Book 4 page 28 CFAI middle page, " For an upward sloping yield curve the immunization target rate of return will be less than the yield to maturity due to the lower reinvestment returns. For a downward sloping yield curve it will result in immunization target rate of return greater than the yield to maturity because of the higher reinvestment return"
Im a bit confused, if we have an upward(downward) sloping yield curve wont we earn more(less) reinvesmtent income since we are reinvesting at higher (lower) rates?
this is something that has been discussed many times, never reached a consensus esp. because of the wording. you might want to search the older posts.
my thought process on this … [may help to read these lines with the GIC example in the book where numbers are put up]
when yield curve is upward sloping - you are earning higher reinvestment income on the coupons [but in terms of returns - since the return is calculated on a growing number - the reinvestment returns are lower] - but lose out on the final price on the bond - so your overall yield on the bond (ITRR) is LOWER than the YTM which by definition assumes that there is no subsequent change. You can also see the price return trumps the reinvestment return.
when yield curve is downward sloping - the reinvestment income on the coupons is lower[but since the return is calculated on a smaller number - returns are higher]. Price return is a much bigger number. Net effect - you end up with a ITRR > YTM.
But in example 5 if we take the 11% yield and compare it to the 10 % yield the interest return (interest income /accumulated value) is higher than the 10% one.
I think that you have stated it backwards .Please see section 4.1.1.3 on page 28 of volume 4 of CFAI text. For a upward sloping yield curve , re-investing cash flows to the horizon earns lower reinvestment return .
If the original term was 10 years and the yield curve stays pretty much same in terms of upward slope throughout , then successive cash flows are invested at lower rates ( as you slide down the slope of the curve down to zero at the horizon , you get lower and lower rates for your investments of cash flows ).
This means that the original assumption of classical immunization which implicitly assumes that cash flows are re-invested at pretty much the original Yield to Maturity will not be correct for upward sloping curve and you will get a lower target.
As an example if the yield curve is +vely sloped you’re shooting for 4% immunized return you have to start with higher YTM , say 5 % or 6% , then you should be able to approximately get 4%.
ignoring the principal, consider reinvestment of annual coupons used to immunize for a liability 4 years from now. If the yield curve is upward sloping then today’s 4,3,2,1 year rates may be 5%,4%,3%,2%. If the yield curve retains its shape then the coupon you get today can be reinvested at 5% but the one you get in a year can only be reinvested at 4% for 3 years, and the one 3 years from now can only get 2% (for the last year.) But when you calculate YTM, you assume that coupons can be reinvested at YTM which would be higher on average.
To restate the above if we have a downward sloping curve and rates are, for years 1 2 3 and 4 , equal to 5% 4% 3% and 2 %. We are reinvesting at the lower rates for a longer term and the higer rates for a shorter term. However since the bond we are immunizing is based on these rates, for example a 4 year bond is based on the 2% (maybe plus risk premiums), we earn more on reinvesting than we are paying out (assuming the 2% plus risk premiums is less than what we are getting). Is this the right way to look at it?
Unfortunately, I cannot say for certain whether the original conclusion is valid or not, because the information given is ambiguous.
When they say “an upward-sloping yield curve”, which yield curve do they mean? The par curve? The spot curve? The forward curve?
If the par curve slopes upward, then the spot curve lies above it (and may be upward sloping everywhere, but not necessarily), so the reinvestment rate for coupons could possibly be higher than the YTM of the original bond.
In short, it’s complicated; I believe that we need more information to reach a definite conclusion.
yield curve usually means the par curve ( when we talk about rates sloping , we speak generally of a treasury curve for different maturities , otherwise there can be no discussion if we cannot even agree on an illustrative instrument )
certainly to illustrate the point we can assume it is par curve…
And we’re talking about immunizing a liability that is known ahead of time and at a horizon . So your statement that “reinvestment rate for coupons could possibly be higher than the YTM of the original bond” doesn’t make sense.
If I had a 10 year schedule originally and the “yield” curve was upwards sloping and stayed that way ( again for illustration! ) , as the years go by , I cannot expect the original YTM to be available for shorter terms than 10 years.
I’m aware of that, and you’re aware of that, but that doesn’t excuse the question writer from being clear.
My point was that if the yield curve stays the same, the coupons will be reinvested at (what are now) the forward rates, which could be _ higher _ than the YTM of the original bond. To conclude that an upward-sloping (par) yield curve necessarily implies that the reinvestment rate for the coupons is lower than the original YTM is incorrect.
I suppose you could invest cash flows in zeroes , but then why talk about YTM of coupon bonds . Again this is a stylized scenario to explain something quite simple : YTM assumption is incorrect in general for a sloping curve, ok for a flat curve.
I think so too. When the yield curve is upward sloping, the reinvestment rate will be higher and price will be lower. So overall total retrun will be lower because price will lose more than compensated by the reinvestment income.
In my mind , and to simplify things for myself, if all things remain equal, and preference theory holds , investors will expect less return for shorter maturity . So as the original bond duration shrinks , coupon income must be invested at lower total return rates , as the time to maturity shrinks .
Not sure what you mean by “reinvestment rate” and “price” . … If you mean that YTM for a coupon reinvestment will be lower , yes , in general this should be so , by preference theory and normally upward sloping curve.
If you take a par curve then price will be constant for each reinvestment while coupon should shrink as YTM shrinks.
Again big assumptions to explain the theory , in practice could get a variety of other factors , as s2000 alluded.
Its not very intuitive, or worded very well, but the point remains true. Key is to understand the idea of the time horizon, where you have a liability due at some point int he future. Example 5 uses the point of the rates staying constant in the future (a flat term structure at 11%, per example).
If you have an upward sloping yield curve, and the term structure remains constant throughout time as it says in the reading, then you receive lower reinvestment income as time goes on. For instance, assuming a 5 year horizon, with 5, 4, 3, 2, 1 rates of 5%, 4%, 3%, 2%, 1%, then in one year from now you will receive a cash flow that you can invest for 4 years at 4% due to the term structure staying constant. After another year passes, with three years until the liability is due, you will be able to invest the coupon for 3 years at 3%, etc. As each year goes on, you are able to reinvest the cash flow from the coupon at a lower rate. So, the higher yields are coming in for longer periods - 4%/year for 4 years, 3%/yr for 3 years. You have a liability due at the end of the original 5 years, so you’re limited as to how long you can invest these coupons for.
Conversely, if the YC is downward sloping, the reverse is true. You invest for 4 years at 1%, 3 years at 2%, 2 years at 3%, etc.
Since the earlier coupons are being reinvested at higher rates in the upward sloping version, less of a “target” return is needed to achieve the target value.
draw a upward sloping line terminating at the horizon . Mark the X-Axis as maturity ( years) and Y axis as YTM ( %) .
Draw a horizontal line at the initial YTM . Now check for yourself how as the years go by and time to maturity shrinks , the reinvestment rate is lower and lower each year than the horizontal line
I did that. I see your point that for reinvestment, we are always on the short end of the curve. So when the yield curve is rising, short end is always lower, but when yield curve is falling, the short end of the curve is higher than long end.
So when the yield curve is rising, reinvestment income is less and price appreciation is more so total return is higher & vice versa?
If you’re reinvesting into par bonds , price return should be zero , and coupon return should be lower than the original YTM , ( look at the curve and line you just drew ). Does that make sense?
So since your total return is less for reinvestments than the original bond , you must plan for a higher YTM to begin with , than the immunization required rate of return.