Mathematics of Finance

Compound InterestFor an original principal of P, the formula:

S = P (1 + r)n

Gives the compound amount S at the end of n interest periods at the periodic rate of r.

Example:1. Suppose you leave an initial amount of $518 in a savings account

for three years. If interest is compounded daily (365 times per year), find the nominal rate of interest so that there is $600 after three years?

2. Suppose you leave an initial amount of $520 in a savings account at an annual rate of 5.2% compounded daily. Find how long it takes for the amount to accumulate to $570!

3. How long will it take for $600 to amount to $900 at an annual rate of 6% compounded quarterly?

Effective Rate• Def : just the rate of simple interest earned over a period of

one year.

• Example:a. What effective rate is equivalent to a nominal rate of 6%

compounded (a) semiannually, (b) quarterly?b. Suppose you have two investment opportunities. You can

invest $10,000 at 11% compounded monthly or you can invest $97,000 at 11.25% compounded quarterly. Which has the better effective rate of interest? Which is the better investment over 20 years?

c. How many years will it take for money to double at the effective rate of r?

11

n

n

rre

Present Value

The principal P that must be invested at the periodic rate of r for n interest period so that the compound amount is S is given by:

P= S (1+r)-n

Example:A trust fund for a child’s education is being set up

by a single payment so that at the end of 15 years there will be $50,000. if the fun earns interest at 7% compounded semiannually, how much money should be paid into the fund?

Equations of ValueExample 1:1. Suppose that Mr. Smith owes Mr. Jones two sums of

money: $1,000 due in two years, and $6,000 due in five years. If Mr. Smith wishes to pay off the total debt now by a single payment, how much should the payment be? (Assume: interest rate 8% compounded quarterly)

2. A debt of $3,000 due in six years from now is instead to be paid off by three payments: $500 now, $1500 in three years, and a final payment at the end of year 5. what this payment be if an interest rate of 6% compounded annually?

Comparing Investments

Suppose that you had the opportunity of investing $5000 in a business such that the value of the investment after five years would be $6300. On the other hand, you could instead put the $5000 in a savings account that pays 6% compounded semiannually. Which investment is better ?

NET PRESENT VALUEThe net present value, denoted NPV, of the cash flows is defined to be the sum of the present value of the cash flow, minus the initial investment.

Example :Suppose that you can invest $20.000 in a business that guarantees you cash flows at the end of years 2, 3, and 5 as indicated in the table t below. Assume an interest rate of 7% compounded annually, and find the net present value of the cash flows.

Year Cash Flow

2 10,000

3 8,000

5 6,000

Interest Compounded Continuously

S = Pert

Gives the compound amount S of a principal of P dollars after t years at an annual interest rate r compounded continuously

Annuities• Def : any finite sequence of payments made at fixed

periods of time over a given interval.• Present Value of an AnnuityDef : is the sum of the present values of all n

payments. It represents the amount that must be invested now to purchase all n of them.

Gives the present value A of an ordinary annuity of R per payment period for n periods at the interest rate of r period

r

rRA

n

)1(1

Annuities• Present Value of an Annuity• Periodic payment of an Annuity :

• An Annuity Due ( payment at the beginning of the year)

nr

rAR

)1(1

r

rRA

n

)1(11

Annuities• Future Value of An AnnuityDef: the sum of the future value of all n payments.

Periodic payment of an Annuity

Future Value of An Annuity Due

r

rRS

n

1)1(

1)1( n

r

SrR

1

1)1(

r

rRS

n

Examples of Annuities

1. Suppose a man purchase a house with an initial down payment of $20000 and then makes quarterly payments: $2000 at the end of each quarter for six years and $3500 at the end of each quarter for eight more years. Given an interest rate of 6% compounded quarterly. Find the amount the man’s debt and the house’s price!

2. A man makes house payment of $1200 at the beginning of every month. If the man wishes to pay one year’s worth of payment in advance, how much should he pay, provided that r= 6.8%?

Examples of Annuities

3. Suppose you invest in an IRA by depositing $2000 at the end of every year for the next 15 years. If the interest rate is 5.7% compounded annually, how much will you have at the end of the year?

4. Suppose you invest in an IRA by depositing $2000 at the beginning of the year for the next 15 years. If the interest rate is 5.7% compounded annually, how much will you have at the end of the year?

Amortization of Loans• Amortization ScheduleSuppose that a bank lends a borrower $1500

and charges interest at the nominal rate of 12% compounded monthly.

period Principal outstanding at the beginning of period

Interest for period

Payment at the end of period

Principal repaid at the end of period

1 $1500 $15 $510.03 $495.03

2 1004.97 10.05 510.03 499.98

3 504.99 5.05 510.03 504.99

Total 30.10 1530.10 1500

Amortization of Loans• Amortization Formula1. Periodic Payment :

2. Principal outstanding at the beginning of the kth period :

3. Interest in kth period : i * Sk-1

4. jumlah periode (n)

nr

rAR

)1(1

r

rRA

kn 1)1(1

)1log(

)*log(log

i

JNSiAAn