What is Immunization and Explain its 3 conditions ?

IMMUNISATION:
Definition:
Immunisation is a strategy of managing a Portfolio of Assets such that business is immune to interest rate fluctuations. In other words, it is a process where an investment manager select an asset portfolio in such a way that his surplus (Present Value of Asset-Present Value of Liabilities) is protected against change in interest rate.

Background:
During early 1900’s we usually saw our portfolio changes due to changes in our cashflows but then there was an increase in interest rate volatility due to which we started seeing the change/impact on our portfolio due to change in interest rate.

3 conditions:
1)VA(i0) = VL(i0) which means Present Value of Assets = Present Value of Liabilities.

Suppose we have to pay Rs.10,000 after 2 years, so what we can we do is purchase 2 year Zero coupon bond whose maturity value will be Rs.10,000, which means that after 2 years proceeds from bond helps in paying our liabilities. So now if discount rate is same in both the cases, our Present Value will be same for both Assets and Liabilities. Thus, our Fund is immune.
Note: Here the Surplus is zero as PVA - PVL = 0 , which further means that at i0 Surplus is zero.
So, question is what will be the impact on Surplus when there is Change in interest rate.

2) VA(i0) = VL(i0) which means Volatilities of asset and liabilities cashflow series are equal or we can say that DMT (or Duration) of both should be same

Now as we know that decrease in interest rate leads to increase in Present Value of Assets and Liabilities. Increase in interest rate leads to decrease in Present Value of Assets and Liabilities.
But the question is which will have more impact. So we are saying here that whatever be the change in interest rate, it will have same impact on both assets and liabilities that’s why we are saying that volatility of both assets and liabilities should be same.

So now present value of both is same and volatilities are also same. Now question is what will happen if there is a cash outflow and then there is cash inflow after some period of time and at the time of cash outflow we don’t have enough money. Now, let’s see the third Condition.

3)VA(i0) >VL(i0) which means Convexity of Assets has to be greater than convexity of Liabilities.

Here we are saying that cash inflow series is more spread out than cash outflow series.
It means that bonds will always have a higher value than the liability, even if the interest rate changes. This means that our portfolio of bonds will always sell for enough money to cover the liability.

Limitations of Immunization:
1.     It is only valid for sufficiently small change in interest rate (as we ignored third-order and higher-order derivatives)
2.     The value of our portfolio of assets changes over time, so we need to rebalance the portfolio to continue satisfying the conditions for Redington immunization.
3.     This theory assumes flat yield curve and requires same change in interest rate at all terms, in practice it is rarely the case.
4.     Immunisation removes likelihood of making profits.