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Chapter 3 Interest

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Simple interest Compound interest Present value Future value Annuity Discounted Cash Flow

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Simple Interest flat rate of interest

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Simple interest Simple interest is when the interest is calculated only on the principal, so the same amount of interest is earned each year.

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$100, 10% p.a. 3 years simple interest I total = P × R × T A = P + I total = P + P × R ×T = P ×(1 + RT) Principal Present Value Total Value Future Value/ Accumulated Value /Maturity Value

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Formula transformation A = P ×(1 + RT) P = R = T = or

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Bills of Exchange Promissory note Used by businesses and government as a form of loan contract over a short period of time. At the end of the period (date of maturity) the principal (face value) of the loan is repayable with interest accrued to that date. Maturity Value(M)=Face Value (F) + Interest(I)

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Bills of Exchange Promissory note Maturity Value(M)=Face Value (F) + Interest(I) I = F × R × T M = F + FRT M = F (1+RT)

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Borrowing Money at Simple Interest $10,000, 10% p.a., simple interest, repay quarterly over two years 1)How much will he pay in total? 2)How much interest is paid together? 3)How much is his quarterly installment? 4)How much interest is paid in each quarter?

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Borrowing Money at Simple Interest $10,000, 10% p.a., simple interest, repay quarterly over two years FV = Payment = I total = $10,000 × 0.1 × 2 I payment = = $1,500 = $2,000 = $250 = $12,000 FV = P (1+RT) I total = P×R×T ① ② ③ ④

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$10,000, 10% p.a., simple interest, repay quarterly over two years

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Compound Interest Interest on Interest

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Compound Interest Paid on the original investment plus any interest previously accrued, and will increase each period as the investment grows.

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$100, 10% p.a. 3 years compound interest compounded annually FV 1 = PV (1+ i) FV 2 = PV (1+ i) (1+ i) FV 3 = PV (1+ i) (1+ i) (1+ i) FV = PV (1 + i) n FV 1 = $100(1+10%) = $110 FV 2 = $100(1+10%)(1+10%) = $121 FV 3 = $100(1+10%)(1+10%)(1+10%)=$133.1 FV 1 = PV(1+i) 1 FV 2 = PV(1+i) 2 FV 3 = PV(1+i) 3

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Interest compounding more than once per annum $5,000 6% p.a. compounding monthly, 2 years FV = PV (1+i) n FV = $5,000 (1+6%/12) 12×2 =$5,635.80 FV = $5,000 (1+6%) 2 =$5,618

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Interest compounding more than once per annum $5,000 6% p.a. compounding monthly, 1 years FV = PV (1+i) n FV = $5,000 (1+6%/12) 12 =$5,308.39(1+6%/12) 12 Nominal interest rate Annual Percentage Rate(APR) Nominal interest rate Annual Percentage Rate(APR) 6% Real interest rate 6%/12 FV = PV (1+i/m) m×n

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Effective Interest Rate (EIR) FV = PV (1+i/m) m (1+i/m) m i e =(1+i/m) m -1 ieie FV = PV (1+i ) 1 (1+i e ) = Effective Annual Rate of Interest(EAR)

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Formula Manipulation FV = PV (1+i) n i = FV = PV (1+i) n (1+i) n = 1 + i = i =

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Formula Manipulation FV = PV (1+i) n n = FV = PV ×(1+i) n lnFV = lnPV + ln(1+i) n lnFV - lnPV = ln(1+i) n lnFV - lnPV = nln(1+i) n = FVIF =

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FV = PV (1+i) n PV = Formula Manipulation FV (1+i) -n

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Further application FV = PV (1+i) n PV = FV (1+i) -n $5,000 now $7,000 in 4 years, 10% p.a., payable quarterly Package 1: Package 2: P1: $5,000 P2: $7,000 × (1+0.1/4) -(4×4) = $4,715.38

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FV = PV (1+i) n t PV = FV (1+i) -n Check Tables Exercises

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Interpolation FVIF = 1.9738 Interest rate = 12% n= 6

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Interpolation FVIF = 3 Interest rate = 10% n?

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Interpolation 0 11 n 1 12 3.1384 3 2.8531 n FVIF

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Interpolation 0 100 x 1 300 600 400 200 x y x 1 = 200

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Interpolation FVIF = 3 Interest rate = 10% n? n = 11.515

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Annuity A series of payments or receipts of a fixed amount for a specific number of periods. Payments are made at fixed intervals.

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Annuity Ordinary annuity Annuity due Deferred annuity Perpetuity

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Ordinary Annuity An ordinary annuity is one in which the payments or cash flows occur at the end of each interest period.

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Deposit $100, end of each month, one year, annual nominal interest of 12% paid per month FVA(Future Value of an Annuity) = … $100(1+1%) 11 +$100(1+1%) 10 + + $100(1+1%) 1 +$100(1+1%) 0

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0 1 2 n-2 n-1 n A A A A A A(1+i) 0 A(1+i) 1 A(1+i) 2 A(1+i) n-2 A(1+i) n-1 + + + + +

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FVA: $100(1+1%) 11 +$100(1+1%) 10 +…+ $100(1+1%) 1 +$100 S ×(1+i) - S = a(1+i) n - a S (1+i-1) = a(1+i) n - a S = S = a + a(1+i) 1 + + a(1+i) n-1 … S ×(1+i) = a(1+i) 1 + + a(1+i) n-1 + a(1+i) n …

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FVA PMT

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FVA Annuity Amount (Sinking Fund) PMT =

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FVA Period(n) n =

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What is the present value of $100 to be received at the end of each month for the next 12 months, nominal interest rate 12% PVA (Present Value of an Annuity)= … $100(1+1%) -1 +$100(1+1%) -2 + +$100(1+1%) -12

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0 1 2 n-1 n A A A A A(1+i) -1 A(1+i) -2 A(1+i) -(n-1) A(1+i) -n + + + +

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S ×(1+i) - S = a - a(1+i) -n S (1+i-1) = a - a(1+i) -n S = S = a(1+i) -1 + a(1+i) -2 + a(1+i) -(n-1) + a(1+i) -n … S ×(1+i) = a + a(1+i) -1 + + a(1+i) -(n-2) + a(1+i) -(n-1) …

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PVA PMT

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Annuity Amount (Periodic repayment) a = PVA

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Period(n) n = -

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Borrowing Money at Compound Interest You borrow $5,000 to be repaid over the next 5 years with equal annual installments. Interest on the loan is 12% p.a. 1)What are the annual repayments? 2)How much will be owing on the loan after the third installment is paid? (principal, interest) 3) If you want to liquidate the loan in the 4th period, how much interest will you save? 4)Calculate the breakdown of interest and principal from the 3rd to the 4th period.

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Borrowing Money at Compound Interest $5,000, 12% p.a., compound interest, repay annually over the next 5 years 1)What are the annual repayments? a= $1,387.05

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$5,000, 12% p.a., compound interest, repay annually over the next 5 years

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Borrowing Money at Compound Interest $5,000, 12% p.a., compound interest, repay annually over the next 5 years 2) How much will be owing on the loan after the third installment is paid? (principal, interest) = $2,344.19 Interest: $2,344.19 ×12% = $281.30 Principal: $1,387.05 - $281.30= $1,105.75

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$5,000, 12% p.a., compound interest, repay annually over the next 5 years

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Borrowing Money at Compound Interest $5,000, 12% p.a., compound interest, repay annually over the next 5 years 3) If you want to liquidate the loan in the 4th period, how much interest will you save? = $2,344.19 Save: $1,387.05×2 - $2,344.19 = $429.91

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$5,000, 12% p.a., compound interest, repay annually over the next 5 years

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Borrowing Money at Compound Interest $5,000, 12% p.a., compound interest, repay annually over the next 5 years More… 4) Calculate the breakdown of interest and principal from the 3rd to the 4th period.

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$5,000, 12% p.a., compound interest, repay annually over the next 5 years

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Annuity Due An annuity due is one in which the payments or cash flows occur at the beginning of each interest period.

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0 1 2 n-2 n-1 n A A A A A A(1+i) 1 A(1+i) 2 A(1+i) n-2 A(1+i) n-1 A(1+i) n FVA (Due) + + + + +

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S = a × FVIFA(i, n) ×(1+i) S = a(1+i) n + a(1+i) n-1 + + a(1+i) 2 + a(1+i) 1 …

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0 1 2 n-1 n A A A(1+i) 0 A(1+i) -1 A(1+i) -2 A(1+i) n-1 PVA (Due) + + + +

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S = a × PVIFA(n, i) × (1+i) S = a(1+i) 0 + a(1+i) -1 + + a(1+i) n-2 + a(1+i) n-1 …

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Deferred Annuity The first payment is deferred for a number of periods. Special case of ordinary annuity

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0 1 2 0 1 2 n-1 n A A A A A(1+i) 0 A(1+i) 1 A(1+i) n-2 A(1+i) n-1 m m+1 m+2 m+n-1 m+n FVA (Deferred) + + + +

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0 1 2 0 1 2 n-1 n A A A A A/(1+i) m+1 m m+1 m+2 m+n-1 m+n A/(1+i) m+2 A/(1+i) m+n-1 A/(1+i) m+n PVA (Deferred) + + + +

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P = P(m+n) –Pm =A × PVIFA(i, m+n) – a × PVIFA(i, m) Pm = A × PVIFA(i, n) = Pm × (1+i) -m Approach 1: Approach 2:

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Perpetuity PVA Where n PVA

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Discounted Cash Flow Discounted cash flow is the result of the effect of time on the outflows and inflows of a financial arrangement (time value of money). NPV (Net Present Value) IRR (Internal Rate of Return Internal Reward Rate)

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Net Present Value It reflects the net income a project can bring.

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End of yearCash ($) 0-$6,000 1$4,000 2$3,000 3-$2,000 4$5,000 Project A is expected to have the following cash flows for it over the next four years. The initial cost is $6,000, followed by an inflow of $4,000 at the end of year 1, then a $3,000 inflow at the end of year 2 and an outflow of $2,000 at the end of year 3 with a final inflow of $5,000 at the end of year 4.

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End of yearCash ($) 0-$6,000 1$4,000 2$3,000 3-$2,000 4$5,000 Given that the cost of capital is 10%, is the project viable?

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CF t = cash flow generated by project in period t (t = 1,2,3, …..,n) I=initial cost of the project n=expected life of the project r=required rate of return (cost of capital) = discount rate

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End of yearCash ($) 0-$20,000 1$11,800 2$13,240 End of yearCash ($) 0-$20,000 1$8,000 2 3 End of yearCash ($) 0-$20,000 1$9,000 2$8,000 3$7,000 Project A: Project B: Project C: Given that the cost of capital is 10%, which project is the most viable?

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Project A: Project B: Project C:

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Internal Rate of Return The highest rate of return a project can reach.

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Company A intends to invest $200,000 to buy cars for rent. The project is expected to have a steady inflow of $122,000 in the coming two years. What is the IRR of the project? Suppose the cost of capital is 10%, is it viable? End of yearCash ($) 0-$200,000 1$122,000 2

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End of yearCash ($) 0-$200,000 1$122,000 2

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Interpolation 0 15% r 1 14% 1.6467 1.6393 1.6257 r PVIFA

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To be specific: 14.35%>10%, the project is viable. Exercise

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