Question 1170648
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.