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Unformatted text preview: Profitability Analysis (including Time Value of Money) Class 15 Time Value of Money Interest the compensation paid for the use of borrowed capital Principal amount of capital on which interest is paid Rate of Interest , i  amount of interest earned by a unit of principal in a unit of time Simple Interest P = Principal; n = number of interest periods; i = interest rate based on length of one interest period; I = amount of interest during n periods; S = principal + interest due after n periods I = (P)(i)(n) S = P + I = P[1 + (i)(n)] Compound Interest Interest is due regularly at the end of each interest period. If payment is not made, the amount due is added to the principal, and interest is charged on this converted principal during the following time unit S = P(1 + i) n Nominal & Effective Interest Rates As expected, the interest rate that a bank would pay using continuous compounding is different from that for quarterly compounding. Similarly, the rate for quarterly compounding is different from the rate for semiannual compounding. Hence, we need to find a way of comparing these various rates Suppose a bank pays 1.5% interest per quarter and compounds the interest 4 times a year. In this case, the nominal interest rate is 6%/year, compounded quarterly. It is essential to include the compounding interval in the description of the interest, because we expect that the effective interest rate on an annual basis will be > 6% r = nominal interest rate m = corresponding interest intervals/year Effective annual interest rate = i eff = (1 + r/m) m – 1 = i eff = (1 + .06/4) 4 – 1 = 0.0614 or, i = 6.14% Continuous Interest Compounding S = (P)(e rn ) = P(1 + i eff ) n i eff = e r 1 r = ln(i eff + 1) Principal # of years Nominal interest rate If the nominal interest rate is 6%/yr, find the value of a $100 deposit after 10 years with: (a) continuous compounding S = (P)(e rn ) = 100 e 0.06(10) = $182.21 (b) daily compounding S = P(1 + r/m) mn = 100 (1 + 0.06/365) 365(10) = $182.20 (c) semiannual compounding S = P(1 + r/m) mn = 100 (1 + .06/2) 2(10) = $180.61 (d) the effective annual interest rate for continuous compounding i eff = e r 1 = e 0.06 1 = 0.0618 Annuity A series of payments occurring at equal time intervals. Annuities are used to pay off debt, accumulate capital, or receive a lump sum of capital that is due in periodic installments. Encountered by engineers in depreciation calculations, where the decrease in value of equipment is accounted for by an annuity plan Discrete Case (Periodic) (Ordinary Annuity) R = uniform periodic payment n = number of discrete payment periods i = interest rate based on payment period S = amount of annuity; or F, future worth (S)(i) = R(1 + i) n R S = R [ ] (1 + i) n 1 i [Discrete uniform series compound amount factor ] Continuous Compounding R = total of all ordinary annuity payments occurring regularly and uniformly r = nominal interest rate m = interest periods per year...
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This note was uploaded on 11/14/2011 for the course CHEN 4520 taught by Professor Wiemer during the Fall '11 term at Colorado.
 Fall '11
 Wiemer

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