If you invest $$ \$ 1000 $$ for $$ t $$ years at $$ 6.71 \% $$ compounded at the frequencies below, find the following. (a) Suppose you compound at $$ 6.71 \% $$ monthly. i) Report an expression equivalent to the value of $$ \$ 1000 $$ invested for $$ t $$ years at $$ 6.71 \% $$ 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 $$ 6.71 \% $$ continuously. i) You would expect $$ 6.71 \% $$ 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 $$ 6.71 \% $$ 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 $$ \underline{4} $$ 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 6.71 \% compounded monthly is equivalent to $$ \square $$ Number $$ \% $$ compounded annually. ii) From part (b) we have that 6.71 \% compounded continuously is equivalent to $$ \square $$ Number $$ \% $$ compounded annually.

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(a) For monthly compounding, the general formula is
$$
1000\left(1+\frac{0.0671}{12}\right)^{12t}.
$$
(i) We can write this in the form
$$
1000\Bigl[(1+\text{annual effective rate})\Bigr]^t.
$$
That is,
$$
1000\left(1+\frac{0.0671}{12}\right)^{12t} = 1000\bigl( (1.0056)^{12} \bigr)^t,
$$
where 
$$
(1.0056)^{12} \approx 1.0692.
$$
Thus the expression is
$$
1000\,(1.0692)^t.
$$

(ii) The effective annual rate (EAR) is given by
$$
\text{EAR}=\left(1+\frac{0.0671}{12}\right)^{12}-1\approx 1.0692-1=0.0692,
$$
which is $$6.92\%$$ (rounded to 2 decimal places).

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(b) For continuous compounding, the formula is
$$
1000\, e^{0.0671\,t}.
$$

(i) Since
$$
e^{0.0671} \approx 1.0694,
$$
which is slightly larger than $$1.0692$$ from (a), continuous compounding gives a higher yield than monthly compounding.

(ii) We rewrite the continuously compounded amount in the form
$$
1000\, e^{0.0671 t} \quad\approx\quad 1000\,(1.0694)^t.
$$
Thus, the equivalent growth factor if compounded annually is approximately $$1.0694$$.

(iii) The effective annual rate for continuous compounding is
$$
e^{0.0671}-1\approx 1.0694-1=0.0694,
$$
or $$6.94\%$$ (rounded to 2 decimal places).

---

(c) Summary

(i) From part (a):  
$$6.71\%$$ compounded monthly is equivalent to an effective annual rate of approximately $$6.92\%$$.

(ii) From part (b):  
$$6.71\%$$ compounded continuously is equivalent to an effective annual rate of approximately $$6.94\%$$.
(a) (i) The expression is $$1000 \times 1.0692^t$$. (ii) The effective annual rate is $$6.92\%$$. (b) (i) Continuous compounding gives a higher yield than monthly compounding. (ii) The expression is $$1000 \times 1.0694^t$$. (iii) The effective annual rate is $$6.94\%$$. (c) (i) $$6.71\%$$ compounded monthly is equivalent to $$6.92\%$$ compounded annually. (ii) $$6.71\%$$ compounded continuously is equivalent to $$6.94\%$$ compounded annually.

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