Your Relationship with the Bank and its Creditors
Since you, as one of the shareholders of the bank, are one of its several owners, bank management works for you .
You, like your fellow shareholders, want a good Return on their capital investment while minimizing their risk. In order to maximize return per risk, bank management needs to allocate your share holder’s equity (also known as equity capital). Changes in the bank’s asset value are reflected in changes in your shareholder’s equity.
Those changes are based on the fundamental principle of accounting:
In order to maximize shareholder return versus risk, one must maximize asset portfolio return versus risk.
The value of your bank portfolio depends on the credit quality of borrowers (creditors) that the bank has lent to. Uncertainty in borrowers’ future credit quality leads to uncertainty in the bank’s future portfolio value.
Bank Portfolio Credit Risk ( ) & Return ( )
Portfolio Value
Expected in 1 Year
Portfolio Value
Today
Probability
Standard Deviation of future portfolio values
Portfolio Risk
Borrowers (creditors) can default on their bank loans causing loss to the Portfolio Value. This is
Portfolio Credit Risk.
Expected Portfolio Return ($)
Expected growth in portfolio value

Bank order: Invest equity capital to maximized Bank Portfolio’s Expected Return /
Risk: Portfolio Sharpe Ratio (Sp):
Objective of Portfolio Optimization and Capital Allocation
Shareholder equity (Capital) can be invested in varying amounts (weights) of loans to various borrowers, e.g. from different countries and industries.
Investing too much in similar borrowers will lead to risk concentration (“too many eggs in a small group of baskets”) and increased risk.
Investing not enough in the high return high risk borrowers may leave money on the table and reduce return (wasted potential). This will also reduce Portfolio Sharpe
Ratio.
The objective of Portfolio Optimization is to calculate the “optimal condition” that maximizes your Portfolio’s Sharpe Ratio.
The objective of Capital Allocation is to incentivize behavior and decisions that will move the bank’s current sub-optimal portfolio towards the Optimal Portfolio, which maximizes
Sp.
The maximum of is where its derivative, with respect to
, the weight of the i th investment, equals zero:

for all i
Therefore, at the maximum
.
If , which means both the risk & the return of the portfolio have been maximized, the following condition must be met for all i (investments) for the portfolio to be optimal:
Simply put, all the Sharpe Ratios for each investment in the portfolio must equal the
Sharpe Ratio of the portfolio.

In this proof, we’ll derive with respect to . Then we’ll use the maximum of (where the partial derivative equals zero) to find the value of .
Chain Rule:
=

Substitute:
To find , we can find work for is easier.
, but plugging in the value we found for back into the
If and then

Now that we have the values of & , we can substitute them to find .
If and
Then
Common denomnator:
We can also find using the values of & yet again.
If and

Then
Common denomnator:
and were equal?
Assume:
=

Again, assume:
=
Substitute:
Then, we need to derive
, set that derivative to equal zero, and find the relative extrema.

If we plug in for and , we can optimize the Sharpe Ratio. Therefore, if you choose to invest equal amounts of money into two facilities that are equal in both risk and return, both the risk and return of the entire portfolio will be optimized.

Just exactly how much is the risk?
Assume
=
Which means, if is the risk of an investment, although
,
= , when optimized, your portfolio has almost 70.7% of the original risk of not diversifying.

But how much is the risk if
Assume
?
= of the original risk before diversification, if

How much of a difference does it make if
Assume is significantly bigger than :
?
=
of the original risk before diversification, if

Or, assume and are almost equal, but not quite:
=
of the original risk before diversification, if

Remember that although diversification does not increase the expected return for the portfolio, it does minimize risk.