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- Explain the relationship between the real interest rate, the nominal interest rate, and inflation.
- Calculate a nominal interest rate, given the real interest rate and an inflation rate, using the Fisher equation and the more common approximation.
- Describe the relationship between bond prices and yields.
- Calculate the risk premium of an asset, given the risk free return and the expected return of the asset.

All investments have a certain amount of risk (even stuffing dollars under your mattress is vulnerable to house fire!), and, in a perfect world, we would be able to completely assess the risk of each investment. There are many employees, companies, and consultants whose entire focus is on evaluating risk. Arguably, all of finance boils down to the study of risk and return.

The value of a bond, like any series of future cash flows, is intrinsically related to the perceived opportunity cost of the investment, as we discussed in Chapter 6 "Time Value of Money: One Cash Flow". This cost, called the real interest rateThe interest rate that only compensates for the perceived opportunity cost of the investment., is determined by the market by causing the price of the bond to rise or fall. When we hear newscasters speaking of “interest rates falling”, what they are really saying is that the relevant bonds have changed in price, exposing a belief by investors that the opportunity costs of holding the bond are going down. So, as a shortcut we can talk about interest rates changing, but we should remember that, ultimately, these rates are derived from the price of the bonds! An investor buying a bond at a certain price point will yield a certain return on her investment; in bond parlance, this expected return is called the bond’s yieldA bond’s expected return.. Specifically, yield to maturity (YTM)The annualized rate of retun on a bond, assuming the bond is held to maturity and all the expected cash flows occur. is the annualized rate of return on a bond, assuming the bond is held to maturity and all the expected cash flows occur.

A bond’s yield is determined by its price, but we often speak of “investors require a yield of 5%”. What we are really saying in this case is that investors will only pay a price for the bond that will give them a 5% return on their money.

If investors determine they need a higher yield, then they won’t be willing to pay as much up front, and bond prices should fall. Thus, as yields rise, prices fall (and vice-versa). Bond prices and yields are inversely related!

Each bond faces a host of additional factors, all of which must be accounted for to properly assess the value of a bond. As time goes by, the purchasing power of a dollar (or whichever unit of currency we are using) could deteriorate; this we call inflationThe deterioration of the purchasing power of a currency.. Opportunity cost and inflation factored together are typically labeled the nominal interest rateThe interest rate that compensates for opportunity cost and inflation factored together..

If both the inflation rate and the real rate of interest are low (that is, < 5% each), then we can safely approximate the nominal interest rate as:

Equation 9.2 Approximate Nominal Interest Rate

We use *r** (pronounced “r-star”) to represent the real rate of interest and π (the Greek letter “pi”), which here is **not** the circle constant of 3.14, to represent inflation. Why have finance scholars chosen to use π? Because we thought using *i* would be too confusing, since both “interest” and “inflation” start with the same letter….

This equation has the benefit of being easy to remember and can be used for fairly accurate results. While historically it has been safe to assume a real interest rate at or below 5%, there have been many occurrences of large rates of inflation much larger than 5% throughout the world. When this is the case, it is more prudent to utilize the more accurate Fisher equation for interest rates:

Equation 9.3 Fisher Equation for Interest Rates

(1 +While a little more cumbersome to remember and use, it is the more accurate, especially if inflation is high.

This combination of the real rate of interest and the inflation rate should be the return required on an asset that is free of any other risks; we call this the risk-free rate of return and designate it: *r*_{RF}. As of the time of this writing, standard practice is to use US government bonds as a proxy for such a risk-free asset.

We already know of one risk, credit risk, that will cause investors to demand higher yields, but there are many other potential sources of risk. For example, if a bond isn’t likely to be easily tradable, then investors could want higher yields to compensate. All of these will contribute to the investment’s risk premiumThe amount of return required by investors over the risk-free asset., or amount of return required by investors over the risk-free asset.

Equation 9.4 Risk Premium

- Real interest rate + Inflation = Nominal Interest Rate (when inflation is reasonable).
- Bond prices go up, yields go down. Bond prices go down, yields go up.
- The risk premium of an asset is the extra amount of yield needed above the risk-free rate to compensate for the higher level of risk.

- If the expected rate of inflation increases, what will happen to the nominal rate of interest? What should happen to bond prices?
- If the real interest rate is 2% and expected inflation is 3%, what is the approximate nominal interest rate? Using the Fisher equation, what is the exact nominal interest rate? Repeat with an expected inflation rate of 15%.
- A company just issued a poor earnings report, throwing into doubt their ability to continue their debt payments. What should happen to their risk premium? What should happen to the price of their bonds?