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Mathematics of Finance[ch5.1-5.4](Sept.13-20).docx

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Raymond Grinnell

Mathematics of Finance(§5.1-5.4) §5.1: Compound Interest Main formula for compound interest is on p.209 S=P(1+r) n P=principal (the amount invested) S=compound interest r=periodic interest rate (amount of interest paid per compounded period) n=number of compounding periods (number of time the interest is paid) -if interest is paid every month it would be 12 We’re often given an APR (Annual % Rate of interest) and a frequency (number of times interest is paid per year, number of compoundings per year). k= frequency a= APR ∴ r= when t is # of years, we have that n=kt ∴ We have this expanded compound and formula S=P(1+ ) kt Ex1) You invest $10K @ 3.05% APR compounding monthly for 5 years. a) Find the compound amount, S. =0.002542 S=10000(1+ )12)(5) =11645.17 b) Compound interest= S-P =11645.17-10000 =1654.17 c) How long to reach $15K We solve for t in the following equation: 15000=10000(1+ )2t ln key (log key) 12t 1.5=(1+ ) 12t ln(1.5)=ln[(1+ ) ] ln(1.5)=12t·ln(1+ ) ∴ t= ≈13.31 years So, we need 13 years and 4 months. ▓ (P>0), (a>0, a decimal),∈IN Ex2) [Example on “Doubling time”, p.210 ex.2&6] We invest $P at an APR compounding k times a year. Solve the compound amount formula for the time to increase m times, m∈IN. So we solve for t where S=mP kt mP=P(1+ ) , t=years ln(m)=ln[(1+ ) ]kt ∴t= Note: this does not depend on P! (I.e. Independent of P) ▓ The Effective Rate Concept (p.211) Consider a period of 1 year. Invest $P @ an APR of r compounding n times for that year. 0 1 year n=4, S=P(1+ ) n The effective rate for interest iser , is the simple rate that gives the same compound amount that would occur above. n re=(1+ ) -1 How did we get this? n n P(1+r e=P(1+ ) P(1+r e=P(1+ ) simple compound Cancel P re=(1+ ) -1 We expect that r er because r hes no intermediate compounding. Ex3) Effective rate is often used to compare interest scheme. Scheme A: 2.95% APR, semi annual Scheme B: 2.93% APR, bi-weekly (a=26) Which is better? Take the larger effective rate! A is better.▓ §5.2: Present and Future Value PV= present value (or past value) kt FV= future value=P(1+ ) P(NOW) t years in the future PV= s(1+ ) = present (or past) value (negative means going back in time) PV is the amount to be unvested t years ago so that in t years we have S We use PV and FV in Equations of Value (p.214-215) An equation of value is an equation that describes payments and debts, both under the assumption of an interest. KEY CONCEPT: At all times, (value of all payments) = (value of all payments) Ex4) You owe $32K and this must be paid in 6 years. You pay 10K now and the balance @ the end of 6 years. Interest is 3% APR, quarterly. Find the ending payment. At time 6 years, (value of all pay)=(value of all debt)=32,000 24 X=10000(1+ ) X≈20,035.86 (Sept 18, Week 2) Equations of Value (cont..) Example: At the end of 7 years, $40,000 debt is to be paid off. We made 3 payments: $10,000 @ the end of year 1, $8,000 @ the end of year 4, and the final payment is made @ the end of year 5. Interest is 2%, compounded semi-annually. Calculate amount of final payment. Let X rep the amount of final payment. Cash-time diagram (10,000) (8,000) (X) 0 1 2 3 4 5 6 (40,000) NOW
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