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Title: estimating risk free rate
Description: What is the riskfree rate? A Search for the Basic Building Block

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1

What is the riskfree rate? A Search for the Basic Building Block
Aswath Damodaran
Stern School of Business, New York University
adamodar@stern
...
edu
www
...
com

December 2008

2

What is the riskfree rate? A Search for the Basic Building Block
In corporate finance and valuation, we start off with the presumption that the riskfree rate
is given and easy to obtain and focus the bulk of our attention on estimating the risk
parameters of individuals firms and risk premiums
...
In this paper, we not only provide a framework for deciding whether to use
short or long term rates in analysis but also a roadmap for what to do when there is no
government bond rate available or when there is default risk in the government bond
...


3
Most risk and return models in finance start off with an asset that is defined as
risk free, and use the expected return on that asset as the risk free rate
...

But what makes an asset risk free? And how do we estimate a riskfree rate? We
will consider these questions in this paper
...
We will also look at cases where estimating a riskfree rate
becomes difficult to do and the mechanisms that we can use to meet the challenges
...


What is a risk free asset?
To understand what makes an asset risk free, let us go back to how risk is
measured in investments
...
The actual returns that they make over
this holding period may by very different from the expected returns, and this is where the
risk comes in
...
For an investment to be risk free in this environment, then, the actual
returns should always be equal to the expected return
...
At the
end of the 1-year holding period, the actual return that this investor would have on this
investment will always be 5%, which is equal to the expected return
...


4
Figure 1: Probability Distribution for Riskfree Investment
Probability = 1

The actual return is
always equal to the
expected return
...

There is a second way in which we can think of a riskfree investment and it is in
the context of how the investment behaves, relative to other investments
...

Note that if we accept the first definition of a riskfree asset as an investment with a
guaranteed return, this property always follows
...


Why do riskfree rates matter?
The riskfree rate is the building block for estimating both the cost of equity and
capital
...
The cost of
debt is estimated by adding a default spread to the riskfree rate, with the magnitude of the
spread depending upon the credit risk in the company
...


5
The level of the riskfree rate matters for other reasons as well
...
Since growth assets deliver cash flows further
into the future, the value of growth assets will decrease more than the value of assets in
place, as riskfree rates rise
...


Existing Assets

Since growth assets deliver
cash flows way into the future,
the effect of the riskfree rate is
much greater
...
Comparing across firms, the
values of growth companeis will
decrease, relative to mature companies
...

Changes in the riskfree rate also have consequences for other valuation inputs
...
In particular, a significant increase in the
riskfree rate will generally result in higher risk premiums, thus increasing the effect on
discount rates
...
Finally, the factors
that cause the shift in riskfree rates – expected inflation and real economic growth – can
also affect the expected cash flows for a firm
...
We will also look at how riskfree rates in nominal
terms can be different for real riskfree rates, and why riskfree rates can vary across
currencies
...



The first is that there can be no default risk
...
The only securities that have a chance of being risk free
are government securities, not because governments are better run than
corporations, but because they control the printing of currency
...
Even this assumption,
straightforward though it might seem, does not always hold up, especially when
governments refuse to honor claims made by previous regimes and when they
borrow in currencies other than their own
...
For an investment to have an actual return equal to its expected return,
there can be no reinvestment risk
...
A six-month treasury bill rate, while default free, will not be risk
free, because there is the reinvestment risk of not knowing what the treasury bill
rate will be in six months
...

The risk free rate for a five-year time horizon has to be the expected return on a
default-free (government) five-year zero coupon bond
...

The Purist Solution
If we accept both requirements – no default risk and no reinvestment risk –as prerequisites for an investment to be riskfree, the risk free rates will be vary with time
horizon
...

In fact, a conventional five-year bond will not yield a riskfree return over 5 years,
even if it is issued by a default free entity, because the coupons every 6 months will have
to be reinvested at uncertain rates
...
Thus, the riskfree rates for each period will be measured by
using the rate on a zero-coupon default-free bond maturing in that period
...
Even if zero coupon bonds are not traded, we can estimate zero coupon rates
for each period by using the rates on coupon bearing bonds
...
We then
progressively can move up the maturity ladder, solving for the zero coupon rates for each
subsequent period
...
5%, 2-year coupon bond = 990
Setting up the one-year coupon bond, we can solve for the one-year rate:
Price of bond = 1000 =

(Principal + Coupon) (1000 + 20)
=
(1+ 1- year zero rate)
(1+ r1 )

Since the bond trades at par, the one-year zero rate = coupon rate on the bond =2%
...
02) (1 + r2 ) 2
Solving for the two-year rate, we get r2=3
...
We can then use the 1-year and 2-year
rates, in conjunction with the 3-year bond to get the three-year rate and so on
...
S
...
50%
1
...
00%
2
...
00
99
...
00
97
...
50%
1
...
04%
2
...
5000%
2
...
7172%
2
...
50%
2
...
00%
3
...
50%
3
...
00
99
...
00
97
...
00
98
...
55%
2
...
06%
3
...
54%
3
...
9543%
2
...
3789%
3
...
7174%
4
...
5% for year 1, 2
...

From a pragmatic standpoint, refining riskfree rates to make them year-specific
may not be worth the effort in mature markets for two reasons
...
The second is that the rest of the parameters that we use in analysis now have
to be defined relative to these riskfree rates; the equity risk premium that we use for the
cost of equity in year 1 has to be defined relative to a one-year riskfree rate rather than
the more conventional computation, which uses ten-year rates
...
For instance, assume that the one-year rate is 2%
and that the ten-year rate is 4% and that the equity risk premium, relative to the ten-year
rate, is 4
...
The cost of equity for an average risk
investment will then be 8% for the one-year cash flow (2%+6%) and 8
...
5%)
...
In market crises, for instance, it is not uncommon to
see big differences (in either direction) between short term and long-term rates
...

1

We use historical norms to define “well behaved”
...


9
A Practical Compromise
If we decide not to estimate year-specific riskfree rates, we have to come up with
one riskfree rate to use on all of the cash flows
...
Put simply, banks that faced interest rate risk in their
assets (generally loans made to corporate and individual borrowers) face two choices
...
The other is to match up the average duration of the assets to the
average duration of the liabilities, resulting in less complete risk hedging, but with far
less cost
...
In capital budgeting, where we may be called upon to analyze short
term as well as wrong term investments, the riskfree rate can vary, depending upon the
duration of the investment being analyzed
...
S&P used the dividend discount model to estimate the
duration of equity in the S&P 500 to be about 16 years in 2004
...
Since the duration of a 10-year coupon bond (with a
coupon rate of about 4%), priced at par, is close to 8 years4, this would lead to use the 10year treasury bond rate as the riskfree rate on all cash flows for most mature firms
...
In valuing these firms, an

2

In investment analysis, where we look at projects, these durations are usually between 3 and 10 years
...
The duration
in these cases is often well in excess of ten years, and increase with the expected growth potential of the
firm
...

4 The duration of a 10-year, 4% coupon bond, trading at par, is 8
...


10
argument can be made that we should be using a 30-year treasury bond rate as the
riskfree rate
...
In
exceptional circumstances, where year-specific rates vary widely across time, we should
consider using riskfree rates that vary across time
...
On October 20, 2008, for instance, the market
interest rate on a ten-year US treasury bond rate was 3
...
On the same date,
the market interest rate on a ten-year Japanese government bond, denominated in yen,
was 1
...
Using the same
logic, Figure 3 lists the two-year and ten-year government bond rates in various
currencies, at least for governments that are rated AAA, and are thus unlikely to default
...

In the US market, which is the only one with a long history of both bonds, the difference between the two
rates has been less than 0
...

6

11

One currency that is missing from this list is the Euro, where at least eleven different
governments, that are part of the European Union, issue 10-year bonds, all denominated
in Euros, but with differences in interest rates
...
However, the market clearly sees more default risk in the
Greek and Portuguese government bonds than it does in the German and French issues
...
81%
would then have been the riskfree rate
...
53%, in Japanese Yen, to 5
...
This gives rise to two follow-up
questions:
1
...
High inflation currencies will have higher riskfree rates than low
inflation currencies
...


13
greater inflation in British pounds than it is in US dollars, and greater inflation in
US dollars than it is in Japanese yen
...
Which riskfree rate should we use in capital budgeting and valuation? If higher
riskfree rates lead to higher discount rates, and holding all else constant, reduce
present value, using a yen riskfree rate seemingly should give a company a higher
value than using a US dollar riskfree rate
...
If we
decide to value a company in Japanese yen, because of the allure of the lower
riskfree rate and lower discount rates, the cash flows will also have to be in
Japanese yen
...
Consequently, whatever we
gain by using a lower yen-based discount rates will be exactly offset by the loss of
having to use yen-based cashflows
...
Thus, if cash flows are
estimated in nominal US dollar terms, the risk free rate will be the US Treasury bond
rate
...
While this may seem illogical, given the higher risk in these
countries, the riskfree rate is not the vehicle for conveying concerns about this risk
...
Thus, Nestle can be valued using cash flows estimated in Swiss Francs,
discounted back at an expected return estimated using a Swiss long term government
bond rate as the riskfree rate, or it can be valued in British pounds, with both the cash
flows and the risk free rate being British pound rates
...
In particular, projects and assets will be valued more
highly when the currency used is the one with low interest rates relative to inflation
...


14
Real versus Nominal Risk free Rates
Under conditions of high and unstable inflation, valuation is often done in real
terms
...
To be consistent, the
discount rates used in these cases have to be real discount rates
...
While government bonds may
offer returns that are risk free in nominal terms, they are not risk free in real terms, since
expected inflation can be volatile
...

Until recently, there were few traded default-free securities that could be used to
estimate real risk free rates, but the introduction of inflation-indexed treasuries has filled
this void
...
Thus, an
inflation-indexed Treasury bond that offers a 3% real return, will yield approximately 7%
in nominal terms if inflation is 4%, and only 5% in nominal terms, if inflation is only 2%
...


15

Note that the difference between the nominal and the real treasury rate can be viewed as a
market expectation of inflation
...
29%
...
The only problem is that real valuations are seldom called for or done in
markets like the United States, which have stable and low expected inflation
...
The real risk free rates in these markets
can be estimated by using one of two arguments:


The first argument is that as long as capital can flow freely to those economies with
the highest real returns, there can be no differences in real risk free rates across
markets
...

The more precise number is obtained as follows:
Expected inflation rate = (1 + Nominal Treasury Rate) −1
(1 + TIPs rate)

In September 2008, for instance, when the nominal rate was 3
...
85%, the
approximate solution would have yielded 1
...
81%
...



The second argument applies if there are frictions and constraints in capital flowing
across markets
...
Thus, the real risk free rate for a mature economy like
Germany should be much lower than the real risk free rate for an economy with
greater growth potential, such as Hungary
...


Issues in estimating riskfree rates
In the last section, we assumed that government bonds were for the most part
default free and that the government bond rate was therefore a riskfree rate
...
In this section, we will consider the tougher cases, where
governments either have no long-term bonds outstanding in the local currency, or even if
they do, are exposed to default risk
...

There are no long term traded government bonds
In the last section, we used the current market interest rate on government bonds
issued by the United States, Japan and the UK as the risk free rates in the respective
currencies
...


17
The Scenario
In many countries (and their associated currencies), the biggest roadblock to
finding a riskfree rate is that the government does not issue long term bonds in the local
currency
...
Quite a
few governments issue long term bonds, but in the currencies of more mature markets,
rather than their own currencies
...
In 2008, only 3
of the 12 South American countries had long-term bonds denominated in local
currencies
...

Common (and dangerous) practices
When there are no long term government bonds in the local currency that are
widely traded, analysts valuing companies in that market often take the path of least
resistance when estimating both cash flows and discount rates, resulting in currency
mismatches in their valuations
...
With cash flows, analysts either stick with
local currency cash flows or convert those cash flows at the current exchange rate into a
mature market currency; with Latin American companies again, the cash flows in the
local currency are converted into US dollars using current exchange rates
...
Consider, for instance, a
Mexican company where the cash flows are estimated in pesos and the discount rate is
estimated in the US dollars
...
Note that converting the peso cash flows into dollar cash flows, using currency
exchange rates, does nothing to alleviate this problem
...
Finally,
assume that the inflation rate in US dollars is 2% and the inflation rate in BR is 6%
...
55 million
(3,579
...

Year
1
2
3
Terminal value
Value of firm =

Cash flow in
BR
100
110
121

Exchange rate
0
...
50
0
...
00
$55
...
50
$2,137
...
87
$46
...
72
$1,650
...
55

Note that the terminal value is computed at the end of year 3:
Terminal value = 60
...
06)/ (
...
06) = $ 2,137
...
In addition, the
terminal value has been computed using a growth rate in nominal BR and a discount rate
in US dollars
...


19
Possible solutions
If the government does not issue long-term local currency bonds (or at least ones that
you can trust to deliver a market interest rate), we have two solutions that preserve
consistency
...
The other is to try to estimate a local currency discount rate, troubles with the
riskfree rate notwithstanding
...
If getting a riskfree rate in Brazilian Reais is too
difficult to do, a Brazilian company can be valued entirely in US dollars or Euros
...
As we noted in the last
section, the right riskfree rate to use will be the US treasury bond rate (and not the tenyear $ denominated Brazilian bond rate, which has an embedded default spread in it)
...
This conversion has to be made using the expected US dollar/
Reai exchange rate and not the current exchange rate
...


Illustration 2: Valuing in Mature Market Currency
Let us revisit the valuation in illustration 2
...
0784

Exchange
Rate ($/BR)
0
...
11

Present
Value
$44
...
1599
2
...
462976148
0
...
93
$53
...
49

$42
...
63
$606
...
17

The higher inflation rate in BR leads to a depreciation in the currency’s value over time
...
91
million in year 3 and an expected growth rate of 2% (reflecting the inflation rate in US
dollars and not in BR):
Terminal value = $53
...
02)/ (,09-
...
49 million
The value that we derive for the firm today is $735
...
35 million BR) and
it reflects more consistent assumptions about inflation in the cash flows and discount
rates and is much lower than the value of $1,789
...

Local Currency valuation
The valuation can be done in the local currency, with the discount rate converted
into a local currency discount rate; the expected cash flows in this case will remain in the
local currency
...
In the first two, we try to
estimate a local currency risk free rate, with estimates of inflation, and in the third, we
convert a foreign currency discount rate, using expected inflation rates
...
To estimate expected inflation, we can start
with the current inflation rate and extrapolate from that to expected inflation in the
future
...
In 2005, for
instance, adding the expected inflation rate of 8%, in India, to the interest rate of
2
...
12% in Indian rupees
...
For instance, the forward
rate between the Thai Baht and the US dollar can be written as follows;
Forward Ratet Baht, $ = Spot RateBaht,$

(1+ Interest Rate Thai Baht ) t
(1 + Interest Rate US dollar ) t

For example, if the current spot rate is 38
...
36 Baht per dollar, and the current ten-year US treasury bond rate

is 5%, the ten-year Thai risk free rate (in nominal Baht) can be estimated as follows:
61
...
10

(1+ Interest Rate Thai Baht )10
(1
...
12%
...

o The discount rate conversion: Since it far easier to estimate the other inputs to the
discount rate computation, such as the equity risk premium and default spreads, in a
mature market currency, the third option is to compute the entire discount rate in the
mature market currency and to convert that discount rate (r) into the local currency at
the last step
...
The Indonesian rupiah cost of
capital can be written as follows:

9

In cases where only a one-year forward rate exists, an approximation for the long term rate can be
obtained by first backing out the one-year local currency borrowing rate, taking the spread over the oneyear treasury bill rate, and then adding this spread on to the long term treasury bond rate
...
95 on the Thai bond, we obtain a one-year Thai baht riskless rate of 9
...
Bill rate of 4%)
...
04% to the ten-year treasury bond rate of 5%
provides a ten-year Thai Baht rate of 10
...


22

Cost of capitalRupiah = (1
...
11)
−1 = 0
...
06%
(1
...
To make this conversion, we still have to

estimate the expected inflation in the local currency and the mature market currency
...

Illustration 3: Valuing in the local currency
In illustration 2, we corrected the inflation mismatch in illustration 1 by doing the
entire valuation in US dollars
...
06)
−1 =
...
27%
(1
...
27% to estimate the value today
...
09) *

Year
1
2
3
Terminal value
Value of firm



Cash flow in BR
100
110
121
1763
...
28111477
85
...
25087292
1213
...
345243

The terminal value is estimated using the nominal growth of 6% in BR and the BR cost
of capital:
Terminal Value = 121 (1
...
1327-
...
1428 million BR
Note that the value of the firm is 1,470
...
Identical to the valuation that we
obtained when we valued the company in US dollars in illustration 2
...
That assumption, reasonable thought it may

23
seem, can be challenged in some countries where investors build in the likelihood that of
default risk into government bonds
...
There are many emerging market
economies where this assumption might not be viewed as reasonable
...
The ratings
agencies capture this potential by providing two sovereign ratings for most countries, one
for foreign currency borrowing and the other for local currency borrowing
...
Table 2 lists local currency and foreign currency ratings for selected emerging
markets (and appendix 1 has the complete listing):
Table 2: Local and Foreign currency ratings for selected markets– October 2008
Country
Brazil
China
India
Russia

Local Currency Rating
Ba1
A1
Baa3
Baa2

Foreign Currency Rating
Ba1
A1
Baa2
Baa2

To the extent that we accept Moody’s assessment of country risk, the long term, local
currency bonds issued by each of these governments will have default risk embedded in
them, with the risk being greater in the Brazilian government bond than it is in the
Chinese government bond
...
To illustrate, the interest rate on long term, rupee denominated bonds
issued by the Indian government in October 2008, which was 10
...

As table 2 shows, India’s local currency rating of Baa3 suggests that there is default risk
in the Indian rupee bond, and that some of the observed interest rate can be attributed to

24
this risk
...
Analysts who use 10
...

The Solution
Since the problem in this case is that the local currency bond rate includes a
default spread, the solution is a fairly simple one
...

Using the Indian rupee bond again as the illustration, we used the local currency rating
for India as the measure of default risk to arrive at a default spread of 2
...
Subtracting
this from the market interest rate yields a riskfree rupee rate of 8
...

Riskfree rate in Indian rupees = Market interest rate on rupee bond – Default SpreadIndia
= 10
...
60% = 8
...
One problem
that we had in estimating the numbers for this table is that relatively few emerging
markets have dollar or Euro denominated bonds outstanding
...
To mitigate this problem, we used spreads from the CDS
market, referenced in the earlier section
...
10 An alternative approach to estimating default spread
is to assume that sovereign ratings are comparable to corporate ratings, i
...
, a Ba1 rated
country bond and a Ba1 rated corporate bond have equal default risk
...
Table 3 also
10

For instance, Turkey, Indonesia and Vietnam all share a Ba3 rating, and the CDS spreads as of
September 2008 were 2
...
15% and 3
...
The average spread across the three countries
is 3
...


25
summarizes the typical default spreads for corporate bonds in different ratings classes in
September 2008
...
15%
0
...
30%
0
...
60%
1
...
80%
1
...
00%
1
...
30%
1
...
40%
1
...
70%
1
...
00%
2
...
25%
2
...
50%
3
...
00%
3
...
25%
4
...
50%
4
...
25%
5
...
00%
6
...
00%
7
...
75%
9
...
50%
11
...

The riskfree rate may change over time
The default-free long-term interest rate in a currency is the riskfree rate that we
use to estimate the costs of equity and capital
...
While this is always true, there may be times when
the current riskfree rate may seem abnormally high or low, relative to history or
fundamentals, and the change over time seems more likely to be in one direction than in
the other
...
The first is that they are volatile and change over time,
more so in some periods than others
...
Figure 6 illustrates both findings by looking at the treasury
bond rate from 1928 to 2007:

While there have been long periods of interest rate stability, they are interspersed with
periods of interest rate volatility
...
In addition, note that
interest rates seem to revert back towards a range of 5-7% over time; this would
correspond to a normal range of rates for the US
...
For instance, the normal interest rate is much higher if we look at
only the last 30 years (average rate = 7
...
70%) or the last 80 years (average rate = 5
...


27
There is less historical data on long-term interest rates outside the United States
but we can safely argue that the volatility in interest rates has been far higher in emerging
markets, especially in Latin America
...
In addition, it is far more difficult to set a normal
range for rates in these markets, where interest rates have been in triple digits in some
periods and single digits in others
...
If
the Treasury bond rate is 3
...
Though this may seem logical, there are three potential problems
...
To provide a simple contrast,
analysts who started working in the late 1980s in the United States, use higher normal
rates than analysts who joined in 2002 or 2003, reflecting their different experiences
...
For instance, using a 5% riskfree rate, when valuing a company,
will lower the value that you attach to the company and perhaps make it over valued
...
e
...
Finally, interest rates generally
change over time because of changes in the underlying fundamentals
...
For example, assume that
the riskfree rate is low currently, because inflation has been unusually low and the
economy is moribund
...
Analysts
who use normal interest rates will then have to also use higher inflation and/or real
growth numbers when valuing companies
...
While
the level of riskfree rates is usually not an explicit factor when comparing PE ratios or

28
EV/EBITDA multiples across companies, changes in riskfree rates can affect companies
differently
...
A careless analyst will tend to find growth companies to be
undervalued in high interest rate scenarios and mature companies to be bargains in low
interest rate scenarios
...
You expect the operating income next year, after taxes, to be $3 billion
...

b
...

c
...

You believe that the treasury bond rate is too low and that it will revert back to its
normalized level, which you estimate to be 5%
...

If you value Dow Chemical, using the cost of capital of 9%, your estimate of
value for the firm is as follows:
After-tax Operating income next year = $3 billion
Reinvestment Rate = Expected growth rate/ Return on capital = 3/15 = 20%
Expected FCFF next year = EBIT (1-t) (1-Reinvestment Rate)
= 3000 (1-
...
09-
...
However, one reason for your lower value

29
was your use of the normalized riskfree rate of 5%, instead of the actual rate of 4%
...
08-
...
In effect, your initial
conclusion that about Dow Chemical being over valued reflected both your assumptions
about the company and your views on interest rates, with the latter being the main reason
for your final conclusion
...

Solutions
As a general rule, it is not a good idea to bring in our idiosyncratic views on
interest rates, no matter how well thought on and reasoned they may be, into individual
company valuations
...
We can still draw on market expectations of
interest rates in valuing companies
...
5%
...
However,
we can use futures or forward markets on treasury bonds to get a sense of what the
market sees as the expected interest rate ten years from now, and use that as the riskfree
rate in the future (perhaps in computing terminal value)
...
In effect, we let the users of our research make
a judgment on what aspect of the research they trust more
...
If, on the other hand, they feel more confident in our
company analyses than in our interest rate views, they will focus on the corporate
valuation and recommendation
...
The steps are captured in figure 7:

30
Figure 7: A Framework for estimating Riskfree Rates
Will you be doing your valuation in real or nominal terms?
Nominal

Real
Riskfree rate = Long term rate
on inflation-indexed bonds
issued by default-free entity
(TIPs)

Pick a currency to do your valuation in

Switch currencies or
do analysis in real
terms

Is there a long term government bond
denominated in that currency?

No
No

Yes
Is the government default free?

No

Does the country have a local currency
rating?

Yes
Riskfree rate = Long term
government bond rate

Yes
Riskfree rate = Long term
government bond rate - Default
spread

Summarizing the key points that we have made over the paper, we would list the
following as the key rules to follow when it comes to riskfree rates
...
A rate that has risk spreads embedded
in it for default or other factors, is not a riskfree rate
...

Rule 2: Choose a riskfree rate that is consistent with how cash flows are defined
...
If the cash flows are in a
specific currency, the riskfree rate has to be defined in that currency
...
When valuing a Russian company in Euros, the riskfree rate should be the Euro
riskfree rate (the German 10-year bond rate)
...
In other words, even if you believe that riskfree rates will rise or
fall over time, it is dangerous to reflect those views in your valuation
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Conclusion
The risk free rate is the starting point for all expected return models
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The first is that there can be no
risk of default associated with its cash flows
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Using these criteria, the appropriate risk free rate to
use to obtain expected returns should be a default-free (government) zero coupon rate
that is matched up to when the cash flow or flows that are being discounted occur
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In corporate finance and valuation, this
will lead us towards long-term government bond rates as risk free rates
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The first is when there are no longterm, traded government bonds in a specific currency
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The second is when the long-term government bond rate has potential
default risk embedded in it, in which case we argued that the riskfree rate in that currency
has to be net of the default spread
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Without passing judgments on the
efficacy of this view, we noted that it is better to separate our views about interest rates
from our assessment of companies
Title: estimating risk free rate
Description: What is the riskfree rate? A Search for the Basic Building Block