5.1

Chapter

5Interest Rates

5.2

Key ConceptsUnderstand different ways interest rates are quotedUse quoted rates to calculate loan payments &

balancesKnow how inflation, expectations, & risk combine to

determine interest rates.Understand link between interest rates in the market

and firm’s cost of capital

5.3

Basis Point

5.4

Annual Percentage Rate (Nominal)This is the annual rate that is quoted by lawBy definition APR = periodic rate times the

number of periods per yearConsequently, to get the periodic rate

Periodic rate = APR / number of periods per year

5.5

Example 5.1a Valuing Monthly Cash FlowsProblem: 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?

5.6

Example 5.1a Valuing Monthly Cash FlowsExecute (cont’d): We can also compute this result using a financial calculator:

Given: 240 0.4074 0 150,000

Solve for: -369.64

Excel Formula: =PMT(RATE,NPER,PV,FV)=PMT(.004074,240,0,150000)

5.7

Converting the APR to a Discount RateProblem: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?

5.8

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:

5.9Example 5.2 Converting the APR to a Discount Rate Execute (cont’d): Using a financial calculator or Excel:

Given: 48 0.5 -4,000 0

Solve for: 170,321.27

Excel Formula: =PV(RATE,NPER, PMT, FV) = PV(0.005,48,-4000,0)

5.10

5.2 Application: Discount Rates and LoansComputing Loan Payments

Consider the timeline for a $30,000 car loan with these terms: 6.75% APR for 60 months

5.11

5.2 Application: Discount Rates and LoansComputing Loan Payments

Alternatively, we can solve for the payment C using a financial calculator or a spreadsheet:

Given: 60 0.5625 30000 0

Solve for: -590.50

Excel Formula: =PMT(RATE,NPER, PV, FV) = PMT(0.005625,60,30000,0)

5.12

Example 5.3 Computing the Outstanding Loan BalanceProblem: 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?

5.13

Example 5.3 Computing the Outstanding Loan BalanceExecute: Using a financial calculator or Excel:

Given: 24 0.5625 -590.50 0

Solve for: 13,222.32

Excel Formula: =PV(RATE,NPER, PMT, FV) = PV(0.005625,24,‑590.50,0)

5.14

Example 5.3a Computing the Outstanding Loan BalanceProblem: 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?

5.15

Example 5.3a Computing the Outstanding Loan BalanceSolution:

Plan: First, we must determine the monthly payment.

Note: 0.055/12 = 0.004583

Given: 48 0.4583 25000 0

Solve for: -581.41

Excel Formula: =PMT(RATE,NPER, PV, FV) = PMT(0.004583,48,25000,0)

5.16

Annual Percentage Rate (Nominal)This is the annual rate that is quoted by lawBy definition APR = periodic rate times the

number of periods per yearConsequently, to get the periodic rate

Periodic rate = APR / number of periods per year

5.17

APR

Annual Percentage Rate=Nominal Rate

5.18

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

5.19

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

5.20

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%

5.21

Things to RememberALWAYS 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

5.22

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? How much are you effectively earning?

APR=NOM= ; P/YR= EFF= ? =

5.23

EAR - Formula

1 m

APR 1 EAR

m

Remember that the APR is the quoted rate

5.24

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?

5.25

Decisions, Decisions II ContinuedLet’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.

5.26

Computing APRs from EARs

If you have an effective rate, how can you compute the APR? Rearrange the EAR equation and you get:

1 - EAR) (1 m APR m

1

5.27

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%

11.39%or

8655152113.1)12.1(12 12 APR

5.28

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 =?=

5.29

Future Values with Monthly CompoundingSuppose 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=? =

5.30

Present Value with Daily CompoundingYou 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 =?=

5.31

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?

5.32

What Determines Interest Rates?

Nominal v. Real interest and inflation effectsReal risk free + inflation + maturity risk +

default risk +liquidity riskYield curves & term structure

5.33

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

5.34

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

5.35

The Fisher EffectThe Fisher Effect defines the relationship between

real rates, nominal rates and inflationApproximation Nominal Rate = real rate + inflation R = r + h Where : R = nominal rate

r = real rateh = expected inflation rate

FISHER EFFECT (1 + Nom) = (1 + Real) x (1 + Inflation)

Or, (1 + R) = (1 + r)(1 + h)

5.36

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.

5.37

Example 5.4Calculating the Real Interest RateProblem: 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?

5.38

Example 5.4Calculating the Real Interest RateExecute:

Thus, the real interest rate in 2000 was:

Nominal Rate Inflation RateReal Rate

1 Inflation Rate

(5.8% 3.4%) / (1.034) 2.32%

In 2003, the real interest rate was:(1% 1.9%) / (1.019) 0.88%.

5.39

Figure 5.2 U.S. Interest Rates and Inflation Rates, 1955–2009

5.40

Term Structure of Interest RatesTerm structure is the relationship between time

to maturity and yields, all else equalIt is important to recognize that we pull out the

effect of default risk, different coupons, etc.Yield curve – graphical representation of the

term structureNormal – upward-sloping, long-term yields are

higher than short-term yieldsInverted – downward-sloping, long-term yields are

lower than short-term yields

5.41Figure 5.3 Term Structure of Risk-Free U.S. Interest Rates, November 2006, 2007, and 2008

5.42

Figure 5.4 Yield Curve Shapes

5.43Figure 5.5 Short-Term versus Long-Term U.S. Interest Rates and Recessions

5.44

5.3 The Determinants of Interest RatesInterest 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

5.45

Factors Affecting Required ReturnMaturity Risk – time frameDefault risk premium – remember bond ratingsTaxability premium – remember municipal

versus taxableLiquidity 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

5.46

Figure 5.6 – Upward-Sloping Yield Curve

A. Upward-sloping term structureInterestrate

Time tomaturity

Nominalinterestrate

Interest raterisk premium

Real rate

Inflationpremium

5.47

Figure 5.6 – Downward-Sloping Yield Curve

Interestrate

B. Downward-sloping term structure

Nominalinterestrate

Time tomaturity

Real rate

Inflationpremium

Interest raterisk premium