# Interest Rate vs Rate of Return

How well a person does by holding a bond or any other security over a particular time period? This is accurately measured by the **return**, or, in more precise terminology, **the rate of return**.

### The Rate of Return

For any security, **the rate of return is defined as the payments to the owner plus the change in its value, expressed as a fraction of its purchase price**.

### example for rate of return

To make this definition clearer. Let us see what the return would look like for a $1,000-face-value coupon bond with a coupon rate of 10%. That is bought for $1,000, held for one year, and then sold for $ 1,200.

- The payments to the owner are the yearly coupon payments of $100, and
- the change in its value is $1,200 – $1,000 = $200.

Adding these together and expressing them as a fraction of the purchase price of $1,000. Gives us **the one-year holding-period return for this bond**:

$100 + $200 / $1000 = $300 / $1000 = 0.30 = 30%

### the return on a bond will not necessarily equal the yield to maturity on that bond

You may have noticed something quite surprising about the return that we have just calculated. It equals 30%, yet as Table 1 indicates, initially the yield to maturity was only 10%. This demonstrates that **the return on a bond will not necessarily equal the yield to maturity on that bond**.

**We now see that the distinction between interest rare and return can be important. Although for many securities the two may be closely related.**

### The Rate of Return Formula

More generally, the return on a bond held from time ** t** to time

**can be written as:**

*t*+ 1**R = C + Pt+1 – Pt / Pt**

where:

- R = return from holding the bond from time
*t*to time*t*+ 1 - Pt = price of the bond ar time
*t* - Pt+ 1 = price of the bond at time
*t*+ 1 - C = coupon payment

### The rate of return formula can be split into two separate terms – current yield and rate of capital gain

**R = C / Pt + Pt+1 – Pt / Pt**

- The first term is the
**current yield**(the coupon payment over the purchase price) :*i*c

** ic** = C / Pt

- The second term is the
**rate of capital gain.**Or the change in the bond’s price relative to the initial purchase price:

**g** = Pt+1 – Pt / Pt

We can rewrite the return formula as :

**R = ic + g**

Which shows that the rate of return on a bond is **the current yield ‘ ic’**, plus

**the rate of capital gain ‘g’**. This rewritten formula illustrates the point we just discovered.

**Even for a bond for which the current yield ‘**.

*i*c’, is an accurate measure of the yield to maturity. The return can differ substantially from the interest rate**Returns will differ from the interest rate. Especially if there are sizable fluctuations in the price of the bond that produce substantial capital gains or losses**.

### what happens to the returns on bonds of different maturities when interest rates rise

To explore this point even further. Let’s look at **what happens to the returns on bonds of different maturities when interest rates rise**. Table 2 calculates the one-year return using the equation above on several 10%-coupon-rate bonds. All purchased at par value when interest rates on all these bonds rise from 10% to 20%.

### Several key findings in this table are generally true of all bonds:

- The only bond whose return equals the initial yield to maturity is one whose time to maturity is the same as the holding period. (see the last bond in Table 2)
- A rise in interest rates is associated with a fall in bond prices. Resulting in capital losses on bonds whose terms to maturity are longer than the holding period.
- The more distant a bond’s maturity, the greater the size of the percentage price change associated with an interest rate change.
- The more distant a bond’s maturity, the lower the rate of return that occurs as a result of the increase in the interest rate.
- Even though a bond has a substantial initial interest rate. Its return can turn out to be negative if interest rates rise.

### A rise in interest rates is associated with a fall in bond prices

At first, it is confusing that a rise in interest rates can mean that a bond has been a poor investment. The trick to understanding this is to recognize that **a rise in the interest rate means that the price of a bond has fallen**. **A rise in interest rates, therefore, means that a capital loss has occurred.** If this loss is large enough, the bond can be a poor investment indeed. For example, we see in Table 2 that the bond that has 30 years to maturity. When purchased has a capital loss of 49.7% when the interest rate rises from 10% to 20%. This loss is so large that it exceeds the current yield of 10%. Resulting in a negative return (loss) of -39.7%.

### Maturity and the Volatility of Bond Returns: Interest-Rate Risk

The finding that the prices of longer-maturity bonds respond more dramatically to changes in interest rates helps explain an important fact about the behavior of bond markets.

**Prices and returns for long-term bonds are more volatile than those for shorter-term bonds**. Price changes of +20% and -20% within a year, with corresponding variations in returns, are common for bonds more than twenty years away from maturity.

We now see that **changes in interest rates make investments in long-term bonds quite risky**. Indeed, the **riskiness of an asset’s return that results from interest-rate changes** is so important that it has been given a special name, **interest-rate risk**.

Interest-rate risk can be quantitatively measured using the concept of *duration*

### there is no interest-rate risk for any bond whose time to maturity matches the holding period

Although long-term debt instruments have substantial interest-rate risk, **short-term debt instruments do not**. Bonds with a maturity that is as short as the holding period have no interest-rate risk (the coupon bond at the bottom of Table 2). Which has no uncertainty about the rate of return because it equals the yield to maturity. Which is known at the time the bond is purchased. The key to understanding why there is no interest-rate risk for any bond whose time to maturity matches the holding period is to recognize that (in this case) **the price at the end of the holding period is already fixed at the face value**. The change in interest rates can then have no effect on the price at the end of the holding period for these bonds. The return will therefore be equal to the yield to maturity known at the time the bond is purchased.

### discount bonds and zero-coupon bonds

The statement that there is no interest-rate risk for any bond whose time to maturity matches the holding period is literally true only for discount bonds and zero-coupon bonds that make no intermediate cash payments before the holding period is over. A coupon bond that makes an intermediate cash payment belore the holding period is over requires that this payment be reinvested. Because the interest rate at which this payment can be reinvested is uncertain, there is some uncertainty about the return on this coupon bond even when the time to maturity equals the holding period. However, the riskiness of the return on a coupon bond from reinvesting the coupon payments is typically quite small, so the basic point that a coupon bond with a time to maturity equaling the holding period has very little risk still holds true.