Yes. I can see that. Thanks.
There is a very simple way to think about this. At the outset, the liability is known and so is the portfolio value. I have a portfolio of $x, and I need to fund a liability of $y in 5 years. There is a YTM I need to achieve.
YTM doesn’t take reinvestment into account - compared to a DS YC, the YTM with an US YC won’t represent the comparatively more reinvestment income we would receive. Because of this, we dont need as high a rate of return on the portfolio, because the coupons are providing us with income that the YTM isnt taking into account.
Example 5 is meant to show us what would happen with a flat yield structure - we get the coupons, and then interest on top of interest with each coupon. If you were you put this into excel and see the future value of all coupons given a constant yield, you can recreate that chart easily. But, if the YC is sloped, the results change.
Kwalew is pretty much right on - At the start of the time horizon, liability is known, rate structure is known so we just calculate the PV of liability using the existing spot rates. (Liability / (1+r)^T)
That (1+r) is the target rate we want… you can also think of it as ‘YTM’ of the liability. This YTM does NOT take consideration of reinvestment of coupons (since it’s a liability with no cash flow) , and remember YTM is more of a 'average return you must achieve each year for x # of years until liability date.
With the bond, you will have the opprotunity to invest the coupon @ variou spot rates at the time. Since the curve is UP SLOPING, you will have the chance to reinvest at rates that are both lower AND higher than the “YTM of the liability”. Since it’s up sloping, you will probabaly be reinvesting more coupons at higher spot rates than the YTM of the liability. This is a more conceptual approach to thinking it…don’t have number to back it up.
Also recall text always said target value is the LOWER bound at end of horizon…i.e target rate 7% we might end up with total return of 7.09% for example.
So when the text says ‘when it’s upsloping the target rate will be lower than ytm of the portfolio due to lower reinvestment return’, I am going out and say it’s because there’s no reinvestment effect factored in when you calculate the target rate in the first place. This statement assumes target rate will always be lower than what you end up achieving (a return > targeted). Because as time passes, reinvestment at higher rate kicks in. Even though at time 0 target rate hypthetically = YTM of portfolio. But by the end… final yield of the piortfoio will > target.
FEEDBACK PLEASE…i am using this to force this to make sense to me.
… agree that spot should be higher than original spot , this is why the forward curve slopes up .
But we don’t invest coupons into the spot , do we? we’re investing to the term . YTM should be lower to the term as time to the liability shrinks , than the original bond
we do I believe - what choice do we have ? if we recieve coupon in yr 3, we invest that at whatever rate is going at the time.
If you start with a term of 10 years , in year 3 you’re re-investing for a term of 7 years.
If all things stayed the same , YTM for the original should be higher than YTM at the 3 year mark , because original bond term is longer . ( preference theory , etc etc. )
Taken to the extreme , the YTM for a zero year term should zero( because the yield curve starts at the origin )
just wondering what do you mean YTM for the original ? At yr 3 you calculate the YTM for the portfolio at that point. Ignore all the coupons that’s been paid out and reinvested in something. YTM at T=0 is more spread out assuming a rate u need to maintain for all years until time horizon ends. YTM at T=3 is a different calculation and likely will be higher given overall yield structure. T=3 is the new T=0.
You still have the original bond you purchased at T=0 to immunize. It will have a market price in yr 3. You still have coupons pending from yr 4, 5, 6, to 10 for example. Now try to find this ‘current’ YTM of this bond by discounting the remaining coupon to equate current price. This current YTM is going to be HIGHER.
do you get my idea? again I am not assuming I am right…i am hoping to have discussion so we both can get it lol
and all this is happening while the target rate is ‘locked’ on the side … it doesn’t change. Hence you can see why YTM of portfolio is higher.