To Save or To Spend | A Dilemma

To save for tomorrow or to spend today is a dilemma that no one seems to have been able to solve. I guess it is one of those problems that competes with the prized unsolved problems that still exists in literature. 

Of course there is no one solution that maps to all the people. So, I will assume a persona 

``You are a salaried individual who earned more than what you could spend and has an Indian mindset of saving for the rainy day and have been focusing on saving. You are married. You have kids who have completed their education and "settled" (in Indian lingo) meaning have started earning their living and are not dependent on you (may or may not be married, yet) and neither your family (parents/siblings) are not dependent on you financially. You have retired. You have a corpus and a monthly pension from your work savings (see Money Matters). Now the dilemma is whether To Save or To Spend.''

A Theory

In general, the majority Indian attitude is to save with the hope that your children will benefit from the saved corpus. But in majority of the cases, the children are quite on their own and in no need of your corpus or the children are of the opinion that they will build things on their own like their parents. Putting this two together, in majority of the cases you die rich, meaning you have a good amount of corpus which remain in the bank. Not that it is wasted, your children will pick it up if you have nominated them or have mentioned it as part of your Will. Even if they don't particularly need it.

The essential idea is that while you build your corpus during your pre-retirement days in the way of monthly wages, along similar lines you plan to spend such that the corpus reduces as you age. The only difference, while you know precisely when you will retire, you do not have a clue when you will die! The rest, how much money you need remain uncertain in both the spending and the saving space. That is, probably the catch. Like retirement if one knew when what the ``end of life'' date. 

Fig.1: Depiction of an ideal scenario. Corpus building till retirement and then spending along the lines of end of life. Shorter, spend faster; longer spend slower. 

Things would be much easier, if you knew how long you would possibly live. If you knew that (short, medium, long life) you could spend along the (red, amber, green) lines depicted in Fig 1 respectively. 

The region of dilemma is clearly captured in Fig 2.  If you are conservative in your spend (spend along the green line) but have a short life span (red dot) you leave behind an ``unspent corpus'', on the other hand, you spend aggressively (along the red line) but live longer you end up with a ``period of no corpus''. Both are possible especially given that no one can yet predict the life span of a person.

Fig. 2: The dilemma. If you are conservative in your spend (green line) but have a short life span you leave behind an unused corpus, on the other hand, you spend fast but live longer you end up with no corpus to live the longer life. 

While there no real solution to this dilemma, the best one can do is try and minimize the corpus left behind.  

Retirement planning is fundamentally a problem of consuming a finite corpus over an uncertain lifetime. Spend too conservatively and wealth remains unused; spend too aggressively and one risks outliving one's savings.

Mathematical Formulation

Let

• ${𝑡}_{𝑅}$​: retirement time
• $𝐸$: random end-of-life (EOL)
• $𝛿$: uncertainty in lifespan
• $𝐸\in [\overline{𝐸}-𝛿,\overline{𝐸}+𝛿]$
  
• $𝐶(𝑡)$: corpus remaining at time $𝑡$
  
• ${𝐶}_{𝑅}=𝐶({𝑡}_{𝑅})$: corpus at retirement
• $𝑆(𝑡)$: essential spending (food, healthcare, housing, etc.)
• $𝑃(𝑡)$: discretionary or "splurge" spending
• $𝑟$: portfolio return rate

Then the corpus dynamics are

$$\frac{𝑑𝐶(𝑡)}{𝑑𝑡}=𝑟𝐶(𝑡)-𝑆(𝑡)-𝑃(𝑡)$$

with

$$𝐶({𝑡}_{𝑅})={𝐶}_{𝑅}\mathrm{.}$$

Ideal Objective:

"I want to enjoy my retirement, but I don't want to die with lots of unused money, and I don't want to run out of money before I die."

That can be expressed as

$$𝐶(𝐸)\approx 0.$$

In other words,

$$\underset{𝑃(𝑡)}{\min }\mathrm{\mid }𝐶(𝐸)\mathrm{\mid }$$

subject to

$$𝐶(𝑡)\ge 0,{𝑡}_{𝑅}\le 𝑡\le 𝐸\mathrm{.}$$

Meaning

• corpus at death should be near zero;
• corpus should never become negative while alive.

However, E is unknown!

Let

$$𝐸={𝐸}^{\ast }\pm 𝛿$$

where,

• ${𝐸}^{\ast }$ = expected end of life
• $𝛿$ = uncertainty in longevity

If retirement spending is planned assuming

$$𝐶({𝐸}^{\ast })=0,$$

then:

• If actual death occurs at ${𝐸}^{\ast }-𝛿$, there will be a positive leftover corpus:

$$𝐶({𝐸}^{\ast }-𝛿)>0$$

This corresponds to the unused corpus region in the Fig. 2.

• If actual death occurs at ${𝐸}^{\ast }+𝛿$, the corpus may be exhausted before death:

$$𝐶({𝐸}^{\ast }+𝛿)<0$$

This corresponds to the outliving savings region in Fig. 2.

Thus, the retirement spending dilemma arises because one must choose $𝑃(𝑡)$ today while the actual end of life is uncertain:

$$𝐸={𝐸}^{\ast }\pm 𝛿\mathrm{.}$$

Let

$$𝐿(𝐸)=\mathrm{\mid }𝐶(𝐸)\mathrm{\mid }$$

be the mismatch between corpus and lifespan.

Solving

$$\underset{𝑃(𝑡)}{\min }\mathbb{𝐸}[𝐿(𝐸)]$$

where the expectation is taken over the probability distribution of lifespan.

Equivalently,

$$\underset{𝑃(𝑡)}{\min }\mathbb{𝐸}[\mathrm{\mid }𝐶(𝐸)\mathrm{\mid }]\mathrm{.}$$

Interpretation:

• Positive $𝐶(𝐸)$ means unused corpus.
• Negative $𝐶(𝐸)$ means outliving savings.

In reality, most people fear running out of money much more than leaving some behind.

We can safely define

$$𝐿(𝐸)=𝛼\max (𝐶(𝐸),0)+𝛽\max (-𝐶(𝐸),0)$$

with

$$𝛽\gg 𝛼\mathrm{.}$$

Then

$$\underset{𝑃(𝑡)}{\min }\mathbb{𝐸}[𝐿(𝐸)]\mathrm{.}$$

Interpretation:

• $𝛼$: penalty for dying with money left over.
• $𝛽$: penalty for being alive after the corpus is exhausted.

Effectively it means one has to choose the discretionary spending path $𝑃(𝑡)$ to minimize

$$\mathbb{𝐸}[𝐿(𝐸)],$$

where $𝐿(𝐸)$ penalizes both leftover corpus at death and corpus exhaustion before death. The optimal solution balances the risk of dying rich against the risk of living poor.

Does this really help? I am still trying to figure this out what should P(t) look like!

NB: All graphics produced and the Mathematical formulation has Copilot (GenAI) assistance based on prompts by the author.