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Brown deposits $5000$ in a $5$-year investment that pays interest quarterly at $8\%$, compounded quarterly. Upon receipt of each interest payment, Brown reinvests the interest in an account that earns $6\%$, compounded quarterly. Determine Brown’s yield rate over the $5$ year period, as a nominal interest rate compounded quarterly.

Solution: We solve the time value of money equation for $i^{(4)}$

$$5000\left(1+\frac{i^{(4)}}{4}\right)^{20} = 5000 + 5000(0.02)s_{\bar{20}|0.015}$$

Can someone explain what the phrase "yield rate ... as a nominal interest rate compounded quarterly" means? And why setting up the equation gives yield rate?

The yield rate as I know it is defined as interest rate $i$ s.t $$ NPV(i) = \sum_{i=0}^n v^{t_i}c_{t_i} $$ where $c_{t_i} = R_{t_i} - C_{t_i}$, the net cash flow for time $t_i$ (in the investor's perspective) How is that definition equivalent to the equation set up?

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    $\begingroup$ For this and future relevant questions, consider adding the "actuarial science" tag. $\endgroup$
    – heropup
    Commented Aug 20, 2020 at 6:24

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"Yield rate" in this context refers to the equivalent annual effective rate of interest compounded quarterly that would result in the same accumulated value for the investment scheme. In other words, if $AV$ is the accumulated total value of the investment (plus reinvestment) after $5$ years, then the yield is the value of $i^{(4)}$ such that $5000(1+i^{(4)}/4)^{20} = AV$.

For a simpler example, suppose I have $12000$ to invest, so I deposit $100$ at the end of every month for $10$ years at a nominal rate of $i^{(12)} = 6\%$ compounded monthly. If instead I had invested the entire principal of $12000$ at the beginning, what is the nominal annual rate of interest compounded quarterly that would yield the same accumulated value? This gives the equation of value $$\require{enclose} 12000 \left(1 + \frac{i^{(4)}}{4}\right)^{40} = 100 s_{\enclose{actuarial}{120} 0.005}.$$ This gives $i^{(4)} \approx 3.12856\%$. The reason why it's so much smaller than $i^{(12)}$ is because the annuity only earns interest on the portion contributed, whereas if invested from the beginning, the entire principal earns interest for the full duration.

Returning to your case, here we have a situation where the cash flow is somewhat different: the $5000$ is invested into a fund that pays back $5000(0.02) = 100$ every quarter in interest. Note that a nominal annual rate of $8\%$ compounded quarterly means $2\%$ interest every quarter, and $2\%$ of $5000$ is $100$. So the investment is giving Brown $100$ in interest at the end of every quarter, which he then reinvests at a nominal rate of $6\%$ compounded quarterly. Thus, the stream of interest payments he is receiving amounts to an annuity-immediate, whose accumulated value at the end of $5$ years is $$\require{enclose} 100 s_{\enclose{actuarial}{20} 0.015}.$$ But this is only the accumulated value of the interest payments when reinvested--it doesn't include the original $5000$ investment. So the total accumulated value is $5000$ plus the annuity.

An important note: you do not write $$\require{enclose} 5000\color{red}{(1.02)^{20}} + 100 s_{\enclose{actuarial}{20} 0.015}.$$ The red portion is wrong because once Brown takes the earned interest and reinvests it, that money is counted in the annuity. It doesn't accrue more interest at the $8\%$ nominal rate. So the principal portion of the total accumulated value does not have any adjustment for accrued interest, because you already took that into account with the reinvestment.

Your formula is for the net present value of a series of cash flows. It is a generalization of the present value of an annuity, in which the time periods over which cash is exchanged can be arbitrary and flows may occur in either direction (positive or negative). For example, you could have the following cash flow:

$$\begin{array}{c|c|c|c|c|c|c|c} i & 0 & 1 & 2 & 3 & 4 & 5 & 6 \\ \hline t_i & 0 & 2.1 & 3.3 & 5 & 7 & 11.2 & 20 \\ R_{t_i} & 100 & 250 & 115 & 0 & 350 & 44 & 500 \\ C_{t_i} & 0 & 0 & 35 & 60 & 0 & 105 & 350 \\ \end{array}$$

And if $i = 0.07$, you could calculate $v = (1.07)^{-1}$ and then input these values into the formula to get the NPV, which would be the present value of all of the revenues minus the costs.

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