A U.S. Treasury bond will pay a lump sum of $1,000 exactly 3 years from today. The nominal interest rate is 6%, semiannual compounding. Which of the following statements is CORRECT?

a. The periodic interest rate is greater than 3%.

b. The periodic rate is less than 3%.

c. The PV of the $1,000 lump sum has a higher present value than the PV of a 3-year, $333.33 ordinary annuity.

d. The present value of the $1,000 would be smaller if interest were compounded monthly rather than semiannually.

e. The present value would be greater if the lump sum were discounted back for more periods.

We are given a U.S. Treasury bond that will pay a lump sum of $1,000 in 3 years, with a nominal interest rate of 6% and semiannual compounding. Let's analyze the statements one by one.

### Step 1: Determine the periodic interest rate
Since the nominal interest rate is 6% per year with semiannual compounding, the periodic interest rate is half of the annual nominal rate:

\[
\text{Periodic Interest Rate} = \frac{6\%}{2} = 3\% \text{ per semiannual period.}
\]

### Step 2: Analyze each statement

- **(a) The periodic interest rate is greater than 3%.**  
  **Incorrect**: The periodic interest rate is exactly 3%, not greater.

- **(b) The periodic rate is less than 3%.**  
  **Incorrect**: The periodic interest rate is 3%, not less.

- **(c) The PV of the $1,000 lump sum has a higher present value than the PV of a 3-year, $333.33 ordinary annuity.**  
  To verify, we need to compare the present value of the lump sum and the annuity.  
  Present value of the lump sum (using semiannual compounding):
  \[
  PV = \frac{1000}{(1 + 0.03)^{6}} \approx 1000 \times 0.8375 = 837.48
  \]
  
  The present value of a 3-year $333.33 ordinary annuity (6 periods) at 3% per period:
  \[
  PV = 333.33 \times \left( \frac{1 - (1 + 0.03)^{-6}}{0.03} \right) \approx 333.33 \times 5.417 = 1805.67
  \]
  
  The PV of the annuity is much larger than the PV of the lump sum.  
  **Incorrect**: The PV of the lump sum is lower than that of the annuity.

- **(d) The present value of the $1,000 would be smaller if interest were compounded monthly rather than semiannually.**  
  With monthly compounding, the effective rate would increase, which decreases the present value of the lump sum.  
  **Correct**: If the interest were compounded monthly, the present value would indeed be smaller.

- **(e) The present value would be greater if the lump sum were discounted back for more periods.**  
  **Incorrect**: Discounting back for more periods results in a lower present value, not a greater one.

### Conclusion: 
The correct statement is **(d)**.

Would you like further details or have any questions?

Here are 5 related questions:
1. How does the compounding frequency affect the present value of future cash flows?
2. What is the difference between nominal and effective interest rates?
3. How do we calculate the present value of an annuity?
4. What would be the PV if the interest rate were compounded quarterly instead of semiannually?
5. How do you compare the present value of lump sums versus annuities?

**Tip:** Remember that more frequent compounding leads to a higher effective interest rate, which reduces the present value of future cash flows.

A U.S. Treasury bond will pay a lump sum of $1,000 exactly 3 years from today. The nominal interest rate is 6%, semiannual compounding. Which of the following statements is CORRECT?

a. The periodic interest rate is greater than 3%.
b. The periodic rate is less than 3%.
c. The PV of the $1,000 lump sum has a higher present value than the PV of a 3-year, $333.33 ordinary annuity.
d. The present value of the $1,000 would be smaller if interest were compounded monthly rather than semiannually.
e. The present value would be greater if the lump sum were discounted back for more periods.

Learn how to calculate the present value of a U.S. Treasury bond with a $1,000 future lump sum, semiannual compounding at a 6% nominal interest rate. Understand key concepts such as periodic interest rates, time value of money, and the impact of different compounding frequencies on present value.

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