0. Chapter 5
Interest Rates

1. Key Concepts Understand different ways interest rates are quoted
Use quoted rates to calculate loan payments & balances
Know how inflation, expectations, & risk combine to determine interest rates.
Understand link between interest rates in the market and firm’s cost of capital
Where m is the number of compounding periods per year
Using the calculator:
The TI BA-II Plus has an I conversion key that allows for easy conversion between quoted rates and effective rates.
2nd I Conv NOM is the quoted rate down arrow EFF is the effective rate down arrow C/Y is compounding periods per year. You can compute either the NOM or the EFF by entering the other two pieces of information, then going to the one you wish to compute and pressing CPT.

2. Basis Point

3. Annual Percentage Rate (Nominal)
This is the annual rate that is quoted by law
By definition APR = periodic rate times the number of periods per year
Consequently, to get the periodic rate
Periodic rate = APR / number of periods per year

4. Example 5.1a Valuing Monthly Cash Flows
Problem:
Suppose your bank account pays interest monthly with an annual rate of 5%. What amount of interest will you earn each month?
If you have no money in the bank today, how much will you need to save at the end of each month to accumulate $150,000 in 20 years? 4

5. Example 5.1a Valuing Monthly Cash Flows
Execute (cont’d):
We can also compute this result using a financial calculator:
Given:
240
0.4074
150,000
Solve for:
Excel Formula: =PMT(RATE,NPER,PV,FV)=PMT( ,240,0,150000) 5

6. Converting the APR to a Discount Rate
Problem:
Your firm is purchasing a new telephone system that will last for four years. You can purchase the system for an upfront cost of $150,000, or you can lease the system from the manufacturer for $4,000 paid at the end of each month. The lease price is offered for a 48-month lease with no early termination—you cannot end the lease early. Your firm can borrow at an interest rate of 6% APR with monthly compounding. Should you purchase the system outright or pay $4,000 per month? 6

7. Converting the APR to a Discount Rate
Solution:
Plan:
The cost of leasing the system is a 48-month annuity of $4,000 per month: 7

8. Example 5.2 Converting the APR to a Discount Rate
Execute (cont’d):
Using a financial calculator or Excel:
Given: 48
0.5
-4,000
Solve for:
170,321.27
Excel Formula: =PV(RATE,NPER, PMT, FV) = PV(0.005,48,-4000,0) 8

9. 5.2 Application: Discount Rates and Loans
Computing Loan Payments
Consider the timeline for a $30,000 car loan with these terms: 6.75% APR for 60 months 9

10. 5.2 Application: Discount Rates and Loans
Computing Loan Payments
Alternatively, we can solve for the payment C using a financial calculator or a spreadsheet:
Given: 60
0.5625
30000
Solve for:
Excel Formula: =PMT(RATE,NPER, PV, FV) = PMT( ,60,30000,0) 10

11. Example 5.3 Computing the Outstanding Loan Balance
Problem:
Let’s say that you are now 3 years into a $30,000 car loan (at 6.75% APR, originally for 60 months) and you decide to sell the car. When you sell the car, you will need to pay whatever the remaining balance is on your car loan. After 36 months of payments, how much do you still owe on your car loan? 11

12. Example 5.3 Computing the Outstanding Loan Balance
Execute:
Using a financial calculator or Excel:
Given: 24
0.5625
Solve for:
13,222.32
Excel Formula: =PV(RATE,NPER, PMT, FV) = PV( ,24,‑590.50,0) 12

13. Example 5.3a Computing the Outstanding Loan Balance
Problem:
Let’s say that you are now 2 years into a $25,000 car loan (at 5.50% APR, originally for 48 months) and you decide to sell the car. When you sell the car, you will need to pay whatever the remaining balance is on your car loan. After 24 months of payments, how much do you still owe on your car loan? 13

14. Example 5.3a Computing the Outstanding Loan Balance
Solution:
Plan:
First, we must determine the monthly payment.
Note: 0.055/12 =
Given: 48
0.4583
25000
Solve for:
Excel Formula: =PMT(RATE,NPER, PV, FV) = PMT( ,48,25000,0) 14

15. Annual Percentage Rate (Nominal)
This is the annual rate that is quoted by law
By definition APR = periodic rate times the number of periods per year
Consequently, to get the periodic rate
Periodic rate = APR / number of periods per year

16. APR
Annual Percentage Rate=Nominal Rate

17. Effective Annual Rate (EAR)
This is the actual rate paid (or received) after accounting for compounding that occurs during the year
To compare two alternative investments with different compounding periods, compute the EAR and use for comparison.
NEVER divide effective rate by number of periods per year – it will NOT give you the period rate
Where m is the number of compounding periods per year
Using the calculator:
The TI BA-II Plus has an I conversion key that allows for easy conversion between quoted rates and effective rates.
2nd I Conv NOM is the quoted rate down arrow EFF is the effective rate down arrow C/Y is compounding periods per year. You can compute either the NOM or the EFF by entering the other two pieces of information, then going to the one you wish to compute and pressing CPT.

18. Nominal (APR) v. Effective (EAR or EFF) Interest Rates
Annual Semi-Annual

19. Computing APRs (Nominal Rates)
What is APR if monthly rate is .5%?
.5% monthly x 12 months per year = 6%
What is APR if semiannual rate is .5%?
.5% semiannually x 2 semiannual periods per year = 1%
Can you divide above APR by 2 to get semiannual rate? NO!!! You need an APR based on semiannual compounding to find semiannual rate.
What is monthly rate if APR is 12% with monthly compounding?
12% APR / 12 months per year = 1%

20. Things to Remember
ALWAYS need to make sure interest rate and time period match.
If looking at annual periods, need an annual rate.
If looking at monthly periods, need a monthly rate.
If have an APR based on monthly compounding, use monthly periods for lump-sum $ amts, or adjust interest rate appropriately if have payments other than monthly

21. Computing EARs - Example
Suppose you can earn 1% per month on $1 invested today.
What is the APR? 1% x 12 monthly periods per year = 12%
How much are you effectively earning?
APR=NOM=12%; P/YR=12 (since Monthly)
EFF= ? =
Suppose if you put it in another account, you earn 3% per quarter.
What is the APR?
APR=NOM= ; P/YR=
Point out that the APR is the same in either case, but your effective rate is different. Ask them which account they should use.

22. EAR - Formula
Remember that the APR is the quoted rate

23. Decisions, Decisions II
You are looking at two savings accounts. One pays 5.25%, with daily compounding. The other pays 5.3% with semiannual compounding. Which account should you use?
First account:
APR= ; P/YR= ; EAR=? =
Second account:
APR= ; P/YR= ; EAR=? =
Which account should you choose and why?
Remind students that rates are quoted on an annual basis. The given numbers are APRs, not daily or semiannual rates.
Calculator:
2nd I conv 5.25 NOM up arrow 365 C/Y up arrow CPT EFF = 5.39%
5.3 NOM up arrow 2 C/Y up arrow CPT EFF = 5.37%

24. Decisions, Decisions II Continued
Let’s verify the choice. Suppose you invest $100 in each account. How much will you have in each account in one year?
First Account:
N= ; I/Y= ; PV=
FV=?=
Second Account:
N= ; I/Y= ; PV=
FV=? =
You have more money in the first account.
It is important to point out that the daily rate is NOT .014, it is

25. Computing APRs from EARs
If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get:

26. APR - Example
Suppose you want to earn an effective rate of 12% and you are looking at an account that compounds on a monthly basis. What APR must they pay?
EAR=EFF=12%; P/YR=12 (since monthly);
APR=NOM=?=11.39%
On the calculator: 2nd I conv down arrow 12 EFF down arrow 12 C/Y down arrow CPT NOM

27. Computing Payments with APRs
Suppose you want to buy a new computer system and the store is willing to sell it to allow you to make monthly payments. The entire computer system costs $3500. The loan period is for 2 years and the interest rate is 16.9% with monthly compounding. What is your monthly payment?
N= ; I/Y=
PV= ; PMT =?=

28. Future Values with Monthly Compounding
Suppose you deposit $50 a month into an account that has an APR of 9%, based on monthly compounding. How much will you have in the account in 35 years? N=
I/Y=
PMT=
FV=? =

29. Present Value with Daily Compounding
You need $15,000 in 3 years for a new car. If you can deposit money into an account that pays an APR of 5.5% based on daily compounding, how much would you need to deposit? N=
I/Y=
FV=
PV =?=

30. Quick Quiz – Part 5 What is the definition of an APR?
What is the effective annual rate?
Which rate should you use to compare alternative investments or loans?
Which rate do you need to use in the time value of money calculations?
APR = period rate * # of compounding periods per year
EAR is the rate we earn (or pay) after we account for compounding
We should use the EAR to compare alternatives
We need the period rate and we have to use the APR to get it

31. What Determines Interest Rates?
Nominal v. Real interest and inflation effects
Real risk free + inflation + maturity risk + default risk +liquidity risk
Yield curves & term structure

32. Inflation and Interest Rates
Real rate of interest – change in purchasing power
Nominal rate of interest – quoted rate of interest, change in purchasing power and inflation
The ex ante nominal rate of interest includes our desired real rate of return plus an adjustment for expected inflation
Be sure to ask the students to define inflation to make sure they understand what it is.

33. Inflation and Interest Rates
Real rate of interest – change in purchasing power
Nominal rate of interest – quoted rate of interest, change in purchasing power and inflation
The ex ante nominal rate of interest includes our desired real rate of return plus an adjustment for expected inflation
Be sure to ask the students to define inflation to make sure they understand what it is.

34. The Fisher Effect
The Fisher Effect defines the relationship between real rates, nominal rates and inflation
Approximation
Nominal Rate = real rate + inflation
R = r + h
Where : R = nominal rate
r = real rate
h = expected inflation rate
FISHER EFFECT
(1 + Nom) = (1 + Real) x (1 + Inflation)
Or, (1 + R) = (1 + r)(1 + h)
The approximation works pretty well with “normal” real rates of interest and expected inflation. If the expected inflation rate is high, then there can be a substantial difference.

35. Example 6.6
If we require a 10% real return and we expect inflation to be 8%, what is the nominal rate?
R = (1.1)(1.08) – 1 = .188 = 18.8%
Approximation: R = 10% + 8% = 18%
Because the real return and expected inflation are relatively high, there is significant difference between the actual Fisher Effect and the approximation.

36. Example 5.4 Calculating the Real Interest Rate
Problem:
In the year 2000, short-term U.S. government bond rates were about 5.8% and the rate of inflation was about 3.4%. In 2003, interest rates were about 1% and inflation was about 1.9%. What was the real interest rate in 2000 and 2003? 36

37. Example 5.4 Calculating the Real Interest Rate
Execute:
Thus, the real interest rate in 2000 was:
In 2003, the real interest rate was: 37

38. Figure 5.2 U.S. Interest Rates and Inflation Rates, 1955–200938

39. Term Structure of Interest Rates
Term structure is the relationship between time to maturity and yields, all else equal
It is important to recognize that we pull out the effect of default risk, different coupons, etc.
Yield curve – graphical representation of the term structure
Normal – upward-sloping, long-term yields are higher than short-term yields
Inverted – downward-sloping, long-term yields are lower than short-term yields

40. Figure 5. 3 Term Structure of Risk-Free U. S
Figure 5.3 Term Structure of Risk-Free U.S. Interest Rates, November 2006, 2007, and 2008 40

41. Figure 5.4 Yield Curve Shapes41

42. Figure 5. 5 Short-Term versus Long-Term U. S
Figure 5.5 Short-Term versus Long-Term U.S. Interest Rates and Recessions 42

43. 5.3 The Determinants of Interest Rates
Interest Rate Determination
Federal Funds Rate
The overnight loan rate charged by banks with excess reserves at a Federal Reserve bank to banks that need additional funds to meet reserve requirements
The Federal Reserve determines very short-term interest rates through its influence on the federal funds rate 43

44. Factors Affecting Required Return
Maturity Risk – time frame
Default risk premium – remember bond ratings
Taxability premium – remember municipal versus taxable
Liquidity premium – bonds that have more frequent trading will generally have lower required returns
Anything else that affects the risk of the cash flows to the bondholders, will affect the required returns

45. Figure 5.6 – Upward-Sloping Yield Curve
A. Upward-sloping term structure
Interest rate
Time to maturity
Nominal interest rate
Interest rate risk premium
Real rate
Inflation premium

46. Figure 5.6 – Downward-Sloping Yield Curve
Interest rate
B. Downward-sloping term structure
Nominal interest rate
Time to maturity
Real rate
Inflation premium
Interest rate risk premium