Chapter 5: The Time Value Of Money
Chapter 5the Time Value Of Money 2017 Pearson Education Inc All Ri
Explain the mechanics of compounding, and bringing the value of money back to the present. Understand annuities. Determine the future or present value of a sum when there are nonannual compounding periods. Determine the present value of an uneven stream of payments and understand perpetuities.
Paper For Above instruction
The concept of the time value of money is fundamental in finance, emphasizing that a sum of money today is worth more than the same sum in the future due to its potential earning capacity. This principle underpins various financial calculations, including compounding, present and future value assessments, annuities, and perpetuities. Understanding these concepts is essential for making informed investment, borrowing, and financial planning decisions.
Mechanics of Compounding
Compounding involves earning interest on previously accrued interest, leading to exponential growth of investments over time. The mechanics can be visualized through timelines, where cash flows are mapped over periods to illustrate how interest accumulates. The core formula for future value (FV) under compounding is:
FV = PV (1 + r)^n
where PV is the present value, r is the interest rate per period, and n is the number of periods. Compounding can occur at various frequencies — annually, semiannually, quarterly, monthly, or even daily — affecting the total amount earned due to different effective interest rates.
Simple vs. Compound Interest
Simple interest is computed only on the original principal, while compound interest considers accumulated interest from prior periods. For example, investing $100 at 6% annual interest yields $6 interest per year under simple interest, whereas compound interest results in increasing interest amounts over time due to interest-on-interest effect, leading to higher total returns.
Future and Present Values
The future value (FV) indicates how much a current sum will grow over time at a specified interest rate, while the present value (PV) discounts a future sum back to today’s dollars, considering the time value of money:
PV = FV / (1 + r)^n
Both calculations are crucial for evaluating investment opportunities, comparing cash flows occurring at different times, and assessing the worth of financial obligations. For instance, if someone expects to receive $500 in ten years, and the discount rate is 6%, the present value is approximately $279.00.
Effect of Nonannual Compounding
When interest is compounded more frequently than annually, we adjust the rate and periods to reflect the number of compounding periods per year. For example, quarterly compounding with a 10% annual rate involves dividing the rate by four and multiplying the number of years by four. This adjustment ensures precise PV and FV calculations, crucial for accurate valuation.
Valuation of Uneven Cash Flows and Perpetuities
Cash flows may not follow predictable patterns; they can be irregular or consist of a mixture of single payments and streams of payments. The present value of such uneven streams is calculated by discounting each cash flow individually.
Perpetuities, which represent infinite series of equal payments, have a simple valuation formula:
PV = PP / r
where PP is the constant payment and r is the discount rate. For example, a perpetuity paying $2,000 annually at a 10% rate is valued at $20,000.
Annuities and Annuities Due
An annuity involves a series of equal payments made at regular intervals. Ordinary annuities assume payments occur at the period's end, while annuities due have payments at the beginning. The future value of an annuity (FVA) and the present value (PVA) can be calculated using specific formulas:
- FVA = PMT [ ((1 + r)^n - 1) / r ]
- PVA = PMT [ 1 - (1 + r)^(-n) ] / r
Adjustments for annuities due involve multiplying the ordinary annuity values by (1 + r), reflecting the earlier occurrence of payments.
Loan Amortization
Amortized loans are repaid through fixed periodic payments that cover interest and reduce principal over time. The amount allocated to interest diminishes with each payment, while the principal repayment increases. An example involves calculating annual payments for a $6,000 equipment loan at 15% over four years, resulting in approximately $2,102 per year.
Comparing Interest Rates and the Effective Annual Rate
Because different loans or investments may have various compounding periods, rates must be standardized using the effective annual rate (EAR) or annual percentage yield (APY). For instance, a quoted rate of 12% compounded monthly yields an EAR of approximately 12.68%, making comparison more accurate across investments.
Practical Applications
These concepts are extensively used in evaluating pension plans, bond investments, loan agreements, and other financial instruments. Accurate valuation of cash flows, consideration of compounding frequency, and understanding of annuities and perpetuities are vital for sound financial decision-making.
Mastery of these tools empowers investors, financial analysts, and corporate managers to forecast growth, assess risks, and optimize financial strategies effectively.
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