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Chapter 5 Interest Rates

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Learning Objectives Discuss how interest rates are quoted, and compute the effective annual rate (EAR) on a loan or investment. Apply the TVM equations by accounting for the compounding periods per year. Set up monthly amortization tables for consumer loans, and illustrate the payment changes as the compounding or annuity period changes. Explain the real rate of interest and the effect of inflation on nominal interest rates. Summarize the two major premiums that differentiate interest rates: the default premium and the maturity premium. Amaze your family and friends with your knowledge of interest rate history.

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**5.1 How Interest Rates Are Quoted: Annual and Periodic Interest Rates**

The most commonly quoted rate is the annual percentage rate (APR). It is the annual rate based on interest being computed once a year. Lenders often charge interest on a non-annual basis. In such a case, the APR is divided by the number of compounding periods per year (C/Y or “m”) to calculate the periodic interest rate. For example: APR = 12%; m=12; i%=12%/12= 1% The EAR is the true rate of return to the lender and the true cost of borrowing to the borrower. An EAR, also known as the annual percentage yield (APY) on an investment, is calculated from a given APR and frequency of compounding (m) by using the following equation:

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**5.1 How Interest Rates Are Quoted: Annual and Periodic Interest Rates (continued)**

Example 1: Calculating an EAR or APY The First Common Bank has advertised one of its loan offerings as follows: “We will lend you $100,000 for up to 3 years at an APR of 8.5% (interest compounded monthly).” If you borrow $100,000 for 1 year, how much interest will you have paid and what is the bank’s APY? Nominal annual rate = APR = 8.5% Frequency of compounding = C/Y = m = 12 Periodic interest rate = APR/m = 8.5%/12 = % = APY or EAR = ( ) = = 8.839% Total interest paid after 1 year = *$100,000 = $8,839.05

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**5.2 Effect of Compounding Periods on the Time Value of Money Equations**

TVM equations require the periodic rate (r%) and the number of periods (n) to be entered as inputs. The greater the frequency of payments made per year, the lower the total amount paid. More money goes to principal and less interest is charged. The interest rate entered should be consistent with the frequency of compounding and the number of payments involved.

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**5.2 Effect of Compounding Periods on the Time Value of Money Equations**

Example 2: Effect of Payment Frequency on Total Payment Jim needs to borrow $50,000 for a business expansion project. His bank agrees to lend him the money over a 5-year term at an APR of 9% and will accept annual, quarterly, or monthly payments with no change in the quoted APR. Calculate the periodic payment under each alternative and compare the total amount paid each year under each option.

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**5.2 Effect of Compounding Periods on the Time Value of Money Equations (Example 2 Answer)**

Loan amount = $50,000 Loan period = 5 years APR = 9% Annual payments: PV = 50000;n=5;i = 9; FV=0; P/Y=1;C/Y=1; CPT PMT = $12, Quarterly payments: PV = 50000;n=20;i = 9; FV=0; P/Y=4;C/Y=4; CPT PMT = $ Total annual payment = $3132.1*4 = $12, Monthly payments: PV = 50000;n=60;i = 9; FV=0; P/Y=12;C/Y=12; CPT PMT = $ Total annual payment = $ *12 = $12,455.04

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**5.2 Effect of Compounding Periods on the Time Value of Money Equations**

Example 3: Comparing Annual and Monthly Deposits Joshua, who is currently 25 years old, wants to invest money into a retirement fund so as to have $2,000,000 saved up when he retires at age 65. If he can earn 12% per year in an equity fund, calculate the amount of money he would have to invest in equal annual amounts and alternatively, in equal monthly amounts starting at the end of the current year or month, respectively.

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**5.2 Effect of Compounding Periods on the Time Value of Money Equations (Example 3 Answer)**

With annual deposits: With monthly deposits: (Using the APR as the interest rate) FV = $2,000,000; FV = $2,000,000; N = 40 years; N = 12*40=480; I/Y = APR = 12%; I/Y = APR = 12%; PV = 0; PV = 0; C/Y=1; C/Y = 12 P/Y=1; P/Y = 12 PMT = $2, PMT = $169.99

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**5.3 Consumer Loans and Amortization Schedules**

Interest is charged only on the outstanding balance of a typical consumer loan. Increases in frequency and size of payments result in reduced interest charges and quicker payoff due to more being applied to loan balance. Amortization schedules help in planning and analysis of consumer loans.

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**5.3 Consumer Loans and Amortization Schedules (continued)**

Example 4: Paying Off a Loan Early! Kay has just taken out a $200,000, 30-year, 5% mortgage. She has heard from friends that if she increases the size of her monthly payment by one-twelfth of the monthly payment, she will be able to pay off the loan much earlier and save a bundle on interest costs. She is not convinced. Use the necessary calculations to help convince her that this is, in fact, true.

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**5.3 Consumer Loans and Amortization Schedules (continued)**

Example 4 (Answer) We first solve for the required minimum monthly payment: PV = $200,000; I/Y=5; N=30*12=360; FV=0; C/Y=12; P/Y=12; PMT = ?$ Next, we calculate the number of payments required to pay off the loan, if the monthly payment is increased by 1/12*$ i.e. by $89.47 PMT = ; PV=$200,000; FV=0; I/Y=5; C/Y=12; P/Y=12; N = ?N= months, or /12 = years.

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**5.3 Consumer Loans and Amortization Schedules (continued)**

Example 4 (Answer—continued) With minimum monthly payments: Total paid = 360*$ = $386, 510.4 Amount borrowed = $200,000.0 Interest paid = $186,510.4 With higher monthly payments: Total paid = *$ = $353,573.53 Amount borrowed = $200,000.00 Interest paid = $153,573.53 Interest saved=$186,510.4-$153, = $32,936.87

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**5.4 Nominal and Real Interest Rates**

The nominal risk-free rate is the rate of interest earned on a risk-free investment such as a bank CD or a treasury security. It is essentially a compensation paid for the giving up of current consumption by the investor. The real rate of interest adjusts for the erosion of purchasing power caused by inflation. The Fisher Effect shown below is the equation that shows the relationship between the real rate (r*), the inflation rate (h), and the nominal interest rate (r): (1 + r) = (1 + r*) x (1 + h) r = (1 + r*) x (1 + h) – 1 r = r* + h + (r* x h)

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**5.4 Nominal and Real Interest Rates (continued)**

Example 5: Calculating Nominal and Real Interest Rates Jill has $100 and is tempted to buy 10 t-shirts, with each one costing $10. However, she realizes that if she saves the money in a bank account, she should be able to buy 11 t-shirts. If the cost of the t-shirt increases by the rate of inflation, i.e., by 4%, how much would her nominal and real rates of return have to be?

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**5.4 Nominal and Real Interest Rates (continued)**

Example 5 (Answer-continued) Real rate of return = (FV/PV)1/n -1 = (11shirts/10shirts)1/1-1 = 10% Price of t-shirt next year = $10(1.04) = $10.40 Total cost of 11 t-shirts = $10.40*11 = $ = FV PV = $100; n=1; I/Y = (FV/PV) -1 = (114.4/100)-1 = 14.4% Nominal rate of return = 14.4% = Real rate + Inflation rate + (real rate*inflation rate) = 10% + 4% + (10%*4%) = 14.4%

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**5.5 Risk-Free Rate and Premiums**

The nominal risk-free rate of interest such as the rate of return on a Treasury bill includes the real rate of interest and the inflation premium. The rate of return on all other riskier investments (r) would have to include a default risk premium (dp)and a maturity risk premium (mp): i.e. r = r* + inf + dp + mp. 30-year corporate bond yield > 30-year T-bond yield Due to the increased length of time and the higher default risk on the corporate bond investment.

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**5.6 A Brief History of Interest Rates**

FIGURE 5.2 Inflation Rates in the United States, 1950–1999

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**5.6 A Brief History of Interest Rates (continued)**

FIGURE 5.3 Interest Rates for the Three-Month Treasury Bill, 1950–1999

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**5.6 A Brief History of Interest Rates (continued)**

TABLE 5.5 Yields on Treasury Bills, Treasury Bonds, and AAA Corporate Bonds, 1953–1999

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**5.6 A Brief History of Interest Rates (continued)**

A fifty year analysis ( ) of the historical distribution of interest rates on various types of investments in the United States shows: Inflation at 4.05% Real rate at 1.18% Default premium of 0.49% (for AAA-rated over government bonds) Maturity premium at 1.28% (for twenty-year maturity differences)

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**ADDITIONAL PROBLEMS WITH ANSWERS Problem 1**

Calculating APY or EAR. The First Federal Bank has advertised one of its loan offerings as follows: “We will lend you $100,000 for up to 5 years at an APR of 9.5% (interest compounded monthly.)” If you borrow $100,000 for 1 year and pay it off in one lump sum at the end of the year, how much interest will you have paid and what is the bank’s APY?

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**ADDITIONAL PROBLEMS WITH ANSWERS Problem 1 (ANSWER)**

Nominal annual rate = APR = 9.5% Frequency of compounding = C/Y = m = 12 Periodic interest rate = APR/m = 9.5%/12 = % = APY or EAR = ( ) = 9.92% Payment at the end of the year = *100,000 $109, Amount of interest paid = $109, $100,000 $9,924.7

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**ADDITIONAL PROBLEMS WITH ANSWERS Problem 2**

EAR with Monthly Compounding If First Federal offers to structure the 9.5%, $100,000, 1 year loan on a monthly payment basis, calculate your monthly payment and the amount of interest paid at the end of the year. What is your EAR?

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**ADDITIONAL PROBLEMS WITH ANSWERS Problem 2 (ANSWER)**

Calculate monthly payment: Total interest paid after 1 year = 12*$8, $100,000 = $105, $100,000 = $5,220.20 EAR is still 9.92%, since the APR and m are the same as #1 above, APY or EAR = ( ) = =9.92%

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**ADDITIONAL PROBLEMS WITH ANSWERS Problem 3**

Monthly versus Quarterly Payments: Patrick needs to borrow $70,000 to start a business expansion project. His bank agrees to lend him the money over a 5-year term at an APR of 9.25% and will accept either monthly or quarterly payments with no change in the quoted APR. Calculate the periodic payment under each alternative and compare the total amount paid each year under each option. Which payment term should Patrick accept and why?

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**ADDITIONAL PROBLEMS WITH ANSWERS Problem 3 (ANSWER)**

Calculate monthly payment: n=60; i/y = 9.25%/12; PV = 70000; FV=0; PMT= -1, Calculate quarterly payment: n=20; i/y = 9.25%/4; PV = 70000; FV=0; PMT= -4, Total amount paid per year under each payment type: With monthly payments = 12* $1, = $17, With quarterly payments = 4*$4, = $17,644.60

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**ADDITIONAL PROBLEMS WITH ANSWERS Problem 3 (ANSWER continued)**

Total interest paid under monthly compounding: Total paid - Amount borrowed = 60*$1, $70,000 = $87, $70,000 = $17,695.4 Total interest paid under quarterly compounding: 20 *$4, $70,000 = $88,223 - $70,000 = $18,223 Since less interest is paid over the 5 years with the monthly payment terms, Patrick should accept monthly rather than quarterly payment terms.

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**ADDITIONAL PROBLEMS WITH ANSWERS Problem 4**

Computing Payment for an Early Payoff: You have just taken on a 30-year, 6%, $300,000 mortgage and would like to pay it off in 20 years. By how much will your monthly payment have to change to accomplish this objective?

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**ADDITIONAL PROBLEMS WITH ANSWERS Problem 4 (ANSWER)**

Calculate the current monthly payment under the 30-year, 6% terms: n=360; i/y = 6%/12; PV = 300,000; FV=0; CPT PMT1,798.65 Next, calculate the payment required to pay off the loan in 15 years or 180 payments: n=180; i/y = 6%/12; PV = 300,000; FV=0; CPT PMT2,531.57 The increase in monthly payment required to pay off the loan in 20 years = $2, $1, = $732.92

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**ADDITIONAL PROBLEMS WITH ANSWERS Problem 5**

You just turned 30 and decide that you would like to save up enough money so as to be able to withdraw $75,000 per year for 20 years after you retire at age 65, with the first withdrawal starting on your 66th birthday. How much money will you have to deposit each month into an account earning 8% per year (interest compounded monthly), starting one month from today, to accomplish this goal?

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**ADDITIONAL PROBLEMS WITH ANSWERS Problem 5 (ANSWER)**

Calculate the amount of money needed to be accumulated at age 65 to provide an annuity of $75,000 for 20 years with the account earning 8% per year (interest compounded monthly) n=20; i/y = 8%; FV=0; PMT=75,000; P/Y = 1; C/Y=12 CPT PV720, Next, calculate the monthly deposit necessary to accumulate a FV of $720, over 35 years or 12*35 = 420 months: n=420; i/y = 8%; FV=720,210.86; P/Y = 12; C/Y=12 CPT PMT313.97

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**TABLE 5.1 Periodic Interest Rates**

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**TABLE 5.2 $500 CD with 5% APR, Compounded Quarterly at 1.25%**

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TABLE 5.3 Abbreviated Monthly Amortization Schedule for $25,000 Loan, Six Years at 8% Annual Percentage Rate

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**TABLE 5.4 Advertised Borrowing and Investing Rates at a Credit Union, January 22, 2007**

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**FIGURE 5.1 Interest rate dimensions**

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