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I'm trying to prove how coupon frequency affects a bond price. I get it intuitively but I have not found a math proof. Could you help me?

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    $\begingroup$ Small hint: How does a cash flow series $\sum_i c \times e^{-r*t_i}\Delta(t_i)$ converge to an integral? And then solve that integral. $\endgroup$ Nov 20 '20 at 14:42
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    $\begingroup$ As an investor, rather than receive $C$ once a year, you would prefer to receive $C/2$ twice a year, or even better receive $C/n$, $n$ times a year. You can re-invest or consume the coupon sooner, a good thing in a world of positive interest rate. So the bond price goes up with $n$. But the effect is small, and becomes ever smaller as $n$ increases. (Think of your salary: how much would you benefit if it was paid weekly rather than monthly: not very much). $\endgroup$
    – noob2
    Nov 20 '20 at 20:29
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    $\begingroup$ @noob2 I would argue that the preference effect should already be found in the time value of money. A bond is a scaled sum of discount factors. As these are non-increasing with time to maturity, you get higher PV with higher frequency (as you suggest). I am wondering whether that’s simply the mean value theorem at work... $\endgroup$ Nov 20 '20 at 21:05
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Let us assume a plain vanilla bond with maturity $T$ and fixed coupon rate $c$ and a unit notional of $1$. The bond pays the coupon at a some frequency $f$ which translates to payments every $\Delta(t_i)=t_i-t_{i-1}=1/f$ years. Further, assume a fixed continuously compounded interest rate $y(t)$ with corresponding discount factor $D(t)=e^{-y(t)t}$.

Then the bond price can be written as:

$$ PV=c\sum_{i=1}^N D(t_i)\Delta(t_i)+D(t_N) $$

Assuming a monotonic discount factor function, i.e.

$$ e^{-y(t)t}\leq e^{-y(t')t'} \quad \forall t>t' $$

the present value of the bond is strictly increasing in the coupon frequency as we are sampling more and more from the higher discount factor values.

As we increase the coupon payment frequency to infinity, $f\to\infty$, the time step becomes infinitesimal, $\Delta(t) \to dt$ and the present value function converges to $$ PV=c\int_0^Te^{-y(t)t}dt+D(t_N) $$

Here, the present value of the bond is maximized.

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  • $\begingroup$ Great! But what if the discount factors D were sometimes allowed to increase with time (negative interest rates)? $\endgroup$ Nov 23 '20 at 12:01
  • $\begingroup$ Hi @DimitriVulis Yes, that would make this discussion more complicated. But since this seems to be a finger exercise in the first place, I would like to assume that discount factors are non-increasing ;-) $\endgroup$ Nov 23 '20 at 12:48
  • $\begingroup$ exactly :) :) :) $\endgroup$ Nov 23 '20 at 13:23
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The frequency is static indicative data. It does not normally change during the life of a bond.

If the frequency changed in a debt restructuring, then it would be highly unlikely that the frequency would be the most material change or that we could isolate its effect,

I recall Indonesian (USD-denominated) floaters where the issuer had the choice at the beginning of each coupon period: either pay 3M USD LIBOR + spread in 3 months, or pay 6M USD LIBOR + spread in 6 months. But this option was known and priced in at issuance.

If you meant to ask: if two bonds have "the same" indicative data, except for the frquency, and trade at "the same" yield, then how does the frequency difference afftect the yield-to-price, then both "the same's" are imprecise:

If you're comparing yields of two bonds having difference frequencies, then you may choose to convert one yield to the other bond's frequency. Or not. Not in the U.S., but in most other bond markets, if you're comparing yields of quarterly, semi-annual, and annual bonds, you first convert them to the same frequency. Yields are are "the same" before conversion are not "the same" after conversion. You may want to work out the price difference if the yields are "the same" after conversion.

If you're quoting "6% a year coupon paid semi-annual" then in most markets the convention is "6/2=3%" coupons, but, for example, in Brazil it means "(1+6%)^(1/2)-1=2.95%" coupons (see, for example, https://sisweb.tesouro.gov.br/apex/f?p=2501:9::::9:P9_ID_PUBLICACAO:27710 , page 8). This Brazil convention is unusual. I'm mentioning it to illustrate that assuming the more common convention might lead to different results. it's safer to explicitly state your assumptions. Many economists are fond of saying "ceteris paribus" without thinking through what that means.

Suppose then that the same positive cash flow amount $C$ can be paid at time $t_1$ or at time $t_2$, $t_1 < t_2$, and that rather than quoted yields (which might not be comparable), you're given discount factors for future cash flows $d_{t_1}$ and $d_{t_2}$. Which choice has higher present value: $d_{t_1}C$ or $d_{t_2}C$? That depends on whether $d_{t_1} > d_{t_2}$. Many older books implicitly assume this (i.e. assume that interest rates are positive). This is usually the case, but sometimes people find themselves in a deflationary environment and would welcome an oppostunity to "lend out" $C$ at zero interest rate from time $t_1$ to time $t_2$.

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The "proof" is simplest for a perpetual bond, where Price = Coupon/Yield.

So an annual coupon of 1 at a 1% yield is worth 100. Pay 0.5 every six months and the same 100 represents a six-monthly yield of 0.5%. (1.005)^2 = 1.0025% compounded annually. Discounted by the same 1% annual rate, the higher frequency will be worth a little more... and the same will be true (to a lesser degree) in any finite-life bond where you DCF these discrete payments.

In reality, most of this will not really affect traded bond prices, because of clean-vs-dirty pricing. On the day after the 6m frequency bond pays its coupon, it will be clean. But anyone trading the 1y equivalent will have that 6m accrued that the seller pays to the buyer. So in reality, it doesn't make any material difference at all.

https://en.wikipedia.org/wiki/Dirty_price

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If coupon rate > yield rate => the more frequent the payment, the higher the bond price

but if coupon rate < yield rate => the more frequent the payment, the lower the bond price

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  • $\begingroup$ I need exactly a proof of the relation you quoted. Can you give me a reference? $\endgroup$
    – user279687
    Jan 28 at 14:05
  • $\begingroup$ @user279687 Actually I came into this conclusion by simply calculating bond prices on an excel sheet and saw this relationship, but when I asked my fixed income professor he said that always increasing payment frequency would lead to an increase in the value of a bond, regardless of if the coupon rate is higher or lower than the yield rate. When I showed him my excel calculation, he rightfully pointed out that when you change the frequency in the excel bond pricing function, it assumes that both yield and coupon would have the same frequency, so it also changes the yield rate. $\endgroup$
    – Sasan
    Jan 29 at 20:33

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