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Chapter 5 Interest Rates

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**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.

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Basis Point

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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**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

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APR Annual Percentage Rate=Nominal Rate

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**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.

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**Nominal (APR) v. Effective (EAR or EFF) Interest Rates**

Annual Semi-Annual

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**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%

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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

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**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.

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EAR - Formula Remember that the APR is the quoted rate

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**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%

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**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

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**Computing APRs from EARs**

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

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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

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**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 =?=

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**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=? =

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**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 =?=

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**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

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**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

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**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.

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**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.

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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.

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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.

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**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

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**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

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

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**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

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**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

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**Figure 5.4 Yield Curve Shapes**

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**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

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**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

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**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

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**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

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**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

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