Contingent Immunisation (source: CFA Reading 20 pp 36-37) – please explain $541.36 million

For example, if a firm has a three-year investment horizon over which it must earn 3 percent and it can immunize its asset portfolio at 4.75 percent, the manager can actively manage part or all of the portfolio until it reaches the safety net rate of return of 3 percent. If the portfolio return drops to this safety net level, the portfolio is immunized and the active management is dropped.

The difference between the 4.75 percent and the 3 percent safety net rate of return is called the cushion spread (the difference between the minimum acceptable return and the higher possible immunized rate).

– If the manager started with a $500 million portfolio, after three years the portfolio needs to grow to $546.72 (n=6, i=1.5%, PV=$500 m)
– At T=0, the portfolio can be immunised at 4.75% therefore PV = $474.90 (n=6, i=2.375%, FV=$546.72)
– The manager therefore has an initial dollar safety margin of $500 million − $474.90 million = $25.10 million.

If the manager invests the entire $500 million in 4.75 percent, 10-year notes at par and the YTM immediately changes, what will happen to the dollar safety margin?

If the YTM suddenly drops to 3.75 percent, the value of the portfolio will be $541.36 million. The initial asset value required to satisfy the terminal value of $546.72 million at 3.75 percent YTM is $489.06 million so the dollar safety margin has grown to $541.36 million − $489.06 million = $52.3 million. ”

*How to derive $541.36 million*??? – please help! @sophie @zee