Optimal Lifecycle Capital
Imagine for a moment that I have perfect information. I am currently 45 years old, I will retire at age 65 and live to age 95. I earn $100,000 a year which will keep pace with inflation running at an annual clip of 2%. And since I have no bequest motives, there is no point in leaving any money past age 95. Yes, not even a single penny! As of today, I have saved $250,000 in my retirement account. I expect a return of 5%. What should my wealth profile look like?
To solve this problem, we have to see how our wealth grows over time. At t=0 (today), I have $250,000 which will grow to 250,000 $\times$ (1 + 0.05). However, to it I will add my salary, $100,000 $\times$ (1 + 0.02) and subtract my consumption. This consumption is determined in a way such that I end up penniless on my deathbed!
We repeat this process for all of the remaining years.
$$ \begin{aligned} W_0 &= 250000 \\ W_1 &= W_0 \times (1+0.05) + 100000 \times (1+0.02) - C^* \\ W_2 &= W_1 \times (1+0.05) + 100000 \times (1+0.02)^2 - C^* \\ &\vdots \\ W_{20} &= W_19 \times (1+0.05) + 100000 \times (1+0.02)^{20} - C^* \end{aligned} $$Once retired, the income stream has ceased (and yes, I’m assuming that I do not receive any social benefits). $$ \begin{aligned} W_{21} &= W_{20} \times (1+0.05) - C^* \\ &\vdots \\ W_{49} &= W_{48} \times (1+0.05) - C^* \\ W_{50} &= 0 \end{aligned} $$
First, I need to determine the value of $C^*$ (i.e. optimal consumption). Before I do that, some financial mathematics is in order…I’ll assume you are familiar with many of the concepts and I will only focus on the building blocks to answer the above question.
A key concept is the time value of money (TVM). TVM allows cash flows to ‘time travel’. I’m interested in the Present Value of a growing cash flow. Retirement Income Recipes in R calls it the Regular Growth Ordinary Annuity (RGOA):
The above is for a $1 cash flow where $g$ is the growth rate and $\nu$ is the valuation or discount rate. As an example, the present value of an annuity having a cash-flow of $100 over a 20 year period, growing at 2% and discounted at 5% would be $100 \times \text{RGOA(0.02, 0.05, 20)}$ = $1,496 (rounded to the nearest dollar).
From an accounting balance sheet perspective, one of the most basic formula that one encounters:
From a personal finance point of view, we alter the formula and rewrite it as:
Assets consist of present wealth and present value of future income (Human Capital).
Recall from my problem statement that I do not wish to leave any bequest. So in my case, ASSETS = LIABILITIES. My assets are my current wealth and my human capital, i.e the present value of my income (for the next 20 years). My liability is the present value of my consumption (for the next 50 years). This then allows me to solve for $C^*$ .
$$ \begin{aligned} &250000 + 100000 \times \text{RGOA(0.02, 0.05, 20)} = C^* \times \text{RGOA(0.0, 0.05, 50)} \\ \\ &C^* = \frac{250000 + 100000 \times \text{RGOA(0.02, 0.05, 20)}}{\text{RGOA(0.0, 0.05, 50)}} = 95,633\\ \end{aligned} $$Now to answer what my wealth profile looks like, let’s split this problem into two phases: pre-retirement and post-retirement. The post-retirement case is trivial, my wealth profile will simply be the present value at time t of the remaining consumption. In other words,
$$W_t = C^* \times \text{RGOA(0, v, 50-t)}$$In the case of pre-retirement, I will compute the present value of the capital including wages earned until time t less any consumption for the duration of time t. These values are then translated to time t by multiplying by a factor $(1+0.05)^t$ . Therefore, in pre-retirement,
$$ W_t = (250000 + 100000 \times \text{RGOA(0.02, 0.05, t)} - C^* \times \text{RGOA(0,0.05,t)}) (1+0.05)^t $$The Optimal Lifecycle Financial Capital Illustration allows you to change some of the parameters and visualize the corresponding wealth profile.