# Contingent Immunization quiz

A Pension Fund has \$100 million to invest and the board feels that a minimum of 5% annualized over next 5 years should be possible and sets this as the minimum expectation .

The hired manager finds that bonds are available with immunization rates of 7% ( with semi-annual coupons ) and a 10 year maturity and selling at par.

He invests the \$100 million. For the duration of the next 5 years a bank is willing to invest coupons from this bond at 7% ( with semi-annual investment schedule ) and the manager agrees.

Consider these independent questions:

A. What is the initial Dollar Safety Margin ?

B. After a year yields go to 8% annualized . What is the new dollar safety margin?

C. If soon after the investment of the \$100 million , rates jump to 10% , what minimum total return is now required , so that the manager does not have to switch to contingent immunization ?

P.s. I haven’t solved this yet

Ugh, I hate these questions:

A: original safety margin of \$9,252,398

1. Terminal Value = 100*(1.025)^10 = 128.0085 Million.

2. Assets:

Since this is a PAR bond => 100 Million.

1. PV (Liabs required) at the end of 5 years: = 128.0085 / (1.035)^10 = 90.7476

A. Initial Dollar Safety Margin = 100 - 90.7476 = 9.2524 Mill \$.

B. After 1 year when rates become 8 %

PV (Liabs required) = 128.0085 / 1.04^8 = 93.5346 Million

Assets Available:

FV=100, PMT=3.5% * 100 = 3.5 N=18 (9 years), I/Y = 4, PV = 93.67 Mill\$

New Dollar Safety = 93.67 - 93.5346 = 0.1354 Million = 135,400

C. FV=100, I/Y=3.5 PV=128.0085/(1.05)^10 = -78.5861, N=20

CPT I/Y = 5.26% -> 5.26*2 = 10.51%

B. New \$ safety margin is:

[100*1.035^2] - [100*1.025^10/1.04^8] = 13.587976 mill

C. I have no idea how to approach.

fin - rate increased and your \$ safety margin went up to more than the initial?

Yields went up, the initial requred return stayed the same. To me that means the terminal value would stay the same but the portfolio could be invested at a higher rate.

Am I interpreting incorrectly?

B. After 1 year when rates become 8 %

PV (Liabs required) = 128.0085 / 1.04^8 = 93.5346 Million

Assets Available:

FV=100, PMT=3.5% * 100 = 3.5 N=18 (9 years), I/Y = 4, PV = 93.67 Mill\$

New Dollar Safety = 93.67 - 93.5346 = 0.1354 Million = 135,400

Thing to remember - Portfolio is also fixed income assets. So if the yield goes up - the assets would come down.

A. VT=100*(1+5%/2)^(5*2) = 128,008,454 V0=VT/(1+7%/2)^10 = 90,747,601 Dollar safety margin = 100,000,000-90,747,601=9,252,399 B. V0=VT/(1+8%/2)^8=128,008,454/1.04^8=93,534,523 The value of bond portfolio: PV + CPNs FV=-100m, I/Y=4, n=8, PMT=-3.5, so PV=96,633,628 CPNs=100m*3.5% + 100m*3.5%*(1+3.5%)=7,122,500 The value of bond portfolio = 96,633,628 + 7,122,500 = 103,756,128 Dollar safety margin = 103,756,128-93,534,523=10,221,605

C. For the bond: FV=-100m, I/Y=5, n=10, PMT=-3.5, so PV=88,417,398

VT=128,008,454 PV=88,417,398, FV=-128,008,454, PMT=0, n=10, so I/Y=3.77 So, the minum total return = 3.77%*2=7.54%

The value of bond: PV + CPNs FV=-100m, I/Y=4, n=8 , PMT=-3.5, so PV=96,633,628

bond changed from a 10 year to a 9 year bond.

so isn’t N=18?

And I have never seen the reinvestment income getting / being added the way tulkuu has above.

For Part 3 -> doesn’t the terminal value need to be discounted back at the 10% new rate?

i agree with CPK on the answer

year later means n = 18

A. VT=100*(1+5%/2)^(5*2) = 128,008,454 V0=VT/(1+7%/2)^10 = 90,747,601 Dollar safety margin = 100,000,000-90,747,601=9,252,399 B. V0=VT/(1+8%/2)^8=128,008,454/1.04^8=93,534,523 The value of bond portfolio: PV + CPNs FV=-100m, I/Y=4, n=18, PMT=-3.5, so PV=93,670,352 CPNs=100m*3.5% + 100m*3.5%*(1+3.5%)=7,122,500 The value of bond portfolio = 93,670,352 + 7,122,500 = 100,792,852 Dollar safety margin = 100,792,852-93,534,523=7,258,329 C. For the bond: FV=-100m, I/Y=5, n=20, PMT=-3.5, so PV=81,306,684 VT=128,008,454 PV=81,306,684, FV=-128,008,454, PMT=0, n=10, so I/Y=4.64 So, the minum total return = 4.64%*2=9.28%

This is an update, but I’m sure I have some other mistakes. Waiting for teh answer.

tul, why are you adding the coupons again when they are already accounted for the the PMT=3.5?

and payments should be positive, not negative

where did you learn this method? i’ve never seen it used

he has FV=-100 and PMT=-3.5 - so his PV becomes positive.

but the coupon part - I agree with - not sure why the reinvestment income component is getting added.

the YTM assumption thro the first year should already account for that piece, in my mind.

and tulkuu - on the Part C - shouldn’t number of years go back to 10 years - so 20 sub periods?

and your Terminal value - 128 Million - should be scaled back at the 10% return … don’t you think?

I haven’t learned this material yet fully, but adding the coupons makes sense to me since they’re both part of the total portfolio (they were received in the first 2 semi-annual periods, and one of them was reinvested). The re-investment was done at the initial 3.5% rate, whereas the remaining coupons to be received are discounted back at the new 4% rate.

Oh but wait, you’re comparing the “Total portfolio” to the 4-yr discounted value of the required terminal value. That doesn’t seem like a fair comparison. Shouldn’t the total portfolio be compared to the 4-yr discounted value of the required terminal value PLUS the received coupons, in which case you can just ignore the coupons in the first place. Ah, maybe that’s why nobody ever bothers with the coupons?

He invests the \$100 million. For the duration of the next 5 years a bank is willing to invest coupons from this bond at 7% ( with semi-annual investment schedule ) and the manager agrees.

they are NOT reinvested at the 8% rate – or at least it does not say so.

Look at the book Pg 36-37 where there is an example solved. And there too they DO NOT do anything with regards to the Coupon reinvestment.

for A , everyone of you is right Initial DSM = \$9.2524 million

for B , PV of the target at year 1 = \$93.53452 million

PV of bond : I/Y=4 , N=18 , FV=100 ,PMT=3.5 , PV = \$93.67035 miliion

PV of first coupon = 3.5 *(1+0.035)=\$3.6225 million

PV of second coupon = \$ 3.5 million

Total investment value = \$100.79285 million

Dollar Safety margin = \$7.25833 million

tulkuu is on the right track about adding the coupon present values in B

for C , PV of bond reduces to :

I/Y=5 , N=20 , FV=100 ,PMT=3.5 , PV = \$81.30668.

so required return from this bond then becomes

PV=-\$81.30668 , N=10 , FV= \$128.008 , PMT=3.5, I/Y=5.88547 or 11.77% per year

I did all this on the fly , it is my own creation