SOLUTION: Calculate the present value of $5,000 received five years from today if your investments pay a. 6 percent compounded annually b. 8 percent compounded annually c. 10 percent com

Question 1170648: Calculate the present value of $5,000 received five years from today if your investments pay
a. 6 percent compounded annually
b. 8 percent compounded annually
c. 10 percent compounded annually
d. 10 percent compounded semiannually
e. 10 percent compounded quarterly
What do your answers to these questions tell you about the relation between present values and interest rates and between present values and the number of compounding periods per year?

Answer by CPhill(1987) (Show Source): You can put this solution on YOUR website!
Let's calculate the present value of $5,000 received five years from today under different compounding scenarios.
**Formula for Present Value (PV):**
PV = FV / (1 + r)^n
Where:
* PV = Present Value
* FV = Future Value ($5,000)
* r = Interest rate per period
* n = Number of periods
**a) 6% Compounded Annually**
* r = 0.06
* n = 5 years
PV = 5000 / (1 + 0.06)^5
PV = 5000 / (1.06)^5
PV = 5000 / 1.3382255776
PV ≈ $3,736.30
**b) 8% Compounded Annually**
* r = 0.08
* n = 5 years
PV = 5000 / (1 + 0.08)^5
PV = 5000 / (1.08)^5
PV = 5000 / 1.4693280768
PV ≈ $3,402.92
**c) 10% Compounded Annually**
* r = 0.10
* n = 5 years
PV = 5000 / (1 + 0.10)^5
PV = 5000 / (1.10)^5
PV = 5000 / 1.61051
PV ≈ $3,104.61
**d) 10% Compounded Semiannually**
* r = 0.10 / 2 = 0.05 (semiannual interest rate)
* n = 5 years * 2 = 10 periods
PV = 5000 / (1 + 0.05)^10
PV = 5000 / (1.05)^10
PV = 5000 / 1.6288946268
PV ≈ $3,069.50
**e) 10% Compounded Quarterly**
* r = 0.10 / 4 = 0.025 (quarterly interest rate)
* n = 5 years * 4 = 20 periods
PV = 5000 / (1 + 0.025)^20
PV = 5000 / (1.025)^20
PV = 5000 / 1.6386164403
PV ≈ $3,051.35
**Observations**
1. **Relation between Present Values and Interest Rates:**
* As the interest rate increases (from 6% to 10%), the present value decreases. This is because a higher interest rate means that money today can grow to a larger amount in the future, so the present value of a future sum is lower.
2. **Relation between Present Values and Compounding Periods:**
* For a given annual interest rate (10% in this case), as the number of compounding periods per year increases (annually, semiannually, quarterly), the present value decreases. This is because more frequent compounding means that the money earns interest on interest more often, making the future value grow faster, and thus requiring a lower present value.
**Summary**
* Higher interest rates lead to lower present values.
* More frequent compounding leads to lower present values.

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