Fin 320 Chapter 5 Interest Rates and Valuing Cash
Adjusting the Discount Rate to Different Time Periods
(1 + r)^0.5 = (1.05)^0.5 = $1.0247, so a yearly rate of 5%, is equivalent to a rate of 2.47% every half of a year.
Growth in Purchasing Power
1 + Real Rate = 1 + Nominal Rate / 1 + Inflation Rate = Growth of Money / Growth of Price
Amortizing Loan
A loan on which the borrower makes monthly payments that include interest on the loan plus some part of the loan balance
Yield Curve
A plot of bond yields as a function of the bonds' maturity date
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?
As Eq. 5.2 shows, the 6% APR with monthly compounding really means 6%/12=0.5% every month. The 12 comes from the fact that there are 12 monthly compounding periods per year. Now that we have the true rate corresponding to the stated APR, we can use that discount rate in the annuity formula Eq. 4.6 to compute the present value of the monthly payments: FV (Annuity) = PV X ( 1 + r) ^ n C/R x 1/r [(1 +r)^N -1] PV = 4,000 x 1 / .005 ( 1 - (1/.005^48) = 170,231.27 N = 48 I/Y = .5 PV = ? PMT = -4000 FV = 0 PV = 170,231.27
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
The Real Interest Rate
Growth in Purchasing Power equation can be rearranged into the real interest rate. Real Rate = (Nominal Rate - Inflation Rate) / (1 + Inflation Rate) = Nominal Rate - Inflation Rate
Which do you prefer: a bank account that pays 4.8% per year (EAR) for three years or
If you deposit $1 into a bank account that pays 4.8% per year for three years: The amount you will receive after three years is $ 1 x (1 + .048) ^ 3 = $1.15102 a. An account that pays 2.5% every six months for 3 years? If you deposit $1 into a bank account that pays 2.5% every six months for three years: 1 x (1 + .025) ^ 6 (=36/6) = 1.15969 Which bank account would you prefer? 2.5 % every six months for three years b. An account that pays 7.2% every 18 months for 3 years? If you deposit $1 into a bank account that pays 7.2% every 18 months for three years: The amount you will receive after three years is $ 1 x (1 + .072) ^ 2 ( = 36 months / 18 months) = 1.14918 Which bank account would you prefer? 4.8% per year for three years c. An account that pays 0.53 %0.53% per month for three years? If you deposit $ 1 into a bank account that pays 0.53 % per month for three years The amount you will receive after three years is $1.20961 Which bank account would you prefer? 0.53% every month for three years
Annual Percentage Rates (APR)
Indicates the amount of interest earned in one year without the effect of compounding
Simple Interest
Interest earned without the effect of compounding
Computing the Outstanding Loan Balance 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?
N = 24 I/Y = .5625 PV = ? PMT = -590.50 FV = 0 PV = 13222.32 You could also compute this as the FV of the original loan amount after deducting payments: N = 36 I/Y = .5625 PV = 30000 PMT = -590.50 FV = ? FV = 13222.32
Nominal Interest Rates
The rate at which your money will grow if invested for a certain period Interest rates quoted by banks and other financial institutions that indicate the rate at which money will grow if invested for a certain period of time
Real Interest Rate
The rate of growth of your purchasing power, after adjusting for inflation
Risk-Free Interest Rate
The interest rate at which money can be borrowed or lent without risk over a given period
Term Structure
The relationship between the investment term and the interest rate
Effective Annual Rate (EAR) Annual Percentage Yield (APY)
The total amount of interest that will be earned at the end of one year
The Effective Annual Rate
With an EAR of 5%, a $100 investment grows to: $100 x (1 + r) = $100 x (1.05) = $105 After two years it will grow to: $100 x (1 + r)2 = $100 x (1.05)2 = $110.25
You are looking to buy a car and you have been offered a loan with an APR of 6%, compounded month
a. What is the true monthly rate of interest? The Monthly rate of interest is Equivalent n-period discount rate = APR / n Equivalent n-period discount rate = .063 / 12 = .525% b. What is the EAR The EAR is EAR = [1 + (APR / m) ^ m] - 1 EAR = [1 + (.063/ 12) ^ 12] - 1 = 6.4851%
Your bank is offering you an account that will pay 18 % interest (an effective two-year rate) in total for a two-year deposit. Determine the equivalent discount rate for the following periods:
a. Six months (1 + .18) ^ 1/4 (= 6 month / 24 months = 2 years) - 1 X 100% The equivalent discount rate for a period length of six months is 4.22% b. one year (1 +.18) ^ 1/2 - 1 X 100% The equivalent discount rate for a period length of six months is 8.63% C. One Month (1 + .18) ^ 1/24 - 1 X 100% The equivalent discount rate for a period length of six months is .693%
You are considering two ways of financing a spring break vacation. You could put it on your credit card, at 14% APR, compounded monthly, or borrow the money from your parents, who want an interest payment of 8% every six months. Which is the lower rate? (Note: Be careful not to round any intermediate steps less than six decimal places.)
a. The effective annual rate for your credit card is EAR = [1 + (APR / m) ^ m] - 1 = [1 + (.14 / 12) ^ 12 ] - 1 = .149342 X 100 = 14.93% b. The effective annual rate for the loan from your parents is Now for the loan from your parents, substitute into the formula but do not divide the rate by two because you were provided with the six-month rate (not the annual rate compounded semiannually): EAR = [( 1 + .08) ^ 2] - 1 = .16640 x 100 = 16.64% The option with the lower effective annual rate is Your credit Card
