# Semi-annual coupon bond vs Annual Coupon bond present value

I am trying to price two bonds with the only difference between the two being the coupon payment frequency. I have copied the calculations below.

I am confused as to why the bond with the semi-annual coupon has a lower present value than the bond with annual coupons. Would semi-annual payments not be preferable as one could reinvest the money for an extra half year and hence have a higher present value?

Bond with annual Coupon

Par value = \$1,000

Coupon rate = 2.5%

maturity = 20 years

discount rate = 4%

PV = (25/0.04) * [1-1/1.04^20]+1000/(1.04^20)

= \$796.14

Bond with semi-annual Coupon

Par value = \$1,000

Coupon rate = 2.5%

maturity = 20 years

discount rate = 4%

PV = (12.5/0.02) * [1-1/(1.02^40)]+1000/(1.02^40)

= \$794.14

I think I may have found the problem. For both bonds I used a nominal rate of 4%. Should I have converted the nominal rate on a yearly basis to a nominal rate on a semi-annual basis?

Effective semi-annual rate = (1+4%)^1/2 - 1 = 1.9803902% This gives a nominal rate on a semi-annual basis as 3.96%.

If I use this to discount the semi-annual bond rather than 2%, I get a present value of \$799.509 which makes more sense to me as it is now higher priced than the annual coupon bond.

Is this right?

Nope. Bonds don’t use effective semi-annual rates. As far as I know the 2% rate for the semi-annual paying bond is the correct rate.

The explanation of why prices differ comes more from mathematics itself. The 20-year bond price is a 20-dimensional function of interest rate (4%). And the 40-semiannual bond price is a 40-dimensional function of interest rate (2%). In other words, they are non-linear functions with different level of non-linearity, thus they will never match.

I did the exercise in excel of a bond that pays weekly coupons (52 coupons per year, so 1040 periods to match the 20-year bond) and the price was below the 40-period bond (1\$ lower).

An economic interpretation of the result would be that having more coupons payable, the investor encounters higher reinvestment risk, therefore the price should be lower to compensate that risk.

Is the discount rate an effective annual rate? When discounting the annual coupon bond, using 4% is ok as the payments are on a yearly basis.

When discounting the semi-annual coupon bond, should the 4% discount rate not be converted to an effective periodic rate for discounting? i.e (1.04)^1/2 -1 = 1.98%. Then discount the periods using this? If I discounted the periodic payments by 2%, this is equivalent to discounting by: (1 + 0.04/2)^2 -1 = 4.04%.

2% is the correct rate. The semi annual bond is worth more when trading at a premium and the difference lies in the reinvestment risk as noted by Harrogath. When the market rate exceeds the bond rate there is reinvestment risk. When the bond rate exceeds the market rate there is no reinvestment risk. K > C trades at a discount and reinvestment risk exists K < C trades at a premium and reinvestment does not exist

If I use the effective semi-annual rate for discounting I get the following.

An example where the market rate is higher than the coupon rate

Par value = \$1,000 Coupon rate = 2.5% maturity = 20 years discount rate = 4%

Annual coupon bond = PV = (25/0.04) * [1-1/1.04^20]+1000/(1.04^20) = \$796.14 Semi-Annual coupon bond = PV = 12.5/sqrt(1.04)-1) * [1-1/1.04^20]+1000/(1.04^20) = \$799.509

Both bonds sell at a discount however with the semi-annual coupons, I can reinvest for an extra 6-months at a higher market rate hence it has a higher present value than the annual bond

An example where the market rate is lower than the coupon rate

Par value = \$1,000 Coupon rate = 2.5% maturity = 20 years discount rate = 1%

Annual coupon bond = PV = (25/0.01) * [1-1/1.01^20]+1000/(1.01^20) = 1270.68 Semi-Annual coupon bond = PV = 12.5/sqrt(1.01)-1) * [1-1/1.01^20]+1000/(1.01^20) = \$1271.81

Both trade at a premium but the semi-coupon has a higher value due to compounding for an additional 6 months at the 1%.

An example where the market rate is lower than the coupon rate Par value = \$1,000 Coupon rate = 2.5% maturity = 20 years discount rate = 1%

Annual coupon bond = PV = (25/0.02) * [1-1/1.02^20]+1000/(1.02^20) = \$1081.75 = = \$1081.75 Semi-Annual coupon bond = PV = 12.5/sqrt(1.02)-1) * [1-1/1.02^20]+1000/(1.02^20) = \$1083.79

The premium is smaller as the coupon rate is closer to the market rate. The difference between the annual and semi-annual is larger as you are compounding for an additional 6 months at a higher rate.

I understand why the bonds trade at a discount or premium based but I do not understand the reinvestment risk aspect. If I received a discount halfway through the year, could I not invest that coupon at the market rate guaranteed and hence earn more through additional compounding?

Wow!

Just . . . wow!

Yes, that’s the problem.

But for goodness sake, get rid of that disgusting “on a <fill-in-your-favorite-adjective> basis” crap. What you mean is that you should have converted the annual, nominal rate to a semiannual nominal rate, and you should just say so. Or, better still, say that you should have converted EAY to BEY.

Yes.

Of course they do: it’s called bond-equivalent yield (BEY): _ twice the semiannual effective rate _.

This really surprises me, because I know you to be much better than to make this sort of silly mistake.

If the 4% is BEY, they you’re correct. But then the 4% used for the annual pay bond is wrong; it should be 4.04%

If the 4% is EAY, then you’re incorrect: the semiannual effective rate is 1.9803903%.

It isn’t clear which one OP meant.

This is all silliness. Assuming that 4% is the EAY, the OP’s second post is correct; it has nothing to do with reinvestment risk or 40-dimensional functions.

Ick.

Only if the 4% yield is BEY. And that isn’t remotely clear.

No, it doesn’t.

Reinvestment risk does not enter into TVM calculations. Period.

When there are interim cash flows there is always reinvestment risk, regardless of the market rate.

When there are no interim cash flows, there is never reinvestment risk, regardless of the market rate.

In short, these two statements are categorically wrong.

The reason you don’t understand it is that some of what the other posters wrote is incorrect.

Yes.

And that’s the reason that your second calculation is correct.

Thanks! That explained it perfectly. I will take heed and never refer to nominal rates on a <fill-in-your-favorite-adjective> basis again.

You’re welcome.

And thanks back atcha.

Wait a sec, this is not the end of this post!

I want to apologize for such a foolish post I made. Indeed I was wrong on my response fueled by a big confusion of mine. I don’t know why I exchanged the terms “discount rate” and “coupon rate”. The annual coupon is indeed a half for semi-annual paying bonds… that’s why I stated 2% as the correct rate… wow such a lame mess up. The OP’s question was really good and honestly had no clear answer for it at first glance, and in the attempt to help I just elaborated a truly weird story.

Honestly speaking, I didn’t go deep in understanding well the BEY and the other measures for bonds at L1, thought it was unnecessary and proved to be wrong until yesterday :’(

Thank you S2000magician for the correction. Appreciate it.

My pleasure, Harrogath.

You’re such a great help to the candidates here. It’s rare for you to make a mistake at all, much less a big one.

Thanks for your kind words S2000, and your support. Really appreciate it.