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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The Deep Dive
To solve your investment questions, let’s break things down! For part (a): i) When you compound monthly at 6.71%, the effective growth factor is calculated using the formula: \[ (1 + \frac{0.0671}{12})^{12t} \] The equivalent expression would be: \(1000(1.0671)^{t}\), which means to round to four decimal places, the growth factor is approximately 1.0686. ii) The effective annual rate (EAR) for monthly compounding will be: \[ (1 + \frac{0.0671}{12})^{12} - 1 \approx 0.0702 \text{ or } 7.02\% \] For part (b): i) Compounding continuously typically yields a higher return than monthly compounding, so we expect a higher result in terms of the effective annual yield. ii) The expression for continuous compounding is given by: \[ 1000 e^{0.0671t} \] In approximate terms, you can express this as: \[ 1000(1.0698)^{t} \quad (\text{rounded to four decimal places}) \] iii) The effective annual rate for continuous compounding can be calculated using: \[ e^{0.0671} - 1 \approx 0.0698 \text{ or } 6.98\% \] For part (c): i) Therefore, from part (a), we find that 6.71% compounded monthly is equivalent to 7.02% compounded annually. ii) From part (b), continuous compounding at 6.71% corresponds to approximately 6.98% compounded annually.
