If you invest $$ \$ 1000 $$ for $$ t $$ years at $$ 6.72 \% $$ compounded at the frequencies below, find the following. (a) Suppose you compound at $$ 6.72 \% $$ monthly. i) Report an expression equivalent to the value of $$ \$ 1000 $$ invested for $$ t $$ years at $$ 6.72 \% $$ compounded monthly by completing the box with the growth factor if compounded annually. 1000( $$ \square $$ Number $$ t $$ (Round to $$ \underline{4} $$ decimal places.) ii) Report the effective annual rate: $$ \square $$ Number \% (Round to $$ \underline{2} $$ decimal places.) (b) Suppose you compound at $$ 6.72 \% $$ continuously. i) You would expect $$ 6.72 \% $$ compounded continuously to give a $$ \square $$ Click for List yield than what is given in part (a). ii) Complete the boxes below to report the expression for the value of $$ \$ 1000 $$ invested for $$ t $$ years at $$ 6.72 \% $$ compounded continuously and the equivalent growth factor if compounded annually. $$ \begin{array}{l} 1000 e^{(\text {Number } t)} \ \approx 1000(\text { Number })^{t} \end{array} $$ (Round to $$ \mathbf{4} $$ decimal places.) iii) Report the effective annual rate: $$ \square $$ Number \% (Round to $$ \underline{2} $$ decimal places.) (c) Complete the boxes to summarize: i) From part (a) we have that $$ 6.72 \% $$ compounded monthly is equivalent to $$ \square $$ Number $$ \% $$ compounded annually. ii) From part (b) we have that $$ 6.72 \% $$ compounded continuously is equivalent to $$ \square $$ Number $$ \% $$ compounded annually.

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**(a) Monthly Compounding**

Let the nominal annual rate be $$ r = 0.0672 $$. When compounded monthly the periodic rate is  
$$
\frac{r}{12} = \frac{0.0672}{12} = 0.0056.
$$
Then the value after $$ t $$ years is given by  
$$
1000\left( 1 + \frac{0.0672}{12} \right)^{12t} = 1000\left( 1.0056 \right)^{12t}.
$$

*i) Expressing the growth factor as if compounded annually:*

We note that an equivalent annual growth factor is  
$$
\left( 1.0056 \right)^{12}.
$$
Calculating numerically,  
$$
\left( 1.0056 \right)^{12} \approx 1.0693,
$$
so the expression becomes  
$$
1000(1.0693)^t.
$$

*Answer for (a)(i):*  
$$
\boxed{1.0693}
$$

*ii) Effective annual rate (EAR):*

The effective annual rate is found from the annual growth factor:  
$$
\text{EAR} = \left( 1.0056 \right)^{12} - 1 \approx 1.0693 - 1 = 0.0693.
$$
Expressed as a percentage,  
$$
6.93\%.
$$

*Answer for (a)(ii):*  
$$
\boxed{6.93\%}
$$

---

**(b) Continuous Compounding**

When compounded continuously at rate $$ r = 0.0672 $$, the value after $$ t $$ years is expressed as  
$$
1000e^{0.0672t}.
$$

*i) Qualitative comparison:*

Since continuous compounding accumulates interest in the limit of infinite compounding periods, it produces a slightly higher yield than monthly compounding.

*Answer for (b)(i):*  
$$
\boxed{\text{higher}}
$$

*ii) Expressing the growth factor as if compounded annually:*

The equivalent annual growth factor satisfies  
$$
e^{0.0672} \approx 1.0695.
$$
Thus, we can express the value as  
$$
1000e^{0.0672t} \approx 1000(1.0695)^t.
$$

*Answers for (b)(ii):*  
- The exponent in the continuous form: $$\boxed{0.0672}$$.  
- The equivalent annual growth factor: $$\boxed{1.0695}$$.

*iii) Effective annual rate:*

The effective annual rate for continuous compounding is given by  
$$
e^{0.0672} - 1 \approx 1.0695 - 1 = 0.0695,
$$
or as a percentage,
$$
6.95\%.
$$

*Answer for (b)(iii):*  
$$
\boxed{6.95\%}
$$

---

**(c) Summary**

*i) From part (a):*

$$6.72\%$$ compounded monthly is equivalent to an effective annual rate of approximately  
$$
\boxed{6.93\%}
$$
compounded annually.

*ii) From part (b):*

$$6.72\%$$ compounded continuously is equivalent to an effective annual rate of approximately  
$$
\boxed{6.95\%}
$$
compounded annually.
**(a) Monthly Compounding** - **i) Growth Factor:** $$ 1000 \times (1.0693)^t $$ - **ii) Effective Annual Rate:** $$ 6.93\% $$ **(b) Continuous Compounding** - **i) Yield Comparison:** Higher than monthly compounding. - **ii) Growth Factor:** $$ 1000 \times (1.0695)^t $$ - **iii) Effective Annual Rate:** $$ 6.95\% $$ **(c) Summary** - **i) Monthly Compounding Equivalent Annual Rate:** 6.93% - **ii) Continuous Compounding Equivalent Annual Rate:** 6.95%

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