1. Chapter 6
Interest Rates

2. Interest Rate History
For a long time in the history –charging interest has been viewed as a “SIN”
Aristotle considered money itself to be "barren", and the charging of interest on loans "unfair".
In Europe throughout the Middle Ages and beyond, charging and paying interest are moral sins. The ban of interest rate is rigorously enforced after the Black Death in the 14th century.
Medici family in Italy get around it by “lending in one currency” and repay in another (or in commodities), where interest rate is built in exchange rate.

3. Determinants of Interest Rates
r = r* + IP + DRP + LP + MRP r = required return on a debt security, nominal interest rate r* = real risk-free rate of interest; represents the “real” risk- free rate of interest. Like a T-bill rate, if there was no inflation. IP = inflation premium DRP = default risk premium LP = liquidity premium MRP = maturity risk premium

4. Nominal versus Real Rate
rRF = represents the rate of interest on Treasury securities. Also often referred to as “nominal risk free rate”
Example: US T-bill; US T-bond
r* represents “real risk free rate”
Inflation premium: average expected inflation over the life of the security
rRF = r* + IP

5. Example: Nominal versus Real Rate
Real risk-free rate r*= 1.7%
Expected inflation premium over the next year: IP = 1%
Rate on one year risk free US T-bill ?
1.7% + 1% =2.7%

6. Inflation and Interest Rates

7. Default Risk Premium
How likely is the corporation default on the bond (not paying back scheduled interest or principal payments)?
Default risk of bond: The difference between a US Treasury bond and a corporate bond of equal maturity and marketability.

8. Default Risk Premium Rating companies evaluate default risk of bonds
Moody’s Investor Service
Standard & Poor’s
Rating Categories
Investment grade
E.g. AAA bond
Speculative grade

9. Liquidity Premium
A premium added to the equilibrium interest rate on a security if that security cannot be converted to cash on short notice and at close to its “fair market value”
What is usually a good measure for asset’s liquidity?

10. Maturity Risk Premium
A premium that reflects interest rate risk—that is, a bond obligation will be more sensitive to interest rate fluctuations the longer to maturity it is.

11. Premiums Added to r* for Different Types of DebtIP
MRP
DRP LP
S-T Treasury 
L-T Treasury
S-T Corporate
L-T Corporate

12. Example
A company’s 5-year bonds are yielding 7.75% per year. Treasury bonds with the same maturity are yielding 5.2% per year, and the real risk-free rate is 2.3%. The average inflation premium is 2.5%; and the maturity risk premium is estimated to be 0.1* (t-1)%, where t=number of years to maturity. If the liquidity premium is 1%, what is the default risk premium on the corporate bonds?

13. Solution MRP = 0.1%(5 – 1) = 0.4% r= r* + IP + DRP + LP + MRP

14. Example
You read in The Wall Street Journal that 30-day T- bills are currently yielding 5.5%. Your brother-in- law, a broker at Safe and Sound Securities, has given the following estimates of current interest rate premiums:
Inflation premium = 3.25%
Liquidity premium =0.6%
Maturity risk premium = 1.8%
Default risk premium = 2.15%
What is the real risk-free rate of return?

15. Solution
T-bill rate = r* + IP
5.5% = r* %
r* = 2.25%.

16. Example
The real risk-free rate of interest is 3%; and it is expected to remain constant over time. Inflation is expected to be 2% per year over the next 3 years and 4% per year from year 4 to year 9. The maturity risk premium is equal to 0.1*(t-1)%, where t=bond’s maturity. The default risk premium for a BBB-rated bond is 1.3%.
What is the average expected inflation rate over the next 4 years?
What is the yield on a 4-year treasury bond?
What is the yield on a 4-year BBB-rated corporate bond with a liquidity premium of 0.5%?

17. Solution Average expected inflation:
(2%*3+4%)/4=2.5%
For T-Bond: r= r* + IP + MRP= 3%+2.5%+0.1*3% =5.8%
For corporate bond: r= r* + IP + DRP + LP + MRP = 5.8% +DRP+LP = 5.8%+ 0.5% + 1.3% = 7.6%

18. What is “Risk-free Rates”
Long term: long term government rate
Short term: short term government security rate
The conventional practice of estimating risk-free rates is to use the government bond rate. In November 2013, for instance, the rate on a ten-year US treasury bond (2.75%) is used as the risk free rate in US dollars.
Since corporate finance generally looks at long term decisions, we will for the most part use the long term risk free rate.

19. When the government is default free: Risk free rates
These are currencies where there is at least one government issuing bonds in that currency that has a Aaa rating. Note, though, that the fact that a ratings agency claims that a rating is Aaa does not necessarily mean that the government is default free.

20. What is the Euro riskfree rate?
Note that there are ten government that issue bonds denominated in Euros… with different rates on each of them. Since they are all in the same currency, the differences have to be attributed to perceptions of default risk. For a riskfree rate in Euros, I would go with the lowest of the rates in this table, which belongs to the German Euro bond; the ten-year rate is 1.75%.
A purist can argue that even the German Euro bond has some default risk embedded in it. The Euro riskfree rate would therefore have to be lower than 1.75%.

21. Term Structure
Term structure: relationship between interest rates (or yields) and maturities.
Yield for US Treasury Bond

22. Yield Curve and the Term Structure of Interest Rates
Yield Curve for US Treasury Bond
The yield curve is a graph of the term structure.
The February Treasury yield curve is shown at the right.

23. How to Construct a Yield Curve?
Construct a yield curve for US Treasury Bond with maturities of 1 year, 10 year and 20 years.
Lets revisit the determinants of interest rates:
r = r* + IP + DRP + LP + MRP
With US T-bond:
r = r* + IP + MRP

24. Constructing the Yield Curve: Inflation
Step 1: Find the average expected inflation rate over Years 1 to N: Assume inflation is expected to be 5% next year, 6% the following year, and 8% thereafter. Must earn these IPs to break even vs. inflation

25. Constructing the Yield Curve: Maturity Risk
Step 2: Find the appropriate maturity risk premium (MRP). For this example, the following equation will be used to find a security’s appropriate maturity risk premium. MRPt = 0.1% (t – 1) Using the given equation: Notice that since the equation is linear, the maturity risk premium is increasing as the time to maturity increases, as it should be.

26. Add the IPs and MRPs to r* to Find the Appropriate Nominal Rates
Step 3: Adding the premiums to r*. rRF, t = r* + IPt + MRPt Assume r* = 3%,

27. Hypothetical Yield Curve
An upward-sloping yield curve.
Upward slope due to an increase in expected inflation and increasing maturity risk premium.
Any other shapes?
Years to
Maturity
Real risk-free rate 5 10 15 1
Interest
Rate (%)
Maturity risk premium
Inflation premium 20

28. Yield Curves
Yields
Maturity
Upward Sloping
Downward Sloping
Flat

29. Yield Curves and Inflation Expectations

30. Pure Expectations Theory
The pure expectations theory contends that the shape of the yield curve depends on investors’ expectations about future interest rates. Observed long‐ term rate is a function of today’s short‐ term rate and expected future rates
Assumptions
Assumes that the maturity risk premium for Treasury securities is zero.
If the pure expectations theory is correct, you can use the yield curve to “back out” expected future interest rates.

31. An Example: Observed Treasury Rates and Pure Expectations
Maturity
Yield
1 year
6.0%
2 years
6.2
3 years
6.4
4 years
6.5
5 years
If the pure expectations theory holds, what does the market expect will be the interest rate on one-year securities, one year from now? Three-year securities, two years from now?

32. One-Year Forward Rate (1.062)2 = (1.060) (1 + X)
6.0% x%
6.2%
(1.062)2 = (1.060) (1 + X)
/1.060 = (1 + X)
6.4004% = X
The pure expectations theory says that one-year securities will yield %, one year from now.

33. Three-Year Security, Two Years from Now
6.2% x%
6.5%
(1.065)5 = (1.062)2 (1 + X) / = (1 + X) % = X
The pure expectations theory says that three-year securities will yield %, two years from now.

34. Example
One-year treasury securities yield 5%. The market anticipates that 1 year from now, 1 year treasury security will yield 6%. If the pure expectations theory is correct, what is the yield today for 2-year treasury securities?

35. Solution
(1.05)*(1.06)=(1+X) 2
X=5.4988%

36. Exercise
The Treasury yield curve shows the following yields to maturity for the next 5 years:
What is the implied 1-year forward rate two years from now?
Maturity 1 2 3 4 5
YTM in %
1.5
2.1
2.6
3.2
3.6

37. Solution
(1+2.1%)2 (1+X) = (1+2.6%)3 ⇔ X = ( )3 / ( )2 - 1 =