A investor is hesitating between two projects. The first will yield steady returns of $X$ every $6$ months for the first $10$ years and $X$ every year after. The second will return $500$ per month for $5$ years, then will yield $500$ per year in perpetuity. If the effective annual rate is $10$%, for what $X$ would the investor be indifferent between the two projects?
I am having trouble starting off the problem.
So far I understand that the cash flows for the first project is $X$ for every $6$ months (semi-annual), for the first $10$ years that means $2(10) = 20$ payments for compounding semi-annually, and $X$ every year going on to infinity. This has "semi-annual" interest so $\frac{0.1}{2} = 0.05$
The second project has a cash flow of $500$ per month for $5$ years so it will be $5(12)=60$ months for compoundings and then there is the $500$ per year in perpetuity (goes to infinity). This project has "monthly" interest rate of $\frac{0.1}{12} = 0.833333$
As for timeline(s), I'm guessing you need two? The first project would have a line going from $0$ to $10$ years with 20 intervals of X, and then infinity years afterward. While the 2nd project would have a line from $0$ to $5$ with subintervals of 12 months for sixty months of 1000 each, then after 5 years is 1000 per year to infinity.
$$\require{enclose}$$ Write out the cash flows in terms of the effective monthly rate of interest, which we will call $j$ for now and calculate its value later. The first flow is $$PV = X(v^6 + v^{12} + v^{18} + \cdots + v^{120} + v^{132} + v^{144} + v^{156} + \cdots),$$ where $v = 1/(1+j)$ is the monthly present value discount factor. The second flow is $$PV = 500(v + v^2 + \cdots + v^{60} + v^{72} + v^{84} + v^{96} + \cdots).$$ It is worth noting that both perpetuities contain the payments $$v^{12} + v^{24} + \cdots + v^{60} + v^{72} + \cdots = a_{\enclose{actuarial}{\infty} i}.$$ So we can instead the first flow as $$PV = X\left((v^6 + v^{18} + \cdots + v^{114}) + a_{\enclose{actuarial}{\infty} i}\right) = X\left((1+j)^6 a_{\enclose{actuarial}{10} i} + a_{\enclose{actuarial}{\infty} i}\right).$$ The second flow can be written as $$\begin{align} PV &= 500\left((v + v^2 + \cdots + v^{11}) + v^{12}(v + v^2 + \cdots + v^{11}) + \cdots v^{48}(v + v^2 + \cdots + v^{11}) + a_{\enclose{actuarial}{\infty} i}\right) \\ &= 500\left((1+v^{12} + v^{24} + v^{36} + v^{48})a_{\enclose{actuarial}{11}j} + a_{\enclose{actuarial}{\infty}i} \right) \\ &= 500\left( (1+i)a_{\enclose{actuarial}{5}i}a_{\enclose{actuarial}{11}j} + a_{\enclose{actuarial}{\infty}i} \right). \end{align}$$ Now we have $$(1+j)^{12} = 1+i,$$ where $i = 0.10$, and employing the usual annuity formulas and equating the present values of these flows, we get $$X \approx 500 \frac{(1.10)(3.79079)(10.4914) + (0.10)^{-1}}{(1.04881)(6.14457) + (0.10)^{-1}} \approx 1634.22.$$
This is not the only possible approach. We can work in terms of $i$, in which case the cash flows look like this: $$PV = X(v^{1/2} + v + v^{3/2} + \cdots + v^{10} + v^{11} + v^{12} + \cdots) = X\left((1+i)^{1/2} a_{\enclose{actuarial}{10}i} + a_{\enclose{actuarial}{\infty}i}\right),$$ and $$\begin{align} PV &= 500(v^{1/12} + v^{2/12} + \cdots + v + v^{13/12} + \cdots + v^{59/12} + v^5 + v^6 + v^7 + \cdots ) \\ &= 500\left( (v^{1/12} + v^{2/12} + \cdots + v^{11/12})(1 + v + v^2 + v^3 + v^4) + a_{\enclose{actuarial}{\infty}i} \right) \\ &= 500\left( (1+i)a_{\enclose{actuarial}{5}i} a_{\enclose{actuarial}{11}j} + a_{\enclose{actuarial}{\infty}i}\right).\end{align}$$ We still get the same result, but the intermediate steps look a little different. In the first way, $v = 1/(1+j)$ and the exponents on $v$ represent months, whereas in the second, $v = 1/(1+i)$ and the exponents on $v$ represent years.
Yet another way to perform the computation is to compute the present values as the sum of a annuity-immediate with finite term, and a deferred perpetuity-immediate whose deferral period equals the term of the annuity component; e.g., $$PV = X\left(a_{\enclose{actuarial}{20}j} + v_i^{10} a_{\enclose{actuarial}{\infty} i}\right),$$ where here $j = (1+i)^{1/2} - 1$ is the effective semiannual or $6$-month rate of interest, and $v_i = 1/(1+i)$ is the effective annual present value discount factor. Correspondingly, the second flow is $$PV = 500\left(a_{\enclose{actuarial}{60}k} + v_i^5 a_{\enclose{actuarial}{\infty} i}\right)$$ where $k = (1+i)^{1/12} - 1$ is the effective monthly rate of interest. This approach may be preferable to you since the annuity calculations are simpler, but it uses three rates (monthly, semiannually, and annually) instead of two.