# Risk Free Rate

Risk free rate (also called risk free interest rate) is the interest rate on a debt instrument that has zero risk, specifically default and reinvestment risk. Risk free rate is the key input in estimation of cost of capital. The capital asset pricing model estimates required rate of return on equity based on how risky that investment is when compared to a totally risk-free asset. Cost of debt is estimated by adding spreads for different risk premia to the risk-free rate.

The interest rate on zero-coupon government securities, such as Treasury bills, notes and bonds in the US, is generally treated as a proxy for the risk-free rate. It is assumed that governments have zero default risk because they can print money to pay back their debt obligation whenever they want. Further, a zero-coupon government bill or bond has zero reinvestment risk.

The choice of risk free asset is relative. It depends on the currency of cash flows, the duration of cash flows and whether they are nominal or real. For nominal long-term cash flows in Japanese Yen, the risk-free rate on long-term Japanese government bonds should be used.

For typical long-term analysis, such as application of the capital asset pricing model to price risk, the interest rate on 10-year government bond is normally considered a valid risk-free rate proxy.

The relationship between real and nominal risk-free rate is as follows:

$$ Nominal\ Risk\ Free\ Rate=(1+{\rm rf}_r)\times(1+i)-1 $$

Where *rf _{r}* is the real risk-free rate and

*i*is the relevant inflation rate.

## Example

You work as corporate financial analyst at the head office of a multinational company. You are analyzing the net present value of a 8-year Japanese project. You need to work out the appropriate discount which is based on the weighted average cost of capital. You estimate the cost of equity using the capital asset pricing model. The cash flows are in real terms, the nominal risk-free rate for the short-term Japanese government bills is 1.5%, the 10-year government bonds rate is 2.5% and inflation rate is 0.7%. US short-term and long-term treasury rates are 1.50% and 2.77% and the inflation rate is 1%.

Work-out the risk-free rate that you must use in the capital asset pricing model if the market return in Japan is 5% and calculate the cost of equity component using the capital asset pricing model assuming a beta of 1.2.

Since the cash flows are in Japanese Yen, you must use the interest rate on the Japanese government bonds. The long-term rate is relevant because the cash flows being discounted have a longer duration. Ideally the risk-free rate duration should match the duration of the cash flows. Further, since the cash flows are in real terms, the real risk-free rate is relevant, which can be calculated as follows:

$$ Real\ risk\ free\ rate\\=\frac{(1+nominal\ risk\ free\ rate)}{(1+inflation\ rate)}\\=\frac{1+\ 2.5\%}{1+0.7\%}-1=1.79\% $$

We can now work out the cost of equity using the capital asset pricing model as follows:

$$ Cost\ of\ Equity\\=risk\ free\ rate\ +\ beta\ \times market\ risk\ premium\\=risk\ free\ rate\ +\ beta\ \times(market\ return\ -\ risk\ free\ rate)\\=1.79\%+1.2\times(5\%-1.79\%)=5.64\% $$

Written by Obaidullah Jan