MATHEMATICS OF FINANCE Adopted from “Introductory Mathematical Analysis for Student of Business and Economics,” (Ernest F. Haeussler, Jr. & Richard S. Paul) Associated Professor Dr. Doğan N. Leblebici

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici WE SHALL USE MATHEMATICS TO MODEL SELECTED TOPICS IN FINANCE THAT DEAL WITH THE TIME-VALUE OF MONEY, SUCH AS INVESTMENTS, LOANS, ETC. PRACTICALLY EVERYONE IS FAMILIAR WITH COMPOUND INTEREST, WHEREBY THE INTEREST EARNED BY AN INVESTED SUM OF MONEY (OR PRINCIPAL -capital sum ) IS REINVESTED SO THAT IT TOO EARNS INTEREST. THAT IS, THE INTEREST IS CONVERTED (OR COMPOUNDED) INTO PRINCIPAL AND HENCE THERE IS "INTEREST ON INTEREST."

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici FOR EXAMPLE, SUPPOSE A PRINCIPAL OF YTL 100 IS INVESTED FOR TWO YEARS AT THE RATE OF 5 PERCENT COMPOUNDED ANNUALLY. AFTER ONE YEAR THE SUM OF THE PRINCIPAL AND INTEREST IS (100) = YTL 105. THIS IS THE AMOUNT ON WHICH INTEREST IS EARNED FOR THE SECOND YEAR, AND AT THE END OF THAT YEAR THE VALUE OF THE INVESTMENT IS (105) = YTL THE YTL REPRESENTS THE ORIGINAL PRINCIPAL PLUS ALL ACCRUED INTEREST; IT IS CALLED THE ACCUMULATED AMOUNT OR COMPOUND AMOUNT.

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici THE DIFFERENCE BETWEEN THE COMPOUND AMOUNT AND THE ORIGINAL PRINCIPAL IS CALLED THE COMPOUND INTEREST. IN THE ABOVE CASE THE COMPOUND INTEREST IS = YTL

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici MORE GENERALLY, IF A PRINCIPAL OF P YTL IS INVESTED AT A RATE OF 100r PERCENT COMPOUNDED ANNUALLY (FOR EXAMPLE, AT 5 PERCENT, r IS.05), THEN THE COMPOUND AMOUNT AFTER ONE YEAR IS P + Pr OR P(1 + r). AT THE END OF THE SECOND YEAR THE COMPOUND AMOUNT IS P(1 +r) + [ P(1 +r)]r = P(1 + r)[1 + r] [FACTORING] = P(1 + r) 2

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici SIMILARLY, AFTER THREE YEARS THE COMPOUND AMOUNT IS P(1 + r) 3. IN GENERAL, THE COMPOUND AMOUNT S OF A PRINCIPAL P AT THE END OF n YEARS AT THE RATE OF r COMPOUNDED ANNUALLY IS GIVEN BY S = P(1 + r) n

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 1 IF YTL 1000 IS INVESTED AT 6 PERCENT COMPOUNDED ANNUALLY, A.FIND THE COMPOUND AMOUNT AFTER TEN YEARS. B.FIND THE COMPOUND INTEREST AFTER TEN YEARS.

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 1 FIND THE COMPOUND AMOUNT AFTER TEN YEARS. WE USE EQ. ( S = P(1 + r) n ) WITH P = 1000, r =.06, AND n = 10. S = 1000(1 +.06) 10 = 1000(1.06) 10. WE FIND THAT (1.06)10 AS THUS, S ≈ 1000( ) ≈ YTL

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 1 FIND THE COMPOUND INTEREST AFTER TEN YEARS. USING THE RESULT FROM PART (A), WE HAVE COMPOUND INTEREST = S — P = = YTL

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 2 SUPPOSE THE PRINCIPAL OF YTL 1000 IN EXAMPLE 1 IS INVESTED FOR TEN YEARS AS BEFORE, BUT THIS TIME THE COMPOUNDING TAKES PLACE EVERY THREE MONTHS (THAT IS, QUARTERLY) AT THE RATE OF 1.5 PERCENT PER QUARTER. THEN THERE ARE FOUR INTEREST PERIODS OR CONVERSION PERIODS PER YEAR, AND IN TEN YEARS THERE ARE 10(4) = 40 CONVERSION PERIODS. THUS THE COMPOUND AMOUNT WITH R =.015 IS 1000(1.015) 40 ≈ 1000( ) ≈ YTL

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 3 THE SUM OF YTL 3000 IS PLACED IN A SAVINGS ACCOUNT. IF MONEY IS WORTH 6 PERCENT COMPOUNDED SEMIANNUALLY, WHAT IS THE BALANCE IN THE ACCOUNT AFTER SEVEN YEARS? (ASSUME NO OTHER DEPOSITS AND NO WITHDRAWALS.)

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 3 HERE P = 3000, THE NUMBER OF CONVERSION PERIODS IN 7(2) = 14, AND THE RATE PER CONVERSION PERIOD IS.06/2 =.03. BY EQ. ( S = P(1 + r) n ) WE HAVE S = 3000(1.03) 14 ≈ 3000( ) ≈ YTL

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 4 HOW LONG WILL IT TAKE FOR YTL 600 TO AMOUNT TO YTL 900 AT AN ANNUAL RATE OF 8 PERCENT COMPOUNDED QUARTERLY?

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 4 THE RATE PER CONVERSION PERIOD IS.08/4 =.02. LET N BE THE NUMBER OF CONVERSION PERIODS IT TAKES FOR A PRINCIPAL OF P = 600 TO AMOUNT TO S = 900. THEN FROM EQ. ( S = P(1 + r) n ), 900 = 600(1.02) n, (1.02) n = 900/600 (1.02) n = 1.5. TAKING THE NATURAL LOGARITHMS OF BOTH SIDES, WE HAVE n ln (1.02) = ln 1.5, (Prop. log b m n =nlog b m) THE NUMBER OF YEARS THAT CORRESPONDS TO QUARTERLY CONVERSION PERIODS IS /4 = , WHICH IS SLIGHTLY MORE THAN 5 YEARS AND 1 MONTH.

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici IF YTL 1 IS INVESTED AT A NOMINAL RATE OF 8 PERCENT COMPOUNDED QUARTERLY FOR ONE YEAR, THEN THE YTL WILL EARN MORE THAN 8 PERCENT THAT YEAR. THE COMPOUND INTEREST IS S - P = 1(1.02) 4 – l ≈ = YTL , WHICH IS ABOUT 8.24 PERCENT OF THE ORIGINAL YTL. THAT IS, 8.24 PERCENT IS THE RATE OF INTEREST COMPOUNDED ANNUALLY THAT IS ACTUALLY OBTAINED, AND IT IS CALLED THE EFFECTIVE RATE.

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici FOLLOWING THIS PROCEDURE, WE CAN SHOW THAT THE EFFECTIVE RATE WHICH CORRESPONDS TO A NOMINAL RATE OF r COMPOUNDED N TIMES A YEAR IS GIVEN BY WE POINT OUT THAT EFFECTIVE RATES ARE USED TO COMPARE DIFFERENT INTEREST RATES, THAT IS, WHICH IS "BEST."

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 5 WHAT EFFECTIVE RATE CORRESPONDS TO A NOMINAL RATE OF 6 PERCENT COMPOUNDED SEMIANNUALLY?

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici THE EFFECTIVE RATE IS THE EFFECTIVE RATE IS 6.09 PERCENT.

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 6 TO WHAT AMOUNT WILL YTL 12,000 ACCUMULATE IN 15 YEARS IF INVESTED AT AN EFFECTIVE RATE OF 5 PERCENT?

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 6 SINCE AN EFFECTIVE RATE IS THE ACTUAL RATE COMPOUNDED ANNUALLY, WE HAVE S = 12,000(1.05) 15 ≈ 12,000( ) ≈ YTL 24,

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 7 HOW MANY YEARS WILL IT TAKE FOR A PRINCIPAL OF P TO DOUBLE AT THE EFFECTIVE RATE OF r ?

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 7 LET N BE THE NUMBER OF YEARS IT TAKES. WHEN P DOUBLES, THEN THE COMPOUND AMOUNT S IS 2P. THUS 2P = P(1 +R) N AND SO 2 = (1 + r) n, ln 2= n ln (1 + r). HENCE, FOR EXAMPLE, IF R =.06, THEN THE NUMBER OF YEARS IT TAKES TO DOUBLE A PRINCIPAL IS APPROXIMATELY

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 8 SUPPOSE THAT YTL 500 AMOUNTED TO YTL IN A SAVINGS ACCOUNT AFTER THREE YEARS. IF INTEREST WAS COMPOUNDED SEMIANNUALLY, FIND THE NOMINAL RATE OF INTEREST, COMPOUNDED SEMIANNUALLY, THAT WAS EARNED BY THE MONEY.

MATHEMATICS OF FINANCE COMPOUND INTEREST Associated Professor Dr. Doğan N. Leblebici EXAMPLE 8 LET r BE THE SEMIANNUAL RATE. THERE ARE SIX CONVERSION PERIODS. THUS, 500(1 + r) 6 = , (1 + r) 6 = /500 THUS THE SEMIANNUAL RATE WAS 2.75 PERCENT, AND SO THE NOMINAL RATE WAS 5.5 PERCENT COMPOUNDED SEMIANNUALLY.

MATHEMATICS OF FINANCE PRESENT VALUE Associated Professor Dr. Doğan N. Leblebici SUPPOSE THAT YTL 100 IS INVESTED FOR ONE YEAR AT A RATE OF 6 PERCENT COMPOUNDED ANNUALLY. THEN THE COMPOUND AMOUNT (OR FUTURE VALUE) OF THE YTL 100 IS YTL 106. EQUIVALENTLY, THE VALUE TODAY (OR PRESENT VALUE) OF THE YTL 106 DUE IN ONE YEAR IS $100. WE CAN GENERALIZE THIS CONCEPT. IF WE SOLVE THE EQUATION THAT GIVES COMPOUND AMOUNT, NAMELY S = P(1 + r) n, FOR P, WE GET P = S/(1 + r) n. P=S (1+r) -n

MATHEMATICS OF FINANCE PRESENT VALUE Associated Professor Dr. Doğan N. Leblebici 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 YTL 24,000. IF THE FUND EARNS INTEREST AT THE RATE OF 7 PERCENT COMPOUNDED SEMIANNUALLY, HOW MUCH MONEY SHOULD BE PAID INTO THE FUND INITIALLY?

MATHEMATICS OF FINANCE PRESENT VALUE Associated Professor Dr. Doğan N. Leblebici WE WANT THE PRESENT VALUE OF YTL 24,000 DUE IN 15 YEARS. FROM EQ. (P=S (1+r) -n ) WITH S = 24,000, r =.07/2 =.035, AND n = 15(2) = 30, WE HAVE P = 24,000(1.035) -30 ≈ 24,000( ) ≈ $

MATHEMATICS OF FINANCE PRESENT VALUE Associated Professor Dr. Doğan N. Leblebici EXAMPLE SUPPOSE MR. SMITH OWES MR. JONES TWO SUMS OF MONEY: YTL 1000 DUE IN TWO YEARS AND YTL 600 DUE IN FIVE YEARS. IF MR. SMITH WISHES TO PAY OFF THE TOTAL DEBT NOW BY A SINGLE PAYMENT, HOW MUCH WOULD THE PAYMENT BE? ASSUME AN INTEREST RATE OF 8 PERCENT COMPOUNDED QUARTERLY.

MATHEMATICS OF FINANCE PRESENT VALUE Associated Professor Dr. Doğan N. Leblebici THE SINGLE PAYMENT X DUE NOW MUST BE SUCH THAT IT WOULD GROW AND EVENTUALLY PAY OFF THE DEBTS WHEN THEY ARE DUE. THAT IS, IT MUST EQUAL THE SUM OF THE PRESENT VALUES OF THE FUTURE PAYMENTS. WE HAVE X = 1000(1.02) (1.02) -20 X=1000( ) + 600( ) = = THUS, THE SINGLE PAYMENT DUE NOW IS YTL

MATHEMATICS OF FINANCE PRESENT VALUE Associated Professor Dr. Doğan N. Leblebici EXAMPLE SUPPOSE THAT YOU HAD THE OPPORTUNITY OF INVESTING YTL 4000 IN A BUSINESS SUCH THAT THE VALUE OF THE INVESTMENT AFTER FIVE YEARS WOULD BE YTL ON THE OTHER HAND, YOU COULD INSTEAD PUT THE YTL 4000 IN A SAVINGS ACCOUNT THAT PAYS 6 PERCENT COMPOUNDED SEMIANNUALLY. WHICH INVESTMENT IS THE BETTER ?

MATHEMATICS OF FINANCE PRESENT VALUE Associated Professor Dr. Doğan N. Leblebici LET US CONSIDER THE VALUE OF EACH INVESTMENT AT THE END OF FIVE YEARS. AT THAT TIME THE BUSINESS INVESTMENT WOULD HAVE A VALUE OF YTL 5300, WHILE THE SAVINGS ACCOUNT WOULD HAVE A VALUE OF 4000(1.03) 10 ≈ YTL CLEARLY THE BETTER CHOICE IS PUTTING THE MONEY IN THE SAVINGS ACCOUNT.

MATHEMATICS OF FINANCE PRESENT VALUE Associated Professor Dr. Doğan N. Leblebici EXAMPLE SUPPOSE THAT YOU CAN INVEST YTL 20,000 IN A BUSINESS THAT GUARANTEES YOU THE FOLLOWING CASH FLOWS AT THE END OF THE INDICATED YEARS; YEARCASH FLOW 2YTL 10,000 3YTL 8,000 5YTL 6,000 ASSUME AN INTEREST RATE OF 7 PERCENT COMPOUNDED ANNUALLY AND FIND THE NET PRESENT VALUE OF THE CASH FLOWS.

MATHEMATICS OF FINANCE PRESENT VALUE Associated Professor Dr. Doğan N. Leblebici SUBTRACTING THE INITIAL INVESTMENT FROM THE SUM OF THE PRESENT VALUES OF THE CASH FLOWS GIVES NPV = 10,000(1.07) (1.07) (1.07) ,000 ≈ 10,000( ) ( ) ( ) - 20,000 = ,000 = -YTL NOTE THAT SINCE NPV < 0, THE BUSINESS VENTURE IS NOT PROFITABLE IF ONE CONSIDERS THE TIME-VALUE OF MONEY. IT WOULD BE BETTER TO INVEST THE YTL 20,000 IN A BANK PAYING 7 PERCENT, SINCE THE BUSINESS VENTURE IS EQUIVALENT TO ONLY INVESTING 20, = YTL 19,

MATHEMATICS OF FINANCE ANNUITIES Associated Professor Dr. Doğan N. Leblebici THE SEQUENCE OF NUMBERS 3, 6, 12, 24, 48 IS CALLED A (FINITE) GEOMETRIC SEQUENCE. THIS IS A SEQUENCE OF NUMBERS, CALLED TERMS, SUCH THAT EACH TERM AFTER THE FIRST CAN BE OBTAINED BY MULTIPLYING THE PRECEDING TERM BY THE SAME CONSTANT. IN OUR CASE THE CONSTANT IS 2. IF THE FIRST TERM OF A GEOMETRIC SEQUENCE IS a AND THE CONSTANT IS r, THEN A SEQUENCE OF TERMS HAS THE FORM a, ar, ar 2, ar 3,..., ar n-1 NOTE THAT THE RATIO OF EVERY TWO CONSECUTIVE TERMS IS THE CONSTANT r ; THAT IS, ar/a=r, ar 2 /ar = r, ETC.(a ≠ 0). FOR THIS REASON WE CALL r THE COMMON RATIO. NOTE ALSO THAT THE nTH TERM IN THE SEQUENCE IS ar n-1.

MATHEMATICS OF FINANCE ANNUITIES Associated Professor Dr. Doğan N. Leblebici THE SEQUENCE OF N NUMBERS a, ar, ar 2,..., ar n-1, WHERE a ≠ 0, IS CALLED A GEOMETRIC SEQUENCE WITH FIRST TERM a AND COMMON RATIO r.

MATHEMATICS OF FINANCE ANNUITIES Associated Professor Dr. Doğan N. Leblebici THE SUM OF GEOMETRIC SERIES:

MATHEMATICS OF FINANCE ANNUITIES Associated Professor Dr. Doğan N. Leblebici THE NOTION OF A GEOMETRIC SERIES IS THE BASIS OF THE MATHEMATICAL MODEL OF AN ANNUITY. BASICALLY, AN ANNUITY IS A SEQUENCE OF PAYMENTS MADE AT FIXED PERIODS OF TIME OVER A GIVEN TIME INTERVAL. THE FIXED PERIOD IS CALLED THE PAYMENT PERIOD, AND THE GIVEN TIME INTERVAL IS THE TERM OF THE ANNUITY. AN EXAMPLE OF AN ANNUITY IS THE DEPOSITING OF YTL 100 IN A SAVINGS ACCOUNT EVERY THREE MONTHS FOR A YEAR.

MATHEMATICS OF FINANCE ANNUITIES Associated Professor Dr. Doğan N. Leblebici THE PRESENT VALUE OF AN ANNUITY IS THE SUM OF THE PRESENT VALUES OF ALL THE PAYMENTS. IT REPRESENTS THE AMOUNT THAT MUST BE INVESTED NOW TO PURCHASE THE PAYMENTS DUE IN THE FUTURE. UNLESS OTHERWISE SPECIFIED, WE SHALL ASSUME THAT EACH PAYMENT IS MADE AT THE END OF A PAYMENT PERIOD; THAT IS CALLED AN ORDINARY ANNUITY. WE SHALL ALSO ASSUME THAT INTEREST IS COMPUTED AT THE END OF EACH PAYMENT PERIOD.

MATHEMATICS OF FINANCE ANNUITIES Associated Professor Dr. Doğan N. Leblebici LET US CONSIDER AN ANNUITY OF n PAYMENTS OF R (YTL) EACH, WHERE THE INTEREST RATE PER. PERIOD IS r AND THE FIRST PAYMENT IS DUE ONE PERIOD R(1+r) -n R(1+r) -2 R(1+r) -1 RRRRR n210n-13

MATHEMATICS OF FINANCE ANNUITIES Associated Professor Dr. Doğan N. Leblebici FROM NOW. THE PRESENT VALUE A OF THE ANNUITY IS GIVEN BY A = R(1 + r) -1 + R(1 + r) R(1 + r) -n. THIS IS A GEOMETRIC SERIES OF n TERMS WITH FIRST TERM R(1 + r) -1 AND COMMON RATIO (1 +r) -1. HENCE WE HAVE

MATHEMATICS OF FINANCE ANNUITIES Associated Professor Dr. Doğan N. Leblebici THUS GIVES THE PRESENT VALUE A OF AN ANNUITY OF R (YTL) PER PAYMENT PERIOD FOR n PERIODS AT THE RATE OF r PER PERIOD. THE EXPRESSION [1 — (1 + r) -n ]/r IS DENOTED a n r AND (LETTING R = 1) REPRESENTS THE PRESENT VALUE OF AN ANNUITY OF YTL 1 PER PERIOD. THE SYMBOL a n r IS READ “ a ANGLE n AT r.” THUS, A=Ra n r

MATHEMATICS OF FINANCE ANNUITIES Associated Professor Dr. Doğan N. Leblebici EXAMPLE GIVEN AN INTEREST RATE OF 5 PERCENT COMPOUNDED ANNUALLY, FIND THE PRESENT VALUE OF THE FOLLOWING ANNUITY: YTL 2000 DUE AT THE END OF EACH YEAR FOR THREE YEARS, AND YTL 5000 DUE THEREAFTER AT THE END OF EACH YEAR FOR FOUR YEARS.

MATHEMATICS OF FINANCE ANNUITIES Associated Professor Dr. Doğan N. Leblebici EXAMPLE 2000(1.05) (1.05) (1.05) (1.05) (1.05) (1.05) (1.05) -7 = YTL 20,762.12

MATHEMATICS OF FINANCE AMORTIZATION OF LOANS Associated Professor Dr. Doğan N. Leblebici SUPPOSE A BANK LOANS YOU YTL THIS AMOUNT PLUS INTEREST IS TO BE REPAID BY EQUAL PAYMENTS OF R YTLs AT THE END OF EACH MONTH FOR THREE MONTHS. FURTHERMORE, LET US ASSUME THAT THE BANK CHARGES INTEREST AT THE NOMINAL RATE OF 12 PERCENT COMPOUNDED MONTHLY. ESSENTIALLY, FOR YTL 1500 THE BANK IS PURCHASING AN ANNUITY OF THREE PAYMENTS OF R EACH. USING FORMULA OF THE LAST SECTION (ANNUITIES),

MATHEMATICS OF FINANCE AMORTIZATION OF LOANS Associated Professor Dr. Doğan N. Leblebici THE BANK CAN CONSIDER EACH PAYMENT AS CONSISTING OF TWO PARTS: (1) INTEREST ON THE OUTSTANDING (GERİ ÖDENMEMİŞ) LOAN, AND (2) REPAYMENT OF PART OF THE LOAN. THIS IS CALLED AMORTIZING. A LOAN IS AMORTIZED WHEN PART OF EACH PAYMENT IS USED TO PAY INTEREST AND THE REMAINING PART IS USED TO REDUCE THE OUTSTANDING PRINCIPAL. SINCE EACH PAYMENT REDUCES THE OUTSTANDING PRINCIPAL, THE INTEREST PORTION OF A PAYMENT DECREASES AS TIME GOES ON. LET US ANALYZE THE LOAN DESCRIBED IN THE EXAMPLE.

MATHEMATICS OF FINANCE AMORTIZATION OF LOANS Associated Professor Dr. Doğan N. Leblebici AT THE END OF THE FIRST MONTH, YOU PAY YTL THE INTEREST ON THE OUTSTANDING PRINCIPAL IS.01(1500) = YTL 15. THE BALANCE OF THE PAYMENT, = YTL , IS THEN APPLIED TO REDUCE THE PRINCIPAL. HENCE THE PRINCIPAL OUTSTANDING IS NOW = YTL AT THE END OF THE SECOND MONTH, THE INTEREST IS.01( ) ≈ YTL THUS THE AMOUNT OF THE LOAN REPAID IS = YTL , AND THE OUTSTANDING BALANCE IS = YTL

MATHEMATICS OF FINANCE AMORTIZATION OF LOANS Associated Professor Dr. Doğan N. Leblebici THE INTEREST DUE AT THE END OF THE THIRD AND FINAL MONTH IS.01(504.99)≈ YTL 5.05, AND SO THE AMOUNT OF THE LOAN REPAID IS = YTL HENCE THE OUTSTANDING BALANCE IS = YTL ACTUALLY, THE DEBT SHOULD NOW BE PAID OFF, AND THE BALANCE OF YTL 0.01 IS DUE TO ROUNDING. OFTEN, BANKS WILL CHANGE THE AMOUNT OF THE LAST PAYMENT TO OFFSET THIS. IN THE ABOVE CASE THE FINAL PAYMENT WOULD BE YTL AN ANALYSIS OF HOW EACH PAYMENT IN THE LOAN IS HANDLED CAN BE GIVEN IN A TABLE CALLED AN AMORTIZATION SCHEDULE.

MATHEMATICS OF FINANCE AMORTIZATION OF LOANS Associated Professor Dr. Doğan N. Leblebici

MATHEMATICS OF FINANCE AMORTIZATION OF LOANS Associated Professor Dr. Doğan N. Leblebici THE TOTAL INTEREST PAID IS YTL 30.10, WHICH IS OFTEN CALLED THE FINANCE CHARGE. AS MENTIONED BEFORE, THE TOTAL OF THE ENTRIES IN THE LAST COLUMN WOULD EQUAL THE ORIGINAL PRINCIPAL WERE IT NOT FOR ROUNDING ERRORS.