# Why do we get a higher yield when we pay the interest at the end?(bonds)

I have an example where I show that if you pay the tax at the end of the bond period, the yield after tax is higher, but I am wondering if it is possible to give an explanation as to why it is like this? I am looking for both intuitive and mathematical answers, the expressions becomes so messy I am not able to show it myself.

Here is the example.

case no tax Assume first that we look at the bond without tax, then we get the expression:

$$PV = \sum\limits_{i=1}^N\frac{100r}{(1+y)^i}+\frac{100}{(1+y)^N},$$

with $$PV=99, r= 5\%, N = 10$$ we can solve it numerically to get $$y=5,13\%$$ if I have solved it correctly.

case tax immediately

Now the equation becomes

$$PV = \sum\limits_{i=1}^N\frac{100r(1-Tax)}{(1+y_2)^i}+\frac{100-(100-PV)*Tax}{(1+y_2)^N}$$

by using $$PV=99, r=5\%, Tax = 25\%, N=10$$ we get $$y_2=3,85 \%.$$

case deferred tax

Now the equation becomes

$$PV = \sum\limits_{i=1}^N\frac{100r}{(1+y_3)^i}+\frac{100-(100-PV)*Tax-N*100*r*Tax}{(1+y_3)^N},$$ now $$y_3= 4,07 \%.$$

conclusion

We see that with the deferred tax situation the yield after tax is higher. But can one explain this in some way? I suspect one explanation is that since we defer it we are able to get interest interest on the tax in some way, but I am not able to show it, is this the case that makes the return higher?, if so, how is it shown, or is there another explanation?

• You've assumed they have the same present value. Imagine a bond A pays 10 then 90. Imagine bond B pays 0 then 100. Now imagine they have the same present value. Which one has a higher yield? Nov 10, 2021 at 14:23
• @MatthewGunn Thank you for that explanation, it makes some sense now. Nov 10, 2021 at 14:36
• Some intuition: because interest rates are positive, 10 today is worth more than 10 tomorrow. Hence all else equal, you'd expect a bond that pays 10 then 90 to be worth more than a bond that pays 0 then 100. You've assumed though that today's price for either bond is the same. Thus the bond that pays 10 then 90 is in some sense cheaper compared to what you get; it has higher yield. Nov 10, 2021 at 18:28
• In teaching investments, Prof. John Cochrane likes to say that the yield is just another way to quote the price. Taking a positive stream of promised cash flows as given, the yield is just a monotonic transformation of the price. It's a shorthand way of quoting a bond price. It's perhaps akin to quoting the price of a call option by stating the implied vol. Nov 10, 2021 at 18:43