Chapter I Mathematics of Finance

I-1 Interest I-1-01: Simple Interest Let: p = Principal in Riyals r =Interest rate per year t = number of years → The accumulated amount A after t years: A = p + prt = p ( 1 + rt)

Graphing the straight line: A(t) = p + prt

What are the interest and the total accumulated amount after 10 years on a deposit of 2000 Riyals at a simple interest rate of 1% per year? Solution: The accumulated amount A = p + prt The interest paid I = prt where p = 2000, r = 1/100 = 0.01 and t = 10. A = (1/100)(10) = = 2200 I = prt = 200 Example (1)

Graphing the straight line: A(t) = (0.01)t

I-1-02: Compound Interest Let: p = Principal in Riyals r =Interest rate per year t = number of years → The accumulated amount A 1 after 1 year: A 1 = p + pr(1) = p (1 + r) The accumulated amount A 2 after 2 years: A 2 = A 1 + A 1 r(1) = A 1 (1 + r) = p(1 + r) (1 + r)= p(1 + r) 2

The accumulated amount A 3 after 3 years: A 3 = A 2 + A 2 r(1) = A 2 (1 + r) = p(1 + r) 2 (1 + r)=p (1 + r) 3 The accumulated amount A after 4 years: A 4 = A 3 + A 3 r(1) = A 3 (1 + r) = p(1 + r) 3 (1 + r)= p(1 + r) 4 We conclude that the accumulated amount after t years A t = p(1 + r) t

What are the interest and the total accumulated amount after 2 years on a deposit of 2000 Riyals at a compound interest rate of 10% per year? Solution: The accumulated amount A = p(1+ r) t The interest paid I = A – p Where, p = 2000, r = 10/100 = 1 /10 = 0.1 and t = 2 A = 2000[1 + (1/10)] 2 =2000 [ 1 + 2(1)(1/10) + 1/100] = = 2420 The interest paid = 2420 – 2000 = 420 Example (2)

I-1-03: Interest compounded m times a year Let: p = Principal in Riyals r = Interest rate per year m = number of times a year the interest is compounded Conversion period = the period of time between successive interest calculations The interest rate per conversion period = i = r / m t = the number of years (term) n = Number of periods in t years = mt → The accumulated amount A 1 after 1 period: A 1 = p + pi = p (1 + i) The accumulated amount A 2 after 2 periods: A 2 = A 1 + A 1 i = A 1 (1 + i) = p(1 + i) (1 + i)= p(1 + i) 2

The accumulated amount A 3 after 3 periods: A 3 = A 2 + A 2 i = A 2 (1 + i) = p(1 + i) 2 (1 + i)=p (1 + i) 3 The accumulated amount A after 4 periods: A 4 = A 3 + A 3 i = A 3 (1 + i) = p(1 + i) 3 (1 + i)= p(1 + i) 4 We conclude that the accumulated amount after n periods A n = p(1 + i) n = p (1+i) mt = p (1 + r/m ) mt Where, p = the principal r = the interest per year m = the number of times (periods) in a year the interest is compounded

What is the total accumulated amount after 3 years on a deposit of 1000 Riyals at interest rate of 10% per year compounded: 1. semiannually ( 2 periods in a year) 2. quarterly ( 4 periods in a year ) 3. monthly ( 12 periods in a year) 4. daily ( 365 periods in a year) 5. every 4 months ( 3 periods in a year ) 6. every two months ( 6 periods in a year ) 7. annually Solution: In all of these cases, we use the formula A n = p(1 + i) n = p (1+i) mt = p (1 + r/m ) mt Example (3)

1. semiannually ( 2 periods in a year) A = 1000[1 + (0.1)/2 ] 2(3 2. quarterly ( 4 periods in a year ) A = 1000[1 + (0.1)/4 ] 4(3) 3. monthly ( 12 periods in a year) A = 1000[1 + (0.1)/12 ] 12(3)

4. daily ( 365 periods in a year) A = 1000[1 + (0.1)/365 ] 365(3) 5. every 4 months ( 3 periods in a year ) A = 1000[1 + (0.1)/3 ] 3(3) 6. every two months ( 6 periods in a year ) A = 1000[1 + (0.1)/6 ] 6(3) 7. annually ( 1 periods in a year ) A = 1000[ ] 3