Download Chap. 5 Risk and Return Return Basics Compound Returns

Return Basics
Chap. 5 Risk and Return
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Return Basics
Compound Returns
Geometric Mean Return, Variations
Risk Premiums
Historical Returns (ex post)
Inflation and “Real” Rates
• Sept. 2003
by William Pugh
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Some returns only include income:
For Bonds it is called the current yield
For stocks it is called the dividend yield
However, unless otherwise specified,return is a
total return - which includes price changes and
is sometimes called holding-period return HPR
• r = (Income + P1- P0)/ P0 = HPR
• 1+ r = (Income + P1)/ P0 is called a return
relative or wealth index. It is very useful in
calculations involving compound interest.
Compound Returns
Compound Returns
• Effective return reflects compound interest.
• To state a bond’s annual effective yield we
usually have to compound over two six-month
periods: A 10% bond is really a bond that pays
5% every six months.
• reff = (1 + .05)2 - 1 = .1025 = 10.25%
• Stocks usually pay dividends quarterly, bond
funds pay interest monthly.
• Money market funds compound daily.
• The yield is stated as a seven-day average yield.
This “ravg” is simply the arithmetic average of
the simple daily return over the last seven days.
• reff would be found by daily compounding.
• Suppose the stated yield in the WSJ for Fidelity’s
Cash Reserves is 5.13%.
• reff = (1 + .0513/365)365 - 1 = 0.0526 = 5.26 %
• The effective yield today (about 1%) is very
close to the simple yield.
Compound Returns
Geometric Mean Return
• WN would be a wealth index after “n” time
periods (usually years). It is found by
compounding annual returns.
• WN = (1+r1) (1+r2) ... (1+rN) = FVN /P0
• Wealth Indexes are sometimes used to compare
historical returns between different investments.
We would say something like “ if you had
invested $1.00 in the stock market in 1926, it
would be worth $1,678 today.” The 1678 would
be a wealth index.
• Usually, however, we are interested in
comparing investment using annual returns ,
saying “if you had invested in the stock market in
1926, you have earned about 10.5% a year”.
• So we would wish compute an average rate of
return that, if compounded, would give us the
wealth index value. Since WN is the result of
compounding N times, we can “decompound”
the index by taking the Nth root of the index to
find the average gain in wealth (1+ravg.)
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Geometric Mean Return
Geometric Mean Return
• This average rate of compounding is the
Geometric mean return, and is signified by “G”.
• (1+G) = (WN)1/N = [(1+r1) (1+r2) ... (1+rt)] 1/N
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= (FVN /P0) 1/N
• My personal preference is this version:
• G = (FVN /P0) 1/N - 1
• Example: gold was worth $20 an ounce at the
start of 1926, what is the compound rate of return
up to the present? Use the YX key on your
calculator. Y = Gold today/$20 and X = 1/77.
• The usefulness of geometric mean return:
• Suppose you get an IPO at issue price ($10) and
it doubles the first year. The second year it goes
up to $30 and the third year it gets cut in half.
Determine the return for each of the three years
[r1 r2 and r3].
• Now how well you did per year on average?
• One method is to compute the arithmetic mean
return: simply the standard average (called x-bar)
• Another is to compute G = (P3 /P0) 1/3- 1
Geometric Mean Return
Variations
• Which makes most sense? Here it is perhaps not
obvious. However, we say that you
compounded at rate G. The x-bar will almost
always be higher than G (and thus some
managers like to state their results using x-bar).
But G is the preferred measure. To better see
why, consider the fourth year:
• Your IPO goes “belly up”. Compute both G and
x-bar.
• Only G makes any sense.
• Dollar-Weighted Return: only really used adjust
portfolio manager’s records by survivorship and
small size bias. Returns that were earned when
the fund was just starting out and small in size
are de-emphasized.
• D-W return is found using IRR.
• APRs (like on your car financing contracts) are
simple interest rates and must be compounded to
get the effective annual rate.
Characteristics of “Normal” Probability
Distributions
1) Mean: average value or return.
2) Standard deviation: a measure of risk
3) Skewness: Some evidence that stock returns are
positively (rightward) skewed. Skewed
distributions are not normal.
4) Kurtosis: some evidence that stock returns do not
fit the ‘shape’ of a normal curve. Instead we see
‘flat’ bell curves with tails that do not quickly
converge to zero.
• In the text we assume normal returns – in real
life this a dangerous assumption.
Normal Distribution
s.d.
s.d.
r
Symmetric distribution
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Skewed Distribution: Large
Positive Returns Possible
Median
Negative
r
Positive
Annual Returns from Table 5.3 of Text
Start of 1926- End of 2001
Geometric
Security
Mean%
Small Stock
12.19
Large Stock
10.51
Treasury Bonds 5.23
Treasury Notes 5.12
T-Bills
3.80
Inflation
3.06
Arithmetic
Mean%
18.29
12.49
5.53
5.30
3.85
3.15
Standard
Dev.%
39.28
20.30
8.18
6.33
3.25
4.40
Risk Premiums
Historical Returns
• Since all investors are risk averse (they don’t
like risk, but will take on risk if they think they
will get a higher return), the additional return is
usually called a risk premium.
• Risk is usually estimated from historical data based the the security’s or asset class’s standard
deviationor other such measure of volatility like range. (see Table 5.3)
• The three month T-Bill is usually thought of as
the risk-free asset.
• Will history repeat itself? Famous caveat: “Past
performance is no guarantee of future results.”
• We observe the validity of the position that there
is a long-term risk-return tradeoff. There is a
strong correlation between standard deviation
and geometric mean.
• Note: arithmetic mean (x-bar) is mostly used to
compute variation and thus standard deviation.
Inflation and “Real” Rates
Inflation and “Real” Rates
• We often make a distinction between the normal
everyday contract rate of interest, and the
interest rate that is adjusted for any loss of
purchasing power caused by inflation.
• The everyday contract rate is called the nominal
interest rate.
• The rate reduced to reflect any loss of purchasing
power is the real rate.
• Irving Fisher popularized the use of the real rate
and thus the relationship was named after him.
• A simple version of this “Fisher Effect” is
r = R - i, where r is the real rate, R is the
nominal rate, and i is expected inflation.
• Under current conditions this would be: the Tbill YTM is 1.2%, inflation is expected to be 2%,
therefore the expected real rate on the T-bill is
roughly -0.8% (awfully low).
• Note: the real rate is usually the residual of a
given nominal rate and inflation rate.
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Inflation and “Real” Rates
Inflation and “Real” Rates
• The Junk Bond rate is more like 11%, so the real
Junk Bond rate is expected to be 9%.
• Note that each nominal rate has its own
corresponding real rate.
• There is, however, a compounding effect that is
reflected in the full, more correct formula:
• (1+r) = (1 + R)/(1 + i)
• or (1+R) = (1 + r)(1 + i)
• The worked out problem on page 147 uses the
more complex formula.
• Example: Suppose we expect Russia to have
100% inflation next year and we want to increase
our purchasing power by 5% (real return). What
does a Russian bank need to offer you?
Inflation and “Real” Rates
Inflation and “Real” Rates
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Not just 105%
(1+R) = (1 + r)(1 + i)
R = (1 + r)(1 + i) -1
R = (1 + .05)(1 + 1.00) - 1 = 2.10 - 1 = 1.10
= 110%
Note you need 100% to buy what you bought last
year, 5% to buy 5% more goods, and another
5% to pay for the inflation on the extra goods.
• The real rate is useful to investors who want
know if they are keeping ahead of inflation.
Taxes should play a part that analysis, however.
• The Fisher Effect says that the real rate is
somewhat steady over time: at least steadier
than the nominal rate. We find that most of
the changes in interest rates can be explained
by changes in inflation. See fig. 5.4
• Usually the Real RFR is about 1%. Lower
values encourage people to borrow, leading to
inflation. High RRFRs should lower inflation.
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