Morningstar Magazine - October/November 2017 - 54

Strategies

Investing Over the Lifecycle, Part I
Lifecycle models address the problems
of spending and saving.

QUANT U

Paul D. Kaplan

Individual investors need to make many financial
decisions beyond constructing their portfolios.
These include:
1 How much to save and spend during their
working years.
2 How much to spend during their retirement years.
3 When and how much life insurance to buy.
4 When and how much to annuitize financial wealth.
5 How the investment portfolio should change
over time.
Financial economists have developed lifecycle
models to address these issues. In this
series of Quant U, I will present a series of lifecycle
models, each more complete than the previous
one, that addresses these problems for a rational
investor. In this issue, I present a barebones model
in which there are no uncertainties and that
addresses the problems of spending and savings.
The Barebones Model
Assumptions
The barebones model makes the following
assumptions:
1 The investor is going to live T mores years beyond
the current year (0) and knows T.
2 The investor works from the current year (0)
through year L, receiving an annual salary of y.
3 In each year t, the investor spends an amount ct on
consumption. The model solves for c0 , c1 , ... , cT .
4 The investor's preferences regarding
c0 , c1 , ... , cT are described by two parameters:

= the subjective discount rate
= the elasticity of intertemporal
substitution (EOIS)

54

Morningstar October/November 2017

The EOIS parameter, , captures how responsive an
investor is to changes in intertemporal trade-offs,
such as a rise in interest rates. An investor
with a high EOIS would lower current consumption
and raise future consumption in response
to the improved trade-off between consumption
today and consumption tomorrow more than
one with a low EOIS.

5 The capital market provides a single constant rate

of return, r.
The preference parameters, and , require some
explanation. The subjective discount rate, ,
captures the idea that, all other things being equal,
investors prefer consumption today to consumption
in the future. This parameter is a discount
rate, so values for it could be on the order of 2%
to 4%. The assumption that current consumption
is preferable to future consumption is not
only an assumption of this model but is also a
central tenet of economics. The reason that
the owners of capital are willing to forgo current
consumption is the prospect of having greater
consumption in the future. This principle applies to
all intertemporal transactions, including
making loans, buying stocks, and investing in
venture capital.

The parameter is typically between 0 and 1.
EXHIB IT 1 shows a set of indifference curves
for consumption in two periods for three values
of . Each curve shows combinations of
consumption in periods one and two that give the
investor the same level of satisfaction.
At the lowest value for (25%) shown, the
indifference curve is somewhat L-shaped,
indicating that the investor is not very flexible with
regards to the timing of consumption. At
the highest value shown (100%), the indifference
curve is more linear, indicating that the
investor is flexible with regards to the timing
of consumption.
Human Capital
A key concept that emerges from the lifecycle
approach is human capital. The idea of
human capital is that a person's wealth is not just
financial assets but is also the present value of

EXHIBIT 1

Indifference Curves for Different Levels of the Elasticity of Intertemporal
Substitution
= 25%

= 50%

Period 2 Consumption

= 100%

50K
USD
40
30
20
10
10K
20
Period 1 Consumption (USD)
Source: Morningstar.

30

40



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