What is Your Rate of Return: Explanation & Calculation
In short:
Yield to Maturity (YTM) is the expected rate of return on a bond or fixed-rate security that is held to maturity. Since bonds do not always trade at face value, YTM gives investors a method to calculate the yield they can expect to earn on a bond.
Assumptions of YTM are that the investor holds the bond until its maturity date, all coupon payments are made in full and on time, and all coupons are quickly reinvested at the same rate of return. In practice, reinvesting coupon payments at the same rate of return will likely prove difficult so YTM may only give an investor a general yield idea and metric for comparison.
Additionally, YTM accounts for the time value of money. That is, money today is worth more than money tomorrow. This is especially important for long-term bonds since the maturity date could be 10, 20, 30 years from the purchase.
Key Points
- Yield to Maturity is the expected return of a bond held until maturity.
- Assumptions: bond is held to maturity, all coupons are paid on time in full, coupons are reinvested at the same rate of return.
- Yield to Maturity accounts for the time value of money.
- Coupons are “cash payments” | Coupon rate is the rate of return
In-depth:
Yield to Maturity for Bonds without Coupons – Zero-Coupon Bonds
The easiest way to understand yield to maturity is to look at YTM on two different types of bonds as the calculations will be different. The first, which is what we’ll talk about in this section, is a zero-coupon bond or a bond that doesn’t have periodic payments.
These bonds are sold and traded at a deep discount to the value they will be redeemed at on the maturity date. Investors make money with these bonds from the appreciation in value over time much like a stock investor benefits from a rising share price on a stock with no dividend.
Formula:
Example:
Say we have a zero-coupon bond with a face value of $1,000, its current price is $800, and has 3 years left until maturity. We could calculate the yield to maturity by simply entering these values into the formula above. Doing so would look like this:
Step 1: ( $1,000 / $800 )^(1/3) – 1
Step 2: (1.25)^(1/3) – 1
Step 3: 1.077 – 1
Answer: 7.7%
In this case, our YTM for this zero-coupon bond is 7.7%
Yield to Maturity for Bonds with Coupons
The second type of bond to calculate YTM for is a traditional coupon bond. The math to calculate the YTM on these bonds is more difficult than for a zero-coupon bond but the logic and underlying principles are the same.
These bonds often trade much closer to their face value since the investor is receiving a periodic coupon. Investors can still make money from the change in the bonds current price, although it’s not exactly like stocks.
YTM Calculation – Coupon Bonds
Since coupon bonds do have periodic payments, we need to account for them in calculating YTM. As such, the formula we use for these bonds is more complex because we are finding the interest rate that makes all future coupon payments equal to the current price of the bond.
Formula:
What we are doing in this formula is discounting all the coupon payments that will be made by this bond by both an interest rate and the year in which the coupon will be received. This allows us to find the present value, what they are worth if received today, of these coupons and face value.
When the sum of all the coupons and face value are equal to the bond price then the interest rate used to accomplish this is the YTM.
Trial and Error – Example
To solve for YTM by hand will require a process of trial and error and an understanding of the relationship between bond price and yield. For bonds whose current price is equal to their par value, their coupon rate is the interest rate. If bonds trade above their par value, called “at a premium”, then the interest rate must be lower than the coupon rate. Conversely, bonds that trade below their par value, called “at a discount”, must have an interest rate higher than the coupon rate.
Now that we have an understanding of the relationship between bond price and yield, we can consider an example to calculate the YTM via trial and error.
Say we are looking at a bond which is currently trading at $935, has 10 years left until mature, the face value of the bond is $1,000, and pays an annual coupon of $50. What is the YTM for this bond?
Our trial and error approach applying the formula above may look like this.
Since the bond is trading at a discount we know that the interest rate must be higher than the coupon rate. From there we can continue to increment higher the interest rate until the sum of the right side of the equation equals the left. At that point, we have found the YTM.
Calculating YTM In Excel
While calculating YTM by hand through trial and error might help someone understand the process of how YTM is calculated. This guessing and checking method isn’t really practical since it requires multiple iterations to solve for YTM. An alternative is to allow a computer to run these iterations for you which we can do in Microsoft Excel.
YTM Underlying ideas
What’s going on behind the scene during the calculation?
Truly understanding yield to maturity requires an understanding of a few foundational components of finance. If you can get a sense of these concepts then understanding what is going on with YTM should be more intuitive.
Time Value of Money – what’s happening during the calculation
The time value of money is actually a really simple concept, all it means is that money today is worth more than money tomorrow. But how can this be? Isn’t $1 today still $1 tomorrow? Yes, however, time has passed meaning you could have done something with that money over that one day to earn more money. The simplest thing is an interest-bearing savings account.
Time value of money applies to bonds in that the same coupon an investor receives 15 years after they purchase the bond will be less valuable than the same coupon they are receiving today. Yield to maturity gives investors a way to account for this.
Present Value – what’s happening during the calculation
So how do you account for the time value of money issue on bonds? You calculate out what all the coupon’s value would be if they were all received today, called the present value. The fancy way to say this is that you discount future coupon values back to their present value -which is what YTM is doing.
We can take all the present values and set them equal to the current price of the bond to find equilibrium. When we achieve both sides being the same value we have found the discount/interest rate.
Yield to Maturity vs Current Yield
So yield to maturity vs current yield, what’s the difference? Current yield is simply the interest rate an investor would earn if they bought the bond at market price today and held it for one year. This value can be found by dividing the coupon or annual cash inflow by the current market price.
Current Yield = Annual Coupon Amount / Current Market Price
The difference between current yield and yield to maturity is that current yield does not account for the present value of future cash inflows of the bond. This means that current yield is a much shorter snapshot or return on investment than yield to maturity.
In other words, you would not use current yield to find out the return on investment for holding a bond long term.
Yield to Maturity Uses
The main use of yield to maturity is as a comparison metric. Since it is unlikely that an investor will be able to reinvest all the coupons at the same rate of return the YTM value isn’t likely to reflect what the investor will actually receive from the bond. Instead, the investor can use the YTM value to compare between other investments to optimize their risk-reward profile.
Frequently Asked Questions
Yield to Maturity (YTM) is the expected rate of return on a bond or fixed-rate security that is held to maturity. There are two formulas for yield to maturity depending on the bond. The yield to maturity formula for a zero-coupon bond: Yield to maturity = [(Face Value / Current Value)(1 / time periods)] -1. The yield to maturity formula for a coupon bond: Bond Price = [ Coupon x (1 – (1 / (1 + YTM)n) / YTM) ] + [ Face Value x (1 / (1 + YTM)n ) ].
The main importance of yield to maturity is as a comparison metric. This allows investors to compare like type fixed-rate securities and decide which is best suited for their goals.
Current yield is simply the interest rate an investor would earn if they bought the bond at market price today and held it for one year. Yield to maturity is the effective interest rate an investor would earn by holding the bond until maturity. The difference between current yield and yield to maturity is that current yield does not account for the present value of future cash inflows of the bond.