926-2017
 geometric average return? = (1.10 ×
1.12 × 1.03 × .91)1/4 − 1 = .0366

Ex10.5 Calculating the Geometric

Average Return
S&P 500 Product
Returns
13.75% 1.1375  small-companies: outperform
35.70 ×1.3570  US treasury bills: lowest real rate of
45.08 ×1.4508 return
-8.80 × .9120  risk premium on long-term corporate >
-25.13 × .7487 gov
Học thuộc thứ tự Series, tất cả các chỉ
=1.5291 số đều dương
- geometric < arithmetric, magnitude of
1.5291: what our investment is worth the difference varies quite a bit (diffirence
after 5y if we started with a $1 greater, more volatile investments)
investment. - geometric = arithmetic – variance/2
 geometric average return? = 1.52911/5 (ex: .1014 = .121 - .1982/2)
– 1 = .0887 Ex10.6 More Geometric Average
Table 10.4: Estimate of World Tradable Figure 10.4: large-company investment
Stock Market Capitalization, 2016 grew to $7,346.15 over 92y
 37% for the year, 485 stock were down for
the year
- 1926-2008: only return in 1931<2008
(-.43<-.37)
- 11/2007 (decline began) – 1/2009: S&P
500 lost 45% of its value
- S&P 500 Monthly Returns, 2008:
+ highly volatile at end year-more than
generally true historically
+ S&P had 126 up days, 126 down days
 geometric average return?  down days were much worse, on
= $ 7,346.151/92 − 1 = .102 average
 giống với figure 10.3 - China, India, Russia: decline of more than
10.6.3 Arithmetic Average Return or 50%
Geometric Average Return? - Iceland:
Geometric: what you actually earned per + drop by more than 90% for the year.
year on average, compounded annually? + Iceland exchange was temporarily
 actual historical investment experience suspended on 9/10.
Arthimetic: what you earned in a typical + Modern record for a single day: stocks
year? Is an unbiased estimate of true fell by 76% when trading resumed on
mean of distribution?  making estimates 14/10
of the future - Perform well in 2008: US Treasury bonds
10.7 The U.S. Equity Risk + long-term Treasury bond gained 20%,
Premium: Historical and short-term Treasury bonds were up to 13%
International Perspectives + higher quality long-term corporate
SD(R) bonds did less well, but: positive return of
SE = SD( R ) =
√The number of observations about 9%
+ SE: standard error (sai số chuẩn) + rate of inflation (measure by CPI) was
+ SD: standard deviation (độ lệch chuẩn) very close to 0
Confidence interval = historical average - 3/2009-2/2011: a period of about 700
return ± (2 × standard error) days, the S&P 500 doubled in value 
19.8 fastest climb since 1936 (S&P did it in only
Ex: 7.2 ± 2( ) = 7.2 ± 3.76
√111 500d)
estimate of the future equity risk  lesson from this recent, and very
premium will involve assumptions: turbulent, bit of capital market history:
- future risk environment 1. (most obviously) stocks have significant
- amount of risk aversion of future risk
investors 2. Depend on mix, a diversified portfolio of
10.8 2008: A Year of Financial stocks and bonds might have suffered in
Crisis 2008, but the losses much smaller than
- 2008S&P 500 index (track total market those experienced by an all-stock portfolio
value of 500 of largest US co.): decreased