the textbook reads: managing funds against liabilities, for an upward-sloping yield curve, the immunization target rate of return will be less than the yield to maturity because of the lower reinvestment return.
if the duration of the portfolio is matched to liabilities, wouldn’t the target rate of return same as the ytm?
it does not depend on whether the assets are matched to the liabilities or not. That is purely a function of the shape of the yield curve and the fact that when the yield curve is upward sloping the Reinvestment Return obtained by reinvesting the coupons at the new rate is lower + the value obtained by the face value of the bond is also lower.
So that means are we considering the short term rate whrn yield curve is upward?
Try some numbers:
You buy a 2-year, $1,000 par, 5% coupon annual pay bond; the price is $1,000 (because the coupon rate equals the YTM). One year from today you get a coupon payment of $50, which you reinvest at 3% (the 1-year par rate, which is also the 1-year spot rate). Two years from today you get a coupon payment of $50, a principal payment of $1,000, and your reinvested coupon payment of $51.50 (= $50 × 1.03), for a total of $1,101.50. Your effective annual yield is:
($1,101.50 / $1,000)^(½) – 1 = 4.9524%
Note that this is less than the original 5% YTM. Why? Because the reinvestment rate was less than the 5% YTM. Why? Because the yield curve sloped upward.
Thank you. Clear explanation.
My pleasure.
Fantastic !! Lovely !..Thank you for such precise explanation!
My pleasure.
But that 3% spot rate is for today’s rate. The first 50 dollar coupon would be reinvested at the implied 1 yr forward rate one yr from today which in this case is about 7%. (5% x 2)-(3%)=7%…just remember that as a guestimate of a forward rate parity concept in level one I think. So the actual ytm to is higher than 5%.
Which is why the immunization target return is lower than the actual return. Immunization rate assumes a flat yield curve but if it’s upward sloping coupon are being reinvested at higher rates resulting in a higher yield
The last line is important. The yield curve is assumed to be flat!
This, of course, isn’t remotely true.
In fact, the first $50 coupon will be reinvested at whatever the 1-year spot rate is one year from today, and that could be anything. If pure expectations holds, it will be about 7%, but odds are that pure expectations doesn’t hold.
The statement in the text is intended to say that the yield curve slopes upward, and remains unchanged for the holding period of the bond.
To be clear: you have this backwards. If the yield curve slopes upward, you will have a realized yield that is lower than the YTM, not higher. That’s the point that the text is making.
Picking this up again hoping for a clarification.
Example 6 in the curriculum states: “If market yields rise, an accumulated value (total return) higher than the target value (target yield) will be achieved. This result follows because the coupon interest payments can be reinvested at a higher rate than the initial yield to maturity”.
I understand with “markets yields rise” the author is referring to an upwards-sloping yield curve.
Then one page later it says: “(…) for an upward-sloping yield curve, the immunization target rate of return will be less than the yield to maturity because of the lower reinvestment return.”
What am I missing, because currently putting these two statements together, they seem to be only consistent with respect to YTM being greater than the target yield in an upwards-sloping yield curve scenario. They seem to conflict with respect to the reinvestment-rate explanation. But maybe it’s more subtle and I just don’t get it :). Thanks.
Say the YC doesn’t change. You get coupon, invest in shorter bond, get lower rate.
Now take the above, and adjust the YC up. You take coupon, invest in higher rate, get higher coupon.
At least that’s how I read it.