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### Chapter 5

Interest Rates

1. Discuss how interest rates are quoted, and compute the effective annual rate (EAR) on a loan or investment.

2. Apply the TVM equations by accounting for the compounding periods per year.

3. Set up monthly amortization tables for consumer loans, and illustrate the payment changes as the compounding or annuity period changes.

4. Explain the real rate of interest and the impact of inflation on nominal rates.

5. Summarize the two major premiums that differentiate interest rates: the default premium and the maturity premium.

6. Amaze your family and friends with your knowledge of interest rate history.

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

Most common rate quoted 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 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:

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

Example: Calculating 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% with interest compounded monthly.” If you borrow $100,000 for 1 year, how much interest expense will you have accumulated over the first year and what is the bank’s APY? Note you make no payments during the year and the interest accumulates over the year.

Nominal annual rate = APR = 8.5%

Frequency of compounding = C/Y = m = 12

Periodic interest rate = APR/m = 8.5%/12 = 0.70833% = .0070833

APY or EAR = (1.0070833)12 - 1 = 1.0883909 - 1 = 8.83909%

Total interest expense after 1 year = .0883909 x $100,000 = $8,839.09

Proof? Determine the FV of the account with a compounding of 12 times a year but the payment is once a year.

5.2 Impact on the TVM Equations from Compounding Periods

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 usually is the same as the number of payments involved.

5.2 Impact on the TVM Equations from Compounding Periods

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 either 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.

5.2 Impact on the TVM Equations from Compounding Periods

Loan amount = $50,000

Loan period = 5 years

APR = 9%

Annual payments: PV = 50000; n=5; I/Y = 9; FV=0;

P/Y=1; C/Y=1; CPT PMT = $12,854.62

Quarterly payments: PV = 50000;n=20;I/Y = 9; FV=0; P/Y=4;C/Y=4; CPT PMT = $3,132.10

Total annual payment = $3132.1 x 4= $12,528.41

Monthly payments: PV = 50000;n=60;I/Y = 9; FV=0; P/Y=12;C/Y=12; CPT PMT = $1,037.92

Total annual payment = $1037.92 x 12 = $12,455.04

5.2 Impact on the TVM Equations from Compounding Periods

Example: 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.

5.2 Impact on the TVM Equations from Compounding Periods

With annual deposits: With monthly deposit

(Using the APR as the interest rate)

FV = $2,000,000; FV = $2,000,000;

N = 40 years; N = 12 x 40=480;

I/Y = APR = 12%; I/Y = APR = 12%;

PV = 0; PV = 0;

P/Y=1; P/Y = 12;

C/Y=1; C/Y = 12;

PMT = $2,607.25 PMT = $169.99

Total = $169.99 x 12 = $2,039.88

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.

5.3 Consumer Loans and Amortization Schedules (continued)

Example: 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.

5.3 Consumer Loans and Amortization Schedules

We first solve for the required minimum monthly payment:

PV = $200,000; I/Y=5; N=30 x 12=360; FV=0; C/Y=12; P/Y=12; PMT = $1,073.64

Next, we calculate the number of payments required to pay off the loan, if the monthly payment is increased by

1/12 x $1073.64, that is by $89.47

PMT = 1163.11; PV=$200,000; FV=0; I/Y=5; C/Y=12; P/Y=12;

Compute N = 303.13 months or 303.13/12 = 25.26 years.

This reduces the time by nearly five years.

5.3 Consumer Loans and Amortization Schedules

How much did you save by adding 1/12th to each payment?

With minimum monthly payments:

Total paid = 360 x $1073.64 = $386, 510.40

Minus Amount borrowed - $200,000.00

Interest paid = $186,510.40

With higher monthly payments:

Total paid = 303.13 x $1163.11 = $353,573.53

Minus Amount borrowed - $200,000.00

Interest paid = $153,573.53

Interest saved=$186,510.4-$153,573.53 = $32,936.87

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 (reward for waiting)
- The nominal of interest compensates for the erosion of purchasing power caused by inflation and a reward for waiting.
- 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)

5.4 Nominal and Real Interest Rates

Example: 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, 4%, how much would her nominal and real rates of return have to be?

5.4 Nominal and Real Interest Rates

Real rate of return = (FV/PV)1/n - 1

= (11 shirts/10 shirts)1/1 - 1 = 10%

Price of t-shirt next year = $10(1.04)1 = $10.40

Total cost of 11 t-shirts = $10.40 x 11 = $114.40

PV = $100; n=1; FV = $114.40; N = 1; CPT I/Y = 14.4%

I/Y is the Nominal rate of return, 14.4%

= Real rate + Inflation rate + (real rate*inflation rate)

= 10% + 4% + (10% x 4%) = 14.4%

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),

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.

5.6 A Brief History of Interest Rates

- A fifty year analysis (1950-1999) of the historical distribution of interest rates on various types of investments in the USA shows:
- Inflation at 4.05%,
- Real rate at 1.18%,
- Default premium of 0.53% (for AAA-rated over government bonds) and,
- Maturity premium at 1.28% (for twenty-year maturity differences).

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