columbia wrote: ↑Sun Dec 03, 2017 10:07 am

I've seen the phrase bond bubble here many times, in relation to decades of declining interest rates.

Pretend that I don't know anything (and that's not too much of a leap): what is the math behind rising interest rates leading to a lower total return?

They

**don't** lead to a lower total return in dollars. A bond is a contract to pay specific numbers of dollars on specific dates (well, many bonds are) and a change in the market rate doesn't change the contract. Assume in ideal bond (Treasuries are close) which won't default and doesn't have any call provisions or anything like that in it.

If a 10-year 2% coupon Treasury note is scheduled to make nineteen payments of $9.95 each on every January 15th and July 15th, then $1,009.95 on January 15th, 2028. Suppose interest rates rise instantly to 4% the day after you buy the bond. Your Treasury note still pays the same numbers of dollars on the same dates, and the rate of return on the bond (if held to maturity) is still 2%. Nothing changes in absolute terms.

What changes is the market value, which in turn is based on math--the calculation of the present value of that stream of payments at the current interest rate. Your bond, which pays $9.95, $9.95, $9.95, ..... $9.95, $1009.95 is now competing in the market against a newly-bought bond which pays $19.80, $19.80, $19.80, ... $19.80, $1019.80. Your bond is going to pay out a grand total of $1,199.00 over its lifetime. A new bond is going to pay $1,396 over its lifetime.

If the interest rate had remained at 2%, your bond would have a stable market value of $1,000 and you could sell it on the market at that price any time you liked.

But if try to sell your bond on the market to someone who knows they can buy a new bond that is going to pay out $1,396, they are certainly not going to pay you $1,000 for your bond, which will only pay out $1,199.00. In a rough conceptual way you might guess that your bond is only worth $1,199/$1,396 or 86% as much as the new 4% bond, and therefore the market value would be $860 instead of $1,000. The actual bond math is more sophisticated and takes compounding into account, and the actual value is $838.57.

1) For many purposes, what is important to you as an investor is to track the bond's market value now. You can get into interesting Zen questions about the value of a bond if you plan to hold it until maturity and never sell it on the market, which get debated endlessly and inconclusively.

2) To a rough first approximation, notice that in our example--using the value $838.57 the interest rate increased by 2%, and the market value = present value of the bond fell by 16.2%. A rate is a rate per year. If we do our math with units the way we do in high school physics, and divide 16.2% by

*2% per year*, the result is 8.1 years. This number is called the "duration" and it has several interesting aspects, but the basic fact is that the duration tells us the interest rate sensitivity of a bond or bond portfolio. If there is a small

*instantaneous* overnight rise in the interest rate, the market value of the bond

*temporarily* falls by an amount equal to the change in rate times the duration.

Notice that it is not a permanent loss in value, because the bond still makes all of its payments. On the day before it makes its final payment of $1,009.95, its market value will be close to $1,009.95, so the value of the bond must rise from $838.57 to $1,009.95.

3)

*If* the reason for the increase in interest rates in inflation, then regardless of the bond's paying its contractual amounts, it will of course suffer a permanent decrease in real (inflation-adjusted) value, in all ways--the interest payments will be worth less in real value, the final payback of principal will be worth less in real value, and the present value calculation of the bond's market value will be worth less in real value than the number of dollars would indicate.

Annual income twenty pounds, annual expenditure nineteen nineteen and six, result happiness; Annual income twenty pounds, annual expenditure twenty pounds ought and six, result misery.