Example you borrow 10000 from a friend you repay him

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Example: You borrow $10,000 from a friend. You repay him in two installments of $5,500 at the end of each of the next two years. Your cash flows look like this: -1 0 1 2 Periods O ----|--------------------|--------------------------|-------------------------|--------- $10,000 - $5,500 - $5,500 O What do your friend s cash flows look like? 15
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Simple Interest O Interest that is paid only on the amount originally invested but not on any interest that accrues subsequently. O >Not very common in finance< O Interest payments are given by I =r*P each period, where r is the simple rate of interest O Over n periods, the original amount invested grows to FV n = P + r*P+ r*P + r*P = P + n*r*P = P(1 + n*r) O Example: Firm borrows $100 at 5% simple interest due at the end of 4 years. What amount must firm repay after 4 years? O Timeline: 0 1 2 3 4 5 .. Years --------|----------|----------|----------|----------|----------|-- 100 - (4*0.05*100) =-20 Cash Flows +(-100) » FV 4 = -$100+(-$100)*4*0.05 = -$100*(1+0.05+4) = -$120 16
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iClicker: Simple Interest Example O Firm borrows $100 at 10% simple interest due at the end of 4 years. One year later, the firm borrows another $100 at 8% simple interest for three years. What amount must firm repay after 4 years? a) $218 b) $264 c) $270 d) $272 e) None of the above Draw a timeline first! 0 1 2 3 4 5 __|____|____|____|____|____|_ 100 100 ? Correct answer is b) FV(4) = $100*(1+0.1*4) + $100*(1+0.08*3) = $264 17
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Compound Interest O Interest is not only earned on the amount initially invested but also on any accrued interest. You earn interest on interest O With compound interest, money grows exponentially instead of linearly O Small differences in the rate of return can have large impact on profits over long periods of time 18
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How Compound Interest Works O An Example: Assume you invest P = $100 at an annual rate of interest r = 5% for n = 4 years. You take no money out of your account before the end of year 5. O Draw a timeline first! Then : After one year, at t=1, the future value of P, denoted by FV 1 is given by From the end of year 1 to the end of year two, the $105 will grow at 5%, i.e. After 3 years, you will have: By repeated multiplication, you will find the value after 4 years to be: 19 FV 2 = FV 1 *(1 + r ) = P *(1 + r )*(1 + r ) = P *(1 + r ) 2 = $105*(1.05) = $100*1.05*1.05 = $100*1.05 2 = $110.25 FV 1 = P + rP = P *(1 + r ) = $100*(1.05) = $105 FV 3 = FV 2 ! (1.05) = $100 ! (1.05) 2 ! (1.05) = $100 ! (1.05) 3 = $115.76 FV 4 = FV 3 ! (1.05) = $100 ! (1.05) 3 ! (1.05) = $100 ! (1.05) 4 = 121.55
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General Compound Interest Formula O First rule of time travel O An investment of PV that is invested at a periodic rate of return of r grows to FV n after n periods, i.e. O Where: n = number of time periods for which interest is earned (n does not have to be in years’) r = effective periodic interest rate O Note: Total interest earned is I = FV n - PV simple interest is n*r*PV compound interest is total interest minus simple interest, i.e. FV n – PV - n*r*PV = FV n -PV(1+n*r) 20 FV n = PV *(1 + r ) n
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