Holding a Rebalancing Bond Fund Through Interest Rate Changes

# How long to hold a bond fund to be guaranteed a certain return? (mathematical proofs)


# How long to hold a bond fund to be guaranteed a certain return? (mathematical proofs)












Is there a mathematical proof of how long one must hold a bond fund or ETF to be guaranteed a positive return? And of how long to be guaranteed a certain return? For simplicity:

- let's consider accumulating funds/ETFs, which reinvest the coupons

- let's ignore fixed maturity bond funds and consider only those which keep duration and weighted average maturity within a certain range, constantly rebalancing

- let's consider only fixed-rate bonds

- let's assume rates never become negative

- and let's ignore taxes, which change greatly from country to country

Another way to think about it: when would a retail investor be better off buying single bonds, holding them to maturity, and rolling them, vs buying an accumulating bond fund / ETF? Of course, if you want to save to meet a certain liability in one year, buy a zero-coupon bond, but if you have a more mid-term horizon, say 5 years, and don't want to buy a 5-year zero-coupon because you think rates are likely to rise?

Let me explain:

If you buy a bond fund or ETF (other than a fixed-maturity bond ETF), it will keep its weighted average maturity and duration within a certain range; the fund itself will not have a maturity, as it rebalances constantly, buying new eligible bonds as the old ones become ineligible or mature and repay. For example, a bond ETF of bonds with 0-1 year maturity will buy new bonds as the old repay, a bond ETF of bonds with 1 - 3 year maturity will sell the bonds which reach a remaining term < 1 year and buy new ones.

With a single bond, if you hold it till maturity and the borrower doesn't default, you are guaranteed your capital back. If it's a fixed rate bond, not linked to inflation nor anything, the returns are certain if you hold it till maturity.

Not exactly so with a bond fund or ETF which doesn't have a fixed maturity.

Of course, if rates go up, the value of your bonds goes down, but you reinvest the coupons at a higher rate. Is there a mathematical proof, for a constantly rebalancing bond fund and not a single bond, of when the two offset each other, and therefore of how long one must hold a bond fund to be guaranteed a positive return, and of how long to be guaranteed a certain return?

I have some vague recollection of reading some proof which only holds if rates go up constantly, but I can't remember.

I have searched a number of fixed income books and couldn't find anything.

I have found some, well, I'd say marketing material, from asset managers, which basically says "please, please, don't buy single bonds, buy our bond funds" :) but doesn't really get into much detail, e.g.

https://www.northerntrust.com/japan/insights-research/2014/the-myth-of-holding-to-maturity

and

https://advisors.vanguard.com/insights/article/how-the-principal-at-maturity-myth-could-cost-you

There was also an article in this consumer forum, but no mathematical proofs or simulations: https://www.bogleheads.org/wiki/Individual_bonds_vs_a_bond_fund#Major_factors