# Session 2: The Risk Free Rate

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Session 2: The Risk Free Rate
Aswath Damodaran

The risk free rate is the starting point
The risk free rate is the starting point.. For both cost of equity and cost of debt To get to a cost of equity from any risk and return model, you begin with a riskfree rate. In the CAPM: Cost of equity = Riskfree Rate + Beta * Equity Risk Premium In the APM: Cost of equity = Riskfree Rate + j=1j (Risk premium for factor j) In build up models: Cost of equity = Riskfree Rate + Build up factors If the cost of debt is the rate at which you can borrow money today, it has to build off a riskfree rate: Cost of debt = Risk free rate + Default spread Lays out the four basic models and how non-diversifiable risk is measured in each model: The capital asset pricing model makes the most restrictive assumptions (no transactions costs, no private information) and arrives at the simplest model to estimate and use. The arbitrage pricing model and multi-factor model make less restrictive assumptions but yield more complicated models (with more inputs to estimate) The proxy model is dependent upon history and the view that firms that have earned higher returns over long periods must be riskier than firms that have lower returns. The characteristics of the firms that earn high returns - small market cap and low price to book value, for example in the Fama-French study - stand in as measures for risk.

What is risk free? On a riskfree asset, the actual return is equal to the expected return. Therefore, there is no variance around the expected return. For an investment to be riskfree, then, it has to have No default risk No reinvestment risk Time horizon matters: Thus, the riskfree rates in valuation will depend upon when the cash flow is expected to occur and will vary across time. Currency matters: The riskfree rate can vary across currencies. Not all government securities are riskfree: Some governments face default risk and the rates on bonds issued by them will not be riskfree. Treasury bills may be default free but there is reinvestment risk when they are used as riskless rates for longer-term cashflows. A 6-month T.Bill is not riskless when looking at a 5-year cashflow. Would a 5-year treasury be riskfree? Not really. The coupons would still expose you to reinvestment risk. Only a 5-year zero-coupon treasury would be riskfree for a 5-year cash flow. If you were a purist, you would need different riskfree rates for different cashflows. A pragmatic solution would be to estimate the duration of the cashflows in a valuation and use a treasury of similar duration. (Since the duration is the weighted average of when the cashflows come in, this should be fairly long, especially when you count in the fact that the terminal value is the present value of cashflows forever).

Test 1: A riskfree rate in US dollars!
In valuation, we estimate cash flows forever (or at least for very long time periods). The right risk free rate to use in valuing a company in US dollars would be A three-month Treasury bill rate A ten-year Treasury bond rate A thirty-year Treasury bond rate A TIPs (inflation-indexed treasury) rate None of the above Since we are valuing cashflows for the long term, we want a long-term riskfree rate (so, rule out the 6-month T.Bill rate). Since the valuation is in US dollars (a nominal rate), we can also rule out the TIPS rate. In theory, the 30-year rate should be the best choice. In practice, I would go with the 20-year bond for the following reasons: It is the most liquid of the treasuries and there is always a fresh auction rate from the previous Monday, Getting other inputs such as equity risk premiums and default spreads is easier with a 10-year rate than a 30-year rate The term structure flattens out by the time you get to the 10-year rate…..

Test 2: A Riskfree Rate in Euros
The only reason for differences in rates across these securities is default risk. Consequently, it makes sense to use the lowest of the rates (Germany) as the rsikfree rete, if you are valuing a company in Euros.

Test 3: A Riskfree Rate in Brazilian Reais
The Brazilian government had 10-year Nominal R\$ bonds outstanding, with a yield to maturity of about 11% on January 1, 2012. In January 2012, the Brazilian government had a local currency sovereign rating of Baa2. The typical default spread (over a default free rate) for Baa2 rated country bonds in early 2012 was 1.75%. The riskfree rate in Nominal R\$ is The yield to maturity on the 10-year bond (11%) The yield to maturity on the 10-year bond + Default spread (12.75%) The yield to maturity on the 10-year bond – Default spread (9.25%) None of the above The government bond rate is not riskfree. To get to a riskfree rate, you would subtract out the default spread of 2.55% from the bond rate to get to a riskfree rate of 8%.

Sovereign Default Spread: Three paths to the same destination…

Sovereign Default Spreads: End of 2011
Rating Default spread in basis points Aaa Aa1 25 Aa2 50 Aa3 70 A1 85 A2 100 A3 115 Baa1 150 Baa2 175 Baa3 200 Ba1 240 Ba2 275 Ba3 325 B1 400 B2 500 B3 600 Caa1 700 Caa2 850 Caa3 1000

Test 4: A Real Riskfree Rate
In some cases, you may want a riskfree rate in real terms (in real terms) rather than nominal terms. To get a real riskfree rate, you would like a security with no default risk and a guaranteed real return. Treasury indexed securities offer this combination. In January 2012, the yield on a 10-year indexed treasury bond was 1.00%. Which of the following statements would you subscribe to? This (1.00%) is the real riskfree rate to use, if you are valuing US companies in real terms. This (1.00%) is the real riskfree rate to use, anywhere in the world Explain. If there were no barriers to the flow of capital, the real riskfree rate should be the same around the world ). To the extent that there are barriers to capital flowing across markets, it is possible that the real riskfree rate in countries like India and China may be higher than than this value. One possible solutions: Use the long term real growth rate of the economy as your riskfree rate.

Test 5: Matching up riskfree rates
You are valuing Embraer, a Brazilian company, in U.S. dollars and are attempting to estimate a riskfree rate to use in the analysis (in August 2004). The riskfree rate that you should use is The interest rate on a Brazilian Reais denominated long term bond issued by the Brazilian Government (11%) The interest rate on a US \$ denominated long term bond issued by the Brazilian Government (6%) The interest rate on a dollar denominated bond issued by Embraer (9.25%) The interest rate on a US treasury bond (3.75%) None of the above The correct riskfree rate is the treasury bond rate. All of the other bonds either are non-currency matched or have default risk.

Why do riskfree rates vary across currencies
Why do riskfree rates vary across currencies? January 2012 Risk free rates Riskfree rates should vary primarily because of differences in expected inflation. Whatever you gain by using a low riskfree rate currency in your discount rate, you will lose in your expected growth rate.

One more test on riskfree rates…
In January 2012, the 10-year treasury bond rate in the United States was 1.87%, a historic low. Assume that you were valuing a company in US dollars then, but were wary about the riskfree rate being too low. Which of the following should you do? Replace the current 10-year bond rate with a more reasonable normalized riskfree rate (the average 10-year bond rate over the last 30 years has been about 4%) Use the current 10-year bond rate as your riskfree rate but make sure that your other assumptions (about growth and inflation) are consistent with the riskfree rate Something else… While it is tempting to second guess yourself and replace the current riskfree rate with a more reasonable number, it is exceedingly dangerous for the following reasons: “Normalized” is in the eyes of the beholder. If you change just the riskfree rate and leave your other numbers unchanged, you can end up with internally inconsistent valuations. If your riskfree rate is too low (high), there is a countervailing variable in the valuation that will offset its impact. In particular, if interest rates are low, you should use a low expected inflation rate and low real growth in your valuation (at least in your stable growth phase).