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A detailed explanation of the derivations of various interest formulas used in finance. It covers single-payment compound amount, single-payment present worth amount, equal-payment series compound amount, equal-payment series sinking fund, equal-payment series present worth amount, equal-payment series capital recovery amount, uniform gradient series annual equivalent amount, and effective interest rate. Each formula is explained step-by-step with clear derivations and practical examples to illustrate their application in real-world financial scenarios.
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Deriving financial formulas involves understanding the accumulation and discounting processes based on the time value of money. Below are the derivations for each of the formulas: 3.3.1 Single-Payment Compound Amount Formula: [ FV = PV(1 + i)^n ] Derivation:
Using the formula: [ FV = PV(1 + i)^n = 5000 \times (1 + 0.06)^5 ] [ FV = 5000 \times (1.06)^5 \approx 5000 \times 1.3382 = 6691 ] Future Value = $6,691.
Problem: You want to have $10,000 in your savings account 10 years from now. If the bank offers an annual interest rate of 5%, what is the present value you need to deposit today? Solution: Given: Future Value ((FV)) = $10, Interest Rate ((i)) = 5% = 0. Number of Periods ((n)) = 10 years Using the formula: [ PV = \frac{FV}{(1 + i)^n} = \frac{10000}{(1 + 0.05)^{10}} ] [ PV = \frac{10000}{(1.05)^{10}} \approx \frac{10000}{1.6289} = 6139 ] Present Value = $6,139.
Problem: You make an annual deposit of $1,000 into an account that pays 4% interest compounded annually. How much will you have in the account after 8 years? Solution: Given: Equal Payment ((A)) = $1, Interest Rate ((i)) = 4% = 0. Number of Periods ((n)) = 8 years Using the formula: [ FV = A \left[\frac{(1 + i)^n - 1}{i}\right] = 1000 \left[\frac{(1 + 0.04)^8 - 1} {0.04}\right] ] [ FV = 1000 \left[\frac{(1.04)^8 - 1}{0.04}\right] \approx 1000 \left[\frac{1.3686 - 1}{0.04}\right] = 1000 \times 9.215 ]
Future Value = $9,215.
Problem: You want to accumulate $15,000 in 6 years by making annual payments into an account that pays 3% interest compounded annually. What should be your annual payment? Solution: Given: Future Value ((FV)) = $15, Interest Rate ((i)) = 3% = 0. Number of Periods ((n)) = 6 years Using the formula: [ A = FV \left[\frac{i}{(1 + i)^n - 1}\right] = 15000 \left[\frac{0.03}{(1 + 0.03)^6 - 1}\right] ] [ A = 15000 \left[\frac{0.03}{(1.03)^6 - 1}\right] \approx 15000 \left[\frac{0.03}{1.194 - 1}\right] = 15000 \times 0.142 ] Annual Payment = $710.
Problem: You plan to receive $2,000 annually for 5 years from an investment. If the interest rate is 7% per year, what is the present value of this series of payments? Solution: Given: Equal Payment ((A)) = $2, Interest Rate ((i)) = 7% = 0. Number of Periods ((n)) = 5 years Using the formula: [ PV = A \left[\frac{1 - (1 + i)^{-n}}{i}\right] = 2000 \left[\frac{1 - (1 + 0.07)^{-5}} {0.07}\right] ] [ PV = 2000 \left[\frac{1 - (1.07)^{-5}}{0.07}\right] \approx 2000 \left[\frac{1 - 0.713}{0.07}\right] = 2000 \times 4.1 ] Present Value = $8,200.
Problem: A savings account offers a nominal interest rate of 8% compounded quarterly. What is the effective annual interest rate? Solution: Given: Nominal Interest Rate ((i_{\text{nom}})) = 8% = 0. Compounding Periods ((m)) = 4 (quarterly) Using the formula: [ i_{\text{eff}} = \left(1 + \frac{i_{\text{nom}}}{m}\right)^m - 1 = \left(1 + \frac{0.08} {4}\right)^4 - 1 ] [ i_{\text{eff}} = \left(1 + 0.02\right)^4 - 1 \approx (1.02)^4 - 1 = 1.0824 - 1 = 0.0824 ] Effective Annual Interest Rate = 8.24%.