# Financial math- Calculating Perpetuity problem

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.

• You cannot just divide the effective interest rate by 12 to get the monthly effective rate Feb 15, 2021 at 3:17
• Look at the explanation of the difference between nominal annual rates and effective annual I gave in answer to your previous question. Feb 15, 2021 at 3:40
• I am rather concerned that you are unable to distinguish between effective and nominal rates of interest in a question for which the equations of value are much more sophisticated. This suggests to me that you lack the prerequisite knowledge or foundational understanding to proceed further in actuarial mathematics. Feb 15, 2021 at 4:14
• I'm taking an elective course, introductory to basic principles of financial mathematics. I don't know why I'm making it more confusing to understand. Maybe it's because I am not used to the notation used, differentiate between formulas, or understand the terminology in word problems. Feb 15, 2021 at 6:25
• @comp890 It is not my intention to discourage you. The concern I have is that you seem to be progressing in the complexity of concepts in financial mathematics but apparently continuing to struggle with earlier concepts, which is likely to cause you much more frustration should you begin to study other financial instruments such as bonds and stock option pricing. I recommend review of earlier concepts and additional practice. Feb 15, 2021 at 7:39

$$\require{enclose}$$ 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.

• I thought the nominal interest rate was i=m*(1+i)^(1/m)−1 while the effective interest rate is i= (1+i/m)^(m)−1. investopedia.com/terms/n/…. Feb 15, 2021 at 5:50
• @comp890 Please refer to the following: en.wikipedia.org/wiki/Actuarial_notation#Interest_rates I should also point out that nominal rates are not applicable to this question. What you need are effective rates for different intervals of time; e.g., effective semiannual rates, effective monthly rates, etc. Feb 15, 2021 at 7:31
• I see. Thanks for clarifying Feb 15, 2021 at 22:36
• you mentioned that v = 1/(1+j). how did you get (1+j)^6 ? Feb 15, 2021 at 23:19
• @comp890 \require{enclose} \begin{align}v^6 + v^{18} + v^{30} + \cdots + v^{114} &= v^{-6}(v^{12} + v^{24} + v^{36} + \cdots + v^{120}) \\ &= (1+j)^6 a_{\enclose{actuarial}{10}i} \end{align} Feb 16, 2021 at 0:16

If a payment is made and compounded m times the effective interest rate is $$r=(1+i_m)^m-1$$. We can solve the equation for $$i_m=\sqrt[m]{r+1}-1$$. That means in the case of semianully payments $$i_2=\sqrt[2]{1.1}-1\approx 0.0488088$$. And in the case of monthly payments $$i_{12}=\sqrt[12]{1.1}-1\approx 0.00797414$$

The present value of $$n$$ payments in the amount of $$X$$ and a interest rate of $$i_m$$ (in one period) is

$$PV_n=X\cdot \frac{(1+i_m)^n-1}{i_m\cdot (1+i_m)^n}$$

The present value of a perpetuity in the amount of $$X$$ and a interest rate of $$i_m$$ (in one period) is

$$PV_{\infty}=X\cdot \frac{1}{i_m}$$

This preliminary considerations lead me to the following equation:

$$X\cdot \frac{1.0488088^{20}-1}{0.0488088\cdot 1.0488088^{20}}+\frac{X}{0.1\cdot 1.0488088^{20}}$$ $$=500\cdot \frac{1.007974^{60}-1}{0.007974\cdot 1.007974^{60}}+\frac{500}{0.1\cdot 1.007974^{60}}$$

I get $$X\approx 1634.22$$

• You are calculating the perpetuity portion of the present value as if the payment frequency were unchanged. This is not the case. Once the semiannual and monthly payments are finished after $10$ and $5$ years, respectively, the payment frequency changes to annual. If you simplify your equation, noticing that the sums on each side are over a common denominator, your equation of value is $$\require{enclose} X a_{\enclose{actuarial}{\infty} i_2} = 500 a_{\enclose{actuarial}{\infty} i_{12}},$$ which results in $$X/0.0488088 = 500/0.007974.$$ This is not the intent of the question. Feb 15, 2021 at 8:37
• Yes, I changed the interest rate to the annual interest rate. Thanks for your comment. Feb 15, 2021 at 8:48
• Please check your computation again? I typed exactly what you wrote in your latest edit and still got $X = 1634.22$. Feb 15, 2021 at 8:56
• Thanks again, for controlling my question. Good job. Now our results are the same. I had a typo at the calculator. Feb 15, 2021 at 9:03