With interest rates around the world at historic lows, most have heard about the performance of bond funds over the last few years. Little attention has been paid to an important factor for bond investors going forward - very low yields can increase volatility substantially.

The degree that volatility increases is based on present value, duration and convexity. These calculations measure the sensitivity of a bond to changes in interest rates.

**Present Value**

Present value is simply the value you would pay now for a certain series of payments in the future. In other words, it is the value that you would be willing to sacrifice now for some future series of payments. The return for which you are willing to loan out your money for a certain amount of time is called the discount rate.

This idea can be simply calculated using the following formula:

Where:

n: number of terms passed

N: Number of terms until maturity

i: discount (interest) rate

PMT: payment from bond at time n

FV: principal repayment at term N, or the maturity date

This formula works out so that the market interest rate is equal to the discount rate where the current market price of the bond is the same as the present value with the market interest rate as the discount rate. In other words, if you plugged the market interest rate into the present value formula as the discount rate, the present value of the bond would be the same as the market price of the bond.

**Duration**

Technically, the first derivate of bond price relative to interest rates, duration measures roughly at what point price risk and reinvestment risk will offset. This calculation can be used to determine a holding period, coupled with reinvestment, that will have a guaranteed return no matter how interest rates change in the interim.

For instance, an investor looking for a fixed rate of return, including reinvested coupon payments on a bond with a duration of 4.0 should hold the bond and reinvest all coupon payments for exactly four years. No matter how interest rates change during this time, the final value of the estimate will remain fairly constant.

As a measure of how prices change with interest rates, duration can also be used to estimate how much prices will change for a change in interest rates. Simply, multiply the negative of the rate change by the bond's duration. For instance, if the above bond with a duration of 4.0 saw its market interest rate rise from 2.0% to 3.0%, the price of the bond should be expected to change by -(3.0-2.0)*4.0=-4.0%.

Duration, though, is based on a linear model of interest rate change. For any change of interest rates, whether it be from 1% to 2% or 1% to 15%, duration calculations will lead one to believe that will move constantly as percentage change*duration. In reality, price/yield relationships are *curved.* For larger interest rate moves, the change in price will be more dramatic than the linear approximation suggested by duration.

**Convexity**

Technically, the *second* derivative bond price relative to interest rates, convexity measures how much more severely the price will swing for extreme changes in the bond's yield than duration would suggest. In other words, it measures the curvature, or convexity, of the price/yield relationship for a bond. The convexity of a bond is important to take into account for long-term investors in bonds, for whom interest rates can change substantially.

Convexity will often be a minor concern when the interest rate outlook is relatively stable. Today's outlook is nothing like that, though, with some forecasts showing rates going further negative while others have rates spiking alongside inflation. Quickly rising rates may bring even "ultra-safe" Treasury bonds into the losses column given current prices.

**Putting It All Together**

Both duration and convexity measures signal more dramatic volatility from interest rate changes when the bond has a low market yield - sound familiar? As yields have fallen around the world, bond prices have increasingly become more volatile. Ironically, duration and convexity react similarly to positive and negative yields, so if rates keep dropping, price volatility may cease to be a serious concern in bonds.

Duration is also larger for longer-term bonds. So, bond investors of distant maturity may be opening themselves to more volatility from rate changes than investors of medium- and short-term bonds. It should be pointed out, though, that long-term bonds are less prone to interest rate re-pricings than shorter-maturity complements.

**Disclosure:** I/we have no positions in any stocks mentioned, and no plans to initiate any positions within the next 72 hours.

I wrote this article myself, and it expresses my own opinions. I am not receiving compensation for it (other than from Seeking Alpha). I have no business relationship with any company whose stock is mentioned in this article.