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Chapter 5 – Mathematics of Finance

5.1 – Simple Interest and Discount

Interest is the fee paid for using someone else’s money. You pay if you borrow money. You earn if you lend money.

Principal – The amount of money borrowed

The amount of interest paid depends on: * Interest rate * Time

Time Period | Years | Stated Rate | Decimal Form | 72 days | 72/365 | 6 % | .06 | 4 months | 4/12 | Seven and one eighth % | .07125 | 18 months | 1.5 | Half a % | .005 |

Interest can be computed two ways: * Simple Interest * Compound Interest

Simple Interest Formula – I = Prt where: * I – the amount of interest * P – the amount of principal * r = interest rate per year * t = length of time in years

Simple Interest is normally used for loans or investments of a year or less. An exception is corporate bonds.

See example 1 and 2 from page 260

Future or Maturity Value for Simple Interest

A = P( 1 + rt ) where A is the future or maturity value

See example 3 from page 261

Do problems 4, 8, 12, 16

Present Value with simple interest – the amount deposited today (the principal) that will yield some future value.

-------------------------------------------------
Since A = P(1+ rt) then : P =

See examples 5 and 6 on page 263

Discount

In a simple discount loan the interest is deducted in advance from the amount of the loan and the balance is given to the borrower. The full value of the loan must be paid back at maturity.

Treasury bills are financial instruments issued by governments in order to secure short term funding. Investors that purchase these T-Bills are essentially loaning money to the government. As an investor you can purchase T-Bills at a discount and then receive the face value of the T-Bill at maturity.

Discount = Prt
Price of T-Bill = Face value – Discount
Actual Interest rate paid - use I=Prt and solve for r.

See example 8.

Do problems 28, 29, 30, 36, 37, 38, 39

5.2 Compound Interest

Normally used for loans or investments greater than one year
Interest is paid (or earned) on interest

Type of Compounding | Number of periods per year - m | Annual rater | Rate per periodr/m | Annual | 1 | .12 | .01 | Semi-annual | 2 | .12 | .06 | Quarterly | 4 | .12 | .03 | Monthly | 12 | .12 | .01 | weekly | 52 | .12 | .002308 | Daily | 365 | .12 | .000329 |

Compound Interest Formula

A = P(1 + i)n where:

* A – the compound amount – future value * P – the deposit – present value * I – interest rate per period or r/m * n – total number of compounding periods or m*t

See example 2 page 269 and example 4 page 270

Do problems 11, 17 page 277

Continuous Compounding - A = Pert * A – the compound amount – future value * P – the deposit – present value * e – 2.7182818 * r – the annual rate * t – number of years

See page307

Do problems 1 on page 309

Effective Rate (APY)

Where APY is the Annual Percentage Yield

Since:

100 invested at a simple interest rate of 7% for one year yields a maturity amount of $107, and 7/100 = .07

100 invested at 7% compounded semi-annually for one year yields a maturity amount of $107.1225 for an effective rate of 7.1225/100 = .071225, or 7.1225%

Effective Rate (APY) re = where re = effective rate r = nominal or stated rate m = number of times compounded per year

Do problem 30 on page 277

Present Value with Compound Interest

------------------------------------------------- since A = P(1 + i)n then P =

See example 11 and 12 on page 275
Do problems 35, 44, 52

5.3 – Future Value of an Annuity and Sinking Funds

An annuity is a stream of equal payments made at regular intervals.

Examples: * Rent * Car payment * Mortgage payment * Pension Income

Ordinary annuity – payments are made at the end of each period.

* The frequency of payments usually = the frequency of compounding * Therefore monthly payments indicate monthly compounding i.e. mortgage payments.

Annuity due – the payments are made at the beginning of each period.

Payment period – time between payments

Term of the annuity – beginning of 1st payment period until the end of the last payment period.

The future value of an annuity is the final sum on deposit.

Example: Saving for a vacation by making equal payments at regular intervals into an account so that you will have some lump sum available in the future.

The future value of an annuity

Where: * S = FV of the annuity * R = the payment made at the end of the period * I = interest rate per period = r/m * n = # of periods over the term of the annuity = m*t

See example 2 on page 283

Sinking Fund – A fund set up to receive periodic payments. The payments together with the interest they earn are designed to produce a given sum at some time in the future. If the payments are the same amount and made at the end of a regular period, they form an ordinary annuity.

See example 5 on page 284 and do problems 6, 10, 18, 41

Present Value of an Annuity and Amortization

The Present Value of an Annuity is the amount on hand now that is equivalent to a stream of regular and equal payments into the future.

Where: * P = PV of the annuity * R = the payment made at the end of the period * I = interest rate per period = r/m * n = # of periods over the term of the annuity = m*t

Examples:

Car loan – you receive the amount now, the present value, in agreement to make equal and regular payments some time in to the future.

See examples 2 & 3 on page 293 and problems 2, 11, 47, 49 on page 301.

Amortization – A loan is amortized if both the principal and interest are paid by a sequence of equal periodic payments. This periodic payment is R.

Amortization Payments

Do example 7

Amortization Schedule for example 7

Rate | periods | principal | Payment | | 0.067 | 360 | $ 110,000.00 | $ 709.81 | | | | | | | | | | | | Payment | Amount of | Interest for | Portion to | Principal at | Number | Payment | Period | Principal | end of period | 0 | | | | $ 110,000.00 | 1 | $ 709.81 | $ 614.17 | $ 95.64 | $ 109,904.36 | 2 | $ 709.81 | $ 613.63 | $ 96.17 | $ 109,808.19 | 3 | $ 709.81 | $ 613.10 | $ 96.71 | $ 109,711.48 |

Do problem 64, give students the handout problems

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