We have all heard of bonds, and might have also bought some bonds in the past, however I think it’s not too easy for some of us to understand the terminologies involved in the fixed income, for instance, how the bond price is determined? Let’s give it a shot in this Economics 101.
Discount model
More often we hear about the interest rate, which is established based on a discount model. The discount model tells how much the money from the future is worth today. Taking a $100 bill in future, the price today can be determined by lowering it by a discount:
$100’s price = $100 - A Discount Say the future is in five years, the discount at the end of five years is $5. Then its today’s price is $95.
The discount varies, for five years, it’s $5. For ten years it can be $10. Normally the bigger the discount the more distant into the future. In order for people to wait longer to receive the money in the future, you need to offer a more discounted price today to convince them for the purchase.
A bond then can be derived from the discount modal.
What is a bond?
If you purchase a bond, you enter an agreement to pay the bond price today, and receive $100 back in a future date when it’s matured. During now and the mature date, it also promises to pay you a coupon every six month.
We can see how the bond price is calculated:
Bond's price = $100’s price + Sum(Coupon's price)There’re two parts: the first is the $100’s price we have already learned, because you are promised to get it back at the mature date; the second part is the sum of each coupon’s price.
Say you are purchasing the bond with coupon 4% that matures in a year. So you should get the money back in a year as well as two coupon payments, one after six months, and one at the end of the year.
We already know the discount for a year is $5. So without the coupon payment, the price of bond would be $95.
Let’s calculate the first coupon payment price:
First coupon's price = ($100 - Discount) * (4% / 2)Say the discount for the six months is $2. And the coupon rate is given for a whole year, thus we can get 2% (4% / 2) each six months. Therefore ($100-$2)*2% would get us $1.96 . Essentially we are promised to get $2, but it needs to be discounted to today’s price.
Let’s continue to calculate the second coupon payment price:
Second coupon's price = ($100 - Discount) * (4% / 2)Similarly, the formula is the same for the second coupon except the discount is not $2, instead it’s the same one we used earlier $5. Therefore ($100-$5)*2% would get us $1.9, a bit cheaper than the first coupon price. Making sense ;)
The bond’s price can be a sum of the above three pieces:
Bond's price = $95 + $1.96 + $1.9 = $98.96Wow, we know how the bond price is calculated now. It’s not difficult, is it?
$100 worth $100?
Hopefully by now we understand $100 in the future has to be discounted to be worthwhile today. So if someone offer you $60, it doesn’t really necessarily mean it’s cheap, it could just imply long mature duration, ex. 30 years. Note: there’re other reasons, we’re not discussing them here.
Each coupon payment is an additional income which adds to the price of the bond. So you might wonder can it be higher than $100 after all the coupons? It can be. But one thing is for sure. Don’t take this word “$100” too serious. Of course we are trying to figure out the price for $100 investment. However, either the price is $50 or $130, with one below $100 and one above $100, is not that important. Because the discount could be large or small based on the current market. Essentially $130 doesn’t make it very expensive, $50 doesn’t make it very cheap. If you want to make a profit based on the fluctuation of the bond price from day to day, you should treat the price similar to the stock’s price. What’s more important is the history of values, not the absolute value itself.
Now to answer is $100 worth $100? From the above analysis, I have to admit, in general no. But in the bond case, if the addition coupon can make up the price to be close to $100 or even above it, then I’ll say mathematically yes, it can be still worth $100. Of course the model need to hold from now to the mature date. BTW, even when $100 is worth $100, it doesn’t mean it’s a bargain (or fair), it just means it’s comparable.
Summary
To calculate the price of the future $100 in today’s money, you need to apply a discount. The farther into the future, the larger the discount. To calculate the bond’s price, you can apply the discount to each coupon and your investment and add them together. This might help answer your question: Is $100 worth $100?
