Estimating CAPM Inputs
Where do rf, Beta, and TC Come
From?
Estimating Betas
• Comparable firm or industry method
– Regress rates of return on common stock
against the rates of return on a “market”
index.
• Normally we try to reduce the noise in
betas by calculating industry betas.
– This is simply the average of all of the equity
betas for firms within the industry.
Estimation Details
• Daily, weekly, monthly or annual returns?
– In the ideal world of the CAPM, without transactions costs, it
does not matter.
– In the real world with bid-ask spreads it does.
• Investors buy stocks at the ask and sell them at the bid. The
difference is the bid-ask spread and allows the market maker to
earn a living.
– Even absent any movement in a stock’s value, its price may change
from day to day as the reported closing transaction bounces from the
bid to the ask.
– Bid-ask bounce makes it appear that movements in a stock are
unrelated to the market.
– Monthly returns minimize the impact of bid-ask bounce.
• If your data set is to short to use monthly returns go to weekly. If it
is too short for that . . . Best of luck.
More Estimation Details
– What time period?
• Try to use 60 periods of data. Yes, that would be
five years of monthly data.
– Are dividends and rights included in rates of
return?
• The holding period return should include ALL cash
flows. If you can spend it, you should include it.
– The initial cash outflow is the purchase price.
– The final cash inflow is the sales price.
The Final Estimation Details
• Which is the correct market index?
– The CAPM market index includes all risky assets.
• In theory this means you want the index of everything.
• In practice this index does not exist. Even if it did you cannot
own it as it includes items like real estate (your house), and
human capital (you).
– Wilshire 5000 comes close (^DWC).
– CRSP index is similar (posted on the class web page)
and includes all stocks on the NASDAQ, AMEX, and
NYSE.
One Final Beta Estimator
• Old joke: All betas equal one plus or
minus estimation error.
– There is a lot of truth to this crack.
– Out of sample firms with low beta estimates
tend to have unexpectedly high return
correlations with the market. The reverse is
true too.
– On average betas have to equal one. Why
not use this value absent convincing evidence
to the contrary?
The Risk Free Rate
No Easy Answers
• Technical answer.
– The mathematical models used to generate the
intertemporal CAPM includes a risk free asset that
pays an instantaneous interest rate rf.
– Based on this you want the interest rate on a short
dated treasury instrument.
– But, these are stationary models. Things like inflation
are not expected to change over time.
– Still there is evidence that this is your best estimate of
the long run risk free rate.
• Out of sample economic models do a very poor job of
predicting interest rates. Best model is typically that next
year’s rate will equal this year’s rate.
Risk Free Rate Continued
• Intuitive Answer
– Use a ten year treasury rate.
• Allows for changing returns over time.
– Problems
• Return from holding short dated government bonds over ten
years. Think of buying one year bonds every year for ten
years.
• Return from holding a single ten year bond.
• Single ten year bond has on average a higher return. This
means the long dated bond returns include a risk premium.
We want the risk free rate.
– Partial solution: subtract out the historical risk
premium of about 1%.
Risk Free Rate Continued
• Not intuitive and not easy answer.
– Calculate the forward rates from government bonds.
Use these forward rates as your risk free rate.
– Good
• Allows for time variation in interest rates.
– Bad
• Forward rates more than a year out are likely to include a risk
premium.
• Pain in the neck to do this, often with only a minor impact to
the final calculated value.
– One of Many Possible Compromises
• Use the five year bond’s return for your projected cash flows.
• Use the ten year bond’s return for your terminal value’s cash
flows.
Market Risk Premium
• Another difficult and controversial topic.
– The argument for 4%.
• Given today’s market, and the historical growth in GDP it is
hard to imagine a world in which stocks return more than 4%
forever.
– The argument for 7%.
• This is the historical value.
• The 4% argument could have been made many times in the
past. It has yet to describe the market.
– The argument for values in between.
• I hate fighting, how about we all compromise on . . .
Estimating rD and βD
• Just how accurate a number do you need?
– Not very? Use either:
• Last year’s interest payments divided by the
outstanding debt.
• The current yield to maturity its debt.
• Set βD to zero.
• What can go wrong?
– How risky is the debt?
• Not very? Then not much. Use one of the above
estimates.
Estimating rD
• No public debt? Call the bank!
– Alternatively, go to their financial statements
and use the interest paid divided by the
outstanding debt.
• With public debt.
– Start with the yield to maturity of the bonds.
– Adjust this for the chance the company will
default on the bonds.
• Use historical estimates to do the adjustment.
Risky Debt
• The firm’s debt is very risky?
– If there is a reasonable chance the firm will
default then the interest rate on the firm’s debt
will overestimate rD.
– Investors will demand a higher promised yield
to compensate them for those times when the
firm fails to pay in full.
– βD will have the same sign as βA.
How Risky is Risky?
Bond Ratings
• Major rating agencies:
– Moody’s
– S&P
– Fitch
• General categories
– Investment grade (a.k.a. high grade)
• BBB rating and higher.
– High yield (a.k.a. junk)
• Not investment grade, and not in default.
– Distressed
• In default
Historical Bond Returns and Yields
Averages for 1989-2003
ML 5-10 Yr
Treasury Yld
ML 5-10 Yr
ML 10-15 Yr
Treasury TR Treasury Yld
6.07
ML BB Hi-Yld
Yld
8.91
ML BB HiYld TR
9.38
ML 1-5 Yr C
Rated Hi-Yld
Yld
9.78
6.88
ML B Hi-Yld
Yld
11.77
ML 10-15 Yr
Treasury TR
9.58
ML B Hi-Yld
TR
9.71
LB AAA LB AAA
Corp Yld Corp TR
6.59
8.96
LB CCC
Hi-Yld
Yld
LB CCC
Hi-Yld
TR
18.53
10.12
ML 1-5 Yr C
Rated HiYld TR
21.93
14.19
Key: ML = Merrill Lynch, LB = Lehman Brothers, Yld = Yield to Maturity,
TR = Total Return
Items to Note
• Yield and total return are not the same thing!
– Question: Why is the total return on treasuries higher
than the yield?
– Question: Why is this relationship reversed for CCC
bonds?
• The lower the corporate bond class rating the
higher the yield.
• The lower the corporate bond class rating the
sometimes higher, sometimes lower the total
return.
• A table with the year by year data is on the class
web page.
Adjusting rD for Default Risk
• The return on the firm’s debt is equivalent
the total return in the table.
• Problem: You have yield to maturity data
not expected return data.
• One possible solution: Adjust the YTM
using the historical relationship between
YTM and rD based on the firm’s debt’s
rating class.
A Warning
• The data on the web page is from 19892003, which includes a very long
economic boom.
– During booms high yield debt should produce
a return higher than investors expected.
– During busts the opposite is true.
• Are the 1989-2003 averages reflective of
what investors expected?
Long Run Bond Returns
December 1926 – December 1989
Treasury Yld Treasury TR
(4-4.99 yrs) (4-4.99 yrs)
Avg.
Avg.
Treasury Yld
(5-9.99 yrs)
Treasury TR
(5-9.99 yrs)
6.49
6.61
6.61
6.35
AA Yld
AA TR
BAA Yld
BAA TR
7.59
5.86
8.46
6.82
Key: Yld = Yield to Maturity, TR = Total Return
Long Run Results
• Contrasts with data from 1989-2003
– For the long run data the AA and BAA bonds have
higher YTM’s than TR’s. As expected.
– For the 1989-2003 data the AA and BAA bonds have
lower YTM’s than TR’s. Probably due to the long
economic boom.
– Lower credit quality bonds have higher total returns.
• Another item to note: AA TR’s have been lower
than government TR’s! Very odd.
• The underlying data for the long run table is on
the class web page.
Estimating TC
• Use the average U.S. combined federal
and state rate of 40%.
• Use the value management provides.
• Matt’s never before seen method!
Average U.S. Combined Rate
• The average combined federal and state
tax rate is 40%.
– Good for primarily domestic firms that have
been profitable.
– Good long run value for firms you believe will
become profitable.
– Bad for firms with current or recent losses that
may have tax loss carry forwards.
– Potentially bad for firms with large foreign
operations or overseas “headquarters.”
Management’s Reported Value
• In the firm’s financial reports management
will often provide their firm’s tax rate for
the previous period.
– This is often calculated as:
1- Net Income After Taxes/Net Income Before Taxes.
Management Reports: Good and
Bad
• Often accurate since management has access
to the proprietary information provided to the IRS
for calculating their taxes.
• Good: Net Income numbers may provide an
accurate picture of the marginal tax rate.
• Bad: Net Income numbers are not cash flows
and may not reflect the firm’s marginal or even
average tax rate.
• May not reflect future tax rates for firm’s that
have been unprofitable but are now becoming
profitable.
Matt’s Never Seen Before Method
• Calculate the annual tax rate as the
Cash Taxes Paid/FCF Before Taxes.
• Use the average of the above from at least
five years of data.
Matt’s Method: Good and Bad
• Good: Uses only cash flow numbers.
• Good: Over the “long run” this is the fraction of
the firm’s free cash flow the government keeps.
– Theoretically the number you want if this is the
expected marginal tax rate going forward.
• Bad: Very volatile. Tax rates will appear to jump
dramatically from year to year.
– Averaging helps quite a bit but with only five to ten
data points perhaps not enough.
• Bad: May not reflect future tax rates for firm’s
that have been unprofitable but are now
becoming profitable.
Combining the Best of All Methods
• Goal: Obtain the firm’s tax rate in the years
ahead.
• Problem: Must use past data.
• Solution: Project based on trends.
– Recently unprofitable firms will have relatively low tax
rates for at least the next few years as they use up
their tax loss carry forwards.
• Once a firm uses up its tax loss carry forwards expect its tax
rate to approach 40% or its industry average.
– Recently profitable firms will likely have the same tax
rate going forward.