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# reinvestment risk

#### narayananvenkat

##### cfo
David..

Identify the most accurate statement that is TRUE.

1.reinvestment risk forbonds is:

1. Long term bonds should be purchased if the investor anticipates higher reinvestment rates.

2. If the investor anticipates lower reinvestment rates , higher coupon bonds should be purchased.

3. Unless the reinvestment rate equals the yield to maturity , the holding period return will be less than the YTM.

4. Zero coupon binds have no reinvestment riskover their term.

venkat

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi venkat,

This is a decent question for meditation. The high-level idea (in Tuckman) here is the trade-off between interest rate risk (e.g., duration) and reinvestment risk.
At one extreme, a zero coupon bond has high duration (interest rate risk) but no reinvestment risk: you do not have to worry about re-investing the cash you receive in the meantime
At the other extreme is a high-coupon bond, with lower duration but high reinvesment risk

if i sell you a zero coupon bond, and you hold to maturity, you can be certain of your return.
But if i sell you a coupon bond, you will reinvest the coupons, and your *realized return* will vary depend on the future evolution of spot interest rates; such that, only under the (unlikely) condition that coupons are reinvested at the intial yield (YTM) will the realized return equal the yield.

so number 3
i.e., "Unless the reinvestment rate equals the yield to maturity , the holding period return will be less than the YTM."
is not correct because the holding period (realized) return can be higher or lower

correct is number 4
and, my tip is to be keep in mind the tradeoff between: duration and reinvestment risk. Because this helps us remember that higher coupons (higher reinvestment risk) implies lower duration (in addition to the direct intuition that higher coupons decrease the average time to receipt of cash flows).

David

#### hsuwang

##### Member
Hello David,
Can you please also explain why choices 1 and 2 are wrong? It seems to me that #1 is correct as well? Higher reinvestment rates will give better return. Thanks!

1. Long term bonds should be purchased if the investor anticipates higher reinvestment rates.

2. If the investor anticipates lower reinvestment rates , higher coupon bonds should be purchased.

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Right, i skipped (1) and (2) because i think they are a bit muddy (specifically: what is meaning of "reinvestment rate?") and mabye debatable

my interpretation of the intent is:

(1) longer term delays return of the principal (and in the meantime, higher rates imply loss in price; all other things being equal, if you anticipate higher rates, you want to shorter the duration). So, I think the intent is: with shorter term, you get the principal back sooner for reinvestment ...
(2) you don't want to re-invest more future coupon dollars at lower rates ... and, the anticipation of lower future rates implies you want to increase duration and therefore you'd want lower coupons

back to the tension: duration (interest rate) versus reinvestment risk
they have the opposite reaction to interest rates:

* anticipation of higher interest rates: benefits reinvested coupons but hurts bond price (so if anticipate higher rates: shorten duration and favor reinvestment risk; e.g., higher coupons)
* anticipation of lower interest rates: hurts reinvestment but boost bond price (so, strategy is to lengthen duration and lower reinvestment risk; e.g., lower coupons)

Davd

#### nc27

##### Member
Subscriber
When a 11 years old post helps you to understand and thighten things up, thanks David!

Nevertheless I have a doubt! In my notes (if they are correct) it is stated that :

- When bond yield are in excess of 6% the conversion factor tends to favor the delivery of low coupon high maturity bond.
-When bond yield are less than 6% the conversion factor tends to favor the delivery of high coupon short maturity bond.

Can I imply the following,

For example, in the case that we expect the yield (complex average of the term structure of spot rates) to exceed 6%, then normally, the high valued bond must be the high coupons (reinvestment risk) low duration (interest rate risk) bond.

Then for the short, the cheapest-to-deliver bond, will be the low coupon high maturity bond which suffers the most from our future expectations of the spot rate term structure.

Is my insight correct @David Harper CFA FRM ???

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#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
Hi @nc27 Well those notes refer to the preference of the short position who seeks the cheapest to deliver (CTD) bond in a T-bond futures contract. I'm not keen on "we expect the yield ..." because, at least in this context (if not in general), yield is a current variable or observation implied by the current price (of course we can speak of anticipated yields but anticipation is not necessary here). In the CTD, if the market's yield (i.e., currently, not an expectation) is 9.0%, then because the CF assumes 6.0%, the short will prefer long duration bonds which are generally long-maturity and low-coupon (aka, low reinvestment risk). I wrote about this over here https://www.bionicturtle.com/forum/threads/highest-conversion-factor.5569/post-15926 and included a numerical example, I hope it's helpful! i.e.,
Hi PL,

Those are just Hull's statements of course. Tuckman actually explains it, see my extraction below from Tuckman Chapter 20.

The short position purchases the bond at the quoted bond price (which reflects the actual yield curve; i.e., not a flat yield curve at 6.0%) and receives, in exchange for delivery (in addition to accrued interest): settlement price * conversion factor. But the conversion factor (abstracting the details) "standardizes" the basket of choices by assuming a flat 6.0% yield curve. So, in real shorthand terms, the short is buying the actual yield curve and receiving credit for a flat 6.0% curve; when yields are above the notional 6.0% and above low coupon rates, prices will be below par and the short tends to prefer bonds that MORE responsive to the actual yield (i.e., higher duration. "low coupon, long-maturity" --> higher duration) ... when yields are below the notional and below high coupon rates, prices will be above par and the short tends to prefer bonds that are LESS responsive to the actual yield (i.e., lower duration. "high-coupon, short-maturity" --> low duration).

For example, a 20-year 2% semi-annual pay coupon bond will have a CF of somewhere around = -PV (6%/2, 20*2, $100*2%/2, 100) =$53.77 = about 0.5377 (not exactly the actual mechanics are shown in the learning XLS). That assumes a flat "notional" yield of 6.0%; but if the market's yield is actually 9.0%, the market price will be nearer to the theoretical (model) price of -PV (9%/2, 20*2, $100*2%/2, 100) =$35.59. So, the short "buys" this for only ~36 per 100 but receives for it on a valuation of ~53 per 100. Actual yields above 6% are favoring long duration bonds because their price drop is greater. As yield increases above (below) 6.0% (i.e., the notional coupon rate), CTD favors higher (lower) duration bonds.

"As yield increases above the notional coupon rate the prices of all bonds fall, but the price of the bond with the highest duration, namely the 5s of August 15, 2011, falls relative to the prices of other bonds. But, because conversion factors are fixed, the delivery price of the 5s of August 15, 2011, stays the same relative to that of all other bonds. In other words, as yields increase above the notional coupon rate, the cost of delivering the 5s of August 15, 2011, falls more than that of any other bond. Therefore, while all bonds are equally attractive to deliver at a yield of 6%, as yield increases the 5s of August 15, 2011, become CTD. Graphically, the ratio of the price to conversion factor of the 5s of August 15, 2011, falls below that of all other bonds.
As yield falls below the notional coupon rate, the prices of all bonds increase but the price of the bond with the lowest duration, namely the 4.75s of November 15, 2008, increases the least. At the same time, since the conversion factors are fixed the delivery price of the 4.75s of November 15, 2008, stays the same relative to those of other bonds. Therefore, while all bonds are equally attractive to deliver at a yield of 6%, as yield decreases the 4.75s of November 15, 2008, become CTD.

Figure 20.1 is a stylized example in that it assumes a flat term structure. It is for this reason that the CTD is either the 4.75s of November 15, 2008, or the 5s of August 15, 2011, but never the 6.50s of February 15, 2010, except, of course, at 6% when all bonds are jointly CTD. In reality, of course, the term structure can take on a wide variety of shapes that will affect the determination of the CTD. In general, anything that cheapens a bond relative to other bonds makes that bond more likely to be CTD. If, for example, the curve steepens, then long-duration bonds (e.g., the 5s of August 15, 2011) are more likely to be CTD. On the other hand, if the curve flattens, then short-duration bonds (e.g., the 4.75s of November 15, 2008) are more likely to be CTD. Figure 20.2 depicts a different shift in which the 6.50s of February 15, 2010, cheapen by 4 basis points (i.e., their yield increases by 4 basis points) relative to levels in Figure 20.1. As a result the 6.5s of February 15, 2010, become CTD when the general yield level is between about 5.60% and 6.20%. For lower yields the 4.75s of November 15, 2008, remain CTD, and for higher yields the 5s of August 15, 2011, remain CTD." -- Tuckman pages 431-33

#### nc27

##### Member
Subscriber
Hi @nc27 Well those notes refer to the preference of the short position who seeks the cheapest to deliver (CTD) bond in a T-bond futures contract. I'm not keen on "we expect the yield ..." because, at least in this context (if not in general), yield is a current variable or observation implied by the current price (of course we can speak of anticipated yields but anticipation is not necessary here). In the CTD, if the market's yield (i.e., currently, not an expectation) is 9.0%, then because the CF assumes 6.0%, the short will prefer long duration bonds which are generally long-maturity and low-coupon (aka, low reinvestment risk). I wrote about this over here https://www.bionicturtle.com/forum/threads/highest-conversion-factor.5569/post-15926 and included a numerical example, I hope it's helpful! i.e.,

Thanks for the clarification, I forgot that the yield is implied by the market price...

#### thanhtam92

##### Member
Subscriber
Hi venkat,

This is a decent question for meditation. The high-level idea (in Tuckman) here is the trade-off between interest rate risk (e.g., duration) and reinvestment risk.
At one extreme, a zero coupon bond has high duration (interest rate risk) but no reinvestment risk: you do not have to worry about re-investing the cash you receive in the meantime
At the other extreme is a high-coupon bond, with lower duration but high reinvesment risk

if i sell you a zero coupon bond, and you hold to maturity, you can be certain of your return.
But if i sell you a coupon bond, you will reinvest the coupons, and your *realized return* will vary depend on the future evolution of spot interest rates; such that, only under the (unlikely) condition that coupons are reinvested at the intial yield (YTM) will the realized return equal the yield.

so number 3
i.e., "Unless the reinvestment rate equals the yield to maturity , the holding period return will be less than the YTM."
is not correct because the holding period (realized) return can be higher or lower

correct is number 4
and, my tip is to be keep in mind the tradeoff between: duration and reinvestment risk. Because this helps us remember that higher coupons (higher reinvestment risk) implies lower duration (in addition to the direct intuition that higher coupons decrease the average time to receipt of cash flows).

David
@David Harper CFA FRM why is #3 wrong? I thought the YTM would be realized return if the rate stay flats and the coupon is reinvest at the same rate as YTM

#### David Harper CFA FRM

##### David Harper CFA FRM
Staff member
Subscriber
@thanhtam92 I did not write that question, I don't like it. Your statement looks correct (i.e., an investor realizes the yield only if the coupons are reinvested at the same yield), but I think the question is simply meant to be false because it omits "greater than" as a possibility. I think the question is false because its corresponding true statement is:
"Unless the reinvestment rate equals the [constant] yield to maturity , the holding period return will be less than or greater than (i.e., unequal to) the YTM."
... but, again, I'm not really a fan of this true/false question, it's the sort of question I would have written earlier in my career but today it just looks too slippery to me!