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Chapter09B9-26.pptx

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Classifications of interest rates

Nominal rate (INOM) – also called the quoted or stated rate. An annual rate that ignores compounding effects.

INOM is stated in contracts. Periods must also be given, e.g. 8% Quarterly or 8% Daily interest.

Periodic rate (IPER) – amount of interest charged each period, e.g. monthly or quarterly.

IPER = INOM / M, where M is the number of compounding periods per year. M = 4 for quarterly and M = 12 for monthly compounding.

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Compounding More than Once per Year

Annual Compounding

0 8% 1

|______________________|

Semiannual Compounding

0 4% 1 4% 2

|__________|___________|

Quarterly Compounding

0 2% 1 2% 2 2% 3 2% 4

|_____|_____|_____|_____|

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Effective Annual Rate (EAR)

Effective (or equivalent) annual rate (EAR = EFF%): The annual rate of interest actually being earned, accounting for compounding.

EFF% for 8% semiannual investment

EFF% = ( 1 + )M - 1

= (1 + )2 – 1 = 8.16%

Should be indifferent between receiving 8.16% annual interest and receiving 8% interest, compounded semiannually. EAR is used to compare investment returns.

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Calculator

Use ICONV key

NOM = INOM

EFF = EAR

C/Y = # of compounding per year

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What is the FV of $100 after 3 years under 10% semiannual compounding? Quarterly compounding?

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calculator

10% semi-annual compounding

5 I/Y, 100 PV, 6 N, FV => 134.01

Quarterly compounding

2.5 I/Y, 100 PV, 12 N, FV => 134.49

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What is the future value of an annuity with $100 monthly payments at 7% after 5 years?

FV = PMT

= 100

OR,

100 PMT

( ) I/Y

( ) N

FV =

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GM Incentives

You need $12,000 loan to buy a car.

There are two financing options to choose:

A: 2.9% financing with a 36 month loan

B: A rebate of $1,000 is available and the remaining $11,000 is to be financed at 10% for 36 months.

Which option would you choose?

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Quarterly Compounding

A. If you deposit $1,000 in a bank that pays 8% quarterly compounding, what is the rate of return if you withdraw after 10 months?

B. How much in dollars will you get if you withdraw after 10 months?

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What’s the FV of a 3-year $100 annuity, if the quoted interest rate is 10%, compounded semiannually?

Payments occur annually, but compounding occurs every 6 months.

Cannot use normal annuity valuation techniques.

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Method 1: Compound each cash flow

FV3 = $100(1.05)4 + $100(1.05)2 + $100

FV3 = $331.80

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Method 2: Financial calculator

Find the EAR and treat as an annuity.

EAR = ( 1 + )2 – 1 = 10.25%.

10.25 I/Y, 3 N, -100 PMT, --- FV = 331.80

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Calculator 2

1 P/Y, 2 C/Y, 100 PMT, 3 N, 10 I/Y,

FV => 331.8006 => 331.80

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Find the PV of this 3-year ordinary annuity.

Could solve by discounting each cash flow, or …

Use the EAR and treat as an annuity to solve for PV.

10.25 I/Y, 3 N, 100 PMT, -- PV = 247.59

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Calculator 2

1 P/Y, 2 C/Y, 100 PMT, 3 N, 10 I/Y,

PV => 247.5947 => 247.59

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P/Y, C/Y

What is the future value of a three-year annuity with quarterly payments of $50 each at 7%, monthly compounding?

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Loan amortization

Amortization tables are widely used for home mortgages, auto loans, business loans, retirement plans, etc.

Financial calculators and spreadsheets are great for setting up amortization tables.

EXAMPLE: Construct an one-year amortization table for a $100,000, 8%, semiannual payment, 30-year loan.

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Step 1: Find the required annual payment

All input information is already given.

60 N, 4 I/Y, 100,000 PV,

PMT = 4,420.18

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Step 2: Find the interest paid in Period 1

The borrower will owe interest upon the initial balance at the end of the first period. Interest to be paid in the first period can be found by multiplying the beginning balance by the periodic interest rate.

INTt = Beg balt (I)

INT1 = $100,000 (0.04) = $4,000

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Step 3: Find the principal repaid in Period 1

If a payment of $4,420.18 was made at the end of the first period and $4,000 was paid toward interest, the remaining value must represent the amount of principal repaid.

PRIN REPAYMENT = PMT – INT

= $4,420.18 - $4,000 = $420.18

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Step 4: Find the ending balance after Period 1

To find the balance at the end of the period, subtract the amount paid toward principal from the beginning balance.

END BAL = BEG BAL – PRIN REP.

= $100,000 - $420.18

= $99,579.82

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Constructing an amortization table: Repeat steps 1 – 4 until end of loan

P	BEG BAL	PMT	INT	PRIN REPAY	END BAL
1	$100,000	$4,420.18	$4,000	$420.18	$99,579.82
2	99,579.82	4,420.18	3,983.19	436.99	99,142.83

Interest paid declines with each payment as the balance declines. What are the tax implications of this?

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Illustrating an amortized payment: Where does the money go?

Constant payments.

Declining interest payments.

Declining balance.

4,420.18

Interest

420.18

Principal Repayments

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Continuous Compounding

= ℮

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If compounding takes place continuously,

FVt = PV∙℮It

Alternatively,

PV = = FVt∙℮-It

Example 1: Suppose you invest $200 at 12% continuously compounded for two years. How much are you going to receive at the end of two years?

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Continued …

Answer: It = 0.12×2 = 0.24, e0.24 = 1.2712. So, FVt = FV2 = $200×1.2712 = $254.25

Example 2: What is the PV of $300 in one year’s time if I = 5%, and continuously compounded?

Answer: It = 0.05×1 = 0.05,

e-It = e-0.05= 0.9512,

so, PV = $300×0.9512 = $285.37

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$134.49

(1.025)

$100

$134.01

(1.05)

$100

)

0.10

(

$100

)

(

NOM