Chapter 5. Interest Rates. LEARNING OBJECTIVES. 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.
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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.
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:
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.
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.
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.
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
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.
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
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.
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.
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.
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
(1 + r) = (1 + r*) x (1 + h)
r = (1 + r*) x (1 + h) – 1
r = r* + h + (r* x h)
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?
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%
(r) would have to include a default risk premium (dp)and a maturity risk premium (mp),
r = r*+inf + dp + mp.