"To immunize a portfolio’s target value or target yield against a change in the market yield, a manager must invest in a bond or a bond portfolio whose (1) duration is equal to the investment horizon and (2) initial present value of all cash flows equals the present value of the future liability. Thus, investing in a bond portfolio with a yield to maturity equal to the target yield and a maturity equal to the investment horizon does not assure that the target value will be achieved because of reinvestment risk.

The presence of coupons introduces reinvestment risk. If the duration of the bond is lower than that of the assets, it means you have lower reinvestment income, and your offsetting gain in value is small. So immunization is suboptimal in this case.

Does " initial present value of all cash flows" mean that coupons are discounted at a certain reinvestment rate which could either go up or down in the future?

That is how I understand it. Duration matching assumes that the duration of both the asset and liability doesn’t change over the life of the immunization period, which is certainly not always the case. To the extent interest rates change, your initial net present value amount will also change and may no longer match due to the coupons having to be reinvested at a different rate. The only sure way to get around this is to use cash flow matching (which is more expensive and costly/difficult to implement) or zero-coupon bonds instead of coupon bonds.

Using massive yield numbers for illustration purposes:

If I owe you $150 in one year, I could buy $100 of annual coupon bonds yielding 50% and have enough to pay you back in full (effectively a zero-coupon bond).

However, I could also buy $100 of semiannual bonds yielding 45%, in which case at time t=0.5, I’d receive a coupon of $22.5, which when reinvested would be worth (22.5*(1+(.45/2)) = $27.5 at time t=1.

$27.5 plus my year-end coupon of $22.5 plus $100 = $150

That said, assume my midyear reinvestment rate drops to 10%. Now my $22.5 t=0.5 coupon is only worth $23.6 at time t=1, which combined with my year-end coupon of $22.5 plus my principal of $100 equals only $146.1, and I have a shortfall.

The implicit assumption with any coupon-paying bond is that the coupons received before maturity can be reinvested at the bond’s stated yield. When that assumption is violated, the bond’s duration changes and it is less effectie for duration-matching purposes.

I understand that we talk about i nvestment horizon when we talk about one single liability. So as already answered by Ov25, investment horizon = date when the liability is due. And I would add, investment horizon = liability duration, since it is a single liability here.

In other words, condition #1 can be reworded as “duration of the portfolio must equal the duration of the liabilities”, which is the same condition as for multiple liability immunization.

No such assumption is made. It is just impossible to make such an assumption because as a matter of fact, duration changes from the mere passage of time. This is why a portfolio is only immunized at a specific moment and need rebalancing as soon as time has passed (among others, but also as market yield changes).

Not that they have decreased, just that the yield curve is upward sloping in general. So the reinvestment of your coupons over a shorter term period than your initial investment returns a lower yield.

I think the portfolio we are talking about here has only one future cash flow cause if we had multple liabilities (i.e. cash flows) we should also assume a wider spectrum of distribution of assets compared to the one referring to the correspondent liabilities? So I think the above definition refer to the classical immunization definition?