## Monday, January 23, 2017

### Bond Basics: Part 6

In Part 5 of this series on bond basics, I derived the formula to calculate the price of a one-year bond in terms of its yield. I started by developing a formula to calculate something more familiar: the amount you end up with in a savings account after one year. In this part of the series I'll derive the formula to calculate the price of a bond with a term to maturity of more than one year, and again, I'll start with the more familiar concept of compound interest in a savings account.

We saw in Part 5 that we can calculate the amount we'll end up with in a savings account after one year with this time value of money (TVM) formula:
```  FV = PV * (1 + i)
```
where FV is the Future Value, PV is the Present Value, and i is the annual interest rate (and * is the multiplication operator).

If we left the money in the savings account for an additional year, we could apply this same formula to calculate the value at the end of the second year, but the starting value at the beginning of year 2 would be the value at the end of year 1. In other words, FV from our calculation for year 1 would be PV in our calculation for year 2. To get set up for the year 2 calculation, let's rewrite the TVM formula above as:
```  FV1 = PV * (1+i)     (Year 1)
```
where FV1 indicates the Future Value at the end of year 1.

Since FV1 is the amount we end up with at the end of year 1, it is the amount we start with, the present value, in year 2. So we can use FV1 as PV in the year-2 formula:
```  FV2 = FV1 * (1+i)    (Year 2)
```
We can then substitute the right-hand side of the year-1 formula for FV1 in the year-2 formula as follows (from here on I'll use extra spaces and brackets to help clarify which expression is being substituted for another):
```  FV2 = [    FV1     ] * (1+i)

FV2 = [ PV * (1+i) ] * (1+i)

FV2 =    PV * (1+i) * (1+i)
```
As a numeric example, for a starting year-1 value of 100 (PV = 100 in the year-1 formula), and an interest rate of 1% (0.01), the value of our account at the end of two years (FV2) would be:
```  FV2 =  PV  *  (  1 +  i   ) * ( 1 +  i   )

FV2 = 100  *  (  1 + 0.01 ) * ( 1 + 0.01 )

FV2 = 100  *      1.01      *     1.01

FV2 = 102.01
```
You will remember from elementary arithmetic that we can write a number multiplied by itself as that number squared, for example we can write 2*2 as 22, or using ^ is the exponentiation operator (as used in spreadsheet formulas), we can write it as 2^2. Thus, we can write (1+i) * (1+i) as (1+i)^2, and rewrite the 2-year compound interest TVM formula as:
```  FV2 = PV * [ (1+i) * (1+i) ]

FV2 = PV * [     (1+i)^2   ]

FV2 = PV * (1+i)^2
```
Note that the only difference between the 2-year and 1-year compound interest formulas is that we multiplied by an additional factor of (1+i) in the 2-year formula. We can use the same reasoning to extend the formula to any number of years. For example, the 3-year compound interest formula is:
```  FV3 = PV * (1+i) * (1+i) * (1+i)
```
Just as we can write 2*2 as 22 or 2^2 (two squared), we can write 2*2*2 as 23 or 2^3 (two cubed), which means we can rewrite the 3-year compound interest formula as:
```  FV3 = PV * [ (1+i) * (1+i) * (1+i) ]

FV3 = PV * [         (1+i)^3       ]

FV3 = PV * (1+i)^3
```
We can generalize this formula to any number of years, n, and write it as:
```  FVn = PV * (1+i)^n
```
which usually is written simply as:
```  FV = PV * (1+i)^n
```
We have derived the generic compound interest TVM formula used to calculate the future value after n years given a present value and an interest rate. I have found this to be the single most useful financial formula, whether used directly or used to derive a related TVM formula, as we'll do next.

The formula above gives us future value in terms of present value, but we saw in Part 5 that bond price is related to a present value. So we want to rearrange the above formula, solving for PV in terms of FV, using the same simple algebra we used in Part 5. Dividing both sides of the equation by (1+i)^n, and reversing the order of the equation, we get:
```  PV = FV / (1+i)^n
```
As I explained in Part 5, FV in a TVM formula can be referred to as a future cash flow. So we can say that the present value of a future cash flow is the future cash flow discounted by the discount rate (i). The discounting is done by dividing the future cash flow by the sum of 1 and the interest rate raised to the nth power, where n is the number of years over which we are compounding.

The formula above is for a single future cash flow, FV, at the end of n years. This formula can be extended to calculate the present value of multiple annual cash flows over N years as follows:
```  PV = CF1 / (1+i)^1 + CF2 / (1+i)^2 + ... + CFN / (1+i)^N
```
where CF1 is the cash flow received at the end of year 1, CF2 is the cash flow received at the end of year 2, and CFN is the cash flow received at the end of year N (the last year). The ellipses ( ... ) represent the cash flows between cash flow 2 and the last cash flow. So if we were solving this for a five-year period, there would be five terms in the right-hand side of the equation, one each for cash flow term CF1 through CF5. I'll refer to this formula as the present value of discounted cash flows formula.

In Part 5 we saw that bond price is described in terms of bond yield as follows:
A bond's price is the present value of its future cash flows discounted at a rate equal to the bond's yield.
Looking at the present value of discounted cash flows formula, we see that the present value of each cash flow is the cash flow divided by the factor (1+i)^n, where i is the interest rate and n is the year in which the cash flow is received. For bonds, the analog of interest in a savings account is the bond's yield, so we'll use y to represent yield instead of i for interest rate, and rewrite the discount factor for year n as (1+y)^n.

So in the the present value of discounted cash flows formula, the present value of each bond cash flow for year n can be written as:
```  PVn = CFn / (1+y)^n
```
All future bond cash flows, except the last one, consist of an interest payment, also referred to as a coupon payment. The final cash flow consists of the final interest payment and the principal payment.

Assuming annual interest payments, each interest payment is the interest rate (coupon rate) times the par value (face value) of the bond. For example, for a par value of 100 and a coupon rate of 1%, each annual interest payment would be 100 * 1% = 1. Using a par value of 100 and designating the coupon rate as r, each annual interest payment cash flow is:
```  CFn = 100 * r
```
Substituting the right-hand side of this equation for CFn in the annual interest payment cash flow formula above:
```  PVn = [   CFn   ] / (1+y)^n

PVn = [ 100 * r ] / (1+y)^n
```
This formula gives us the discounted cash flow for the annual interest payment in year n for a bond with par value 100, coupon rate r, and yield y.

The final cash flow also includes the principal payment, which for a bond with par value 100 is 100. This is discounted by the same discount factor, which for the last year, N, is (1+y)^N. We could just add the discounted principal payment as a final term in the discounted cash flow formula; this final term is:
```  Discounted principal payment = 100 / (1+y)^N
```
Using this approach, the final cash flow would consist of two terms, one for the final interest payment and one for the principal payment. Designating the present value of this final cash flow in the last year, N, as PVN:
```  PVN = [ 100 * r / (1+y)^N ] + [ 100 / (1+y)^N ]
```
Alternately, we can combine the final interest payment and the principal payment into a single cash flow, CFN, consisting of the principal payment of 100 plus the final interest payment of 100*r, and use the distributive property to simplify as follows:
```  CFN = 100 + 100 * r

CFN = 100 * (1 + r)
```
We can then rewrite the final discounted cash flow formula in the simpler form:
```  PVN = [      CFN    ] / (1+y)^N

PVN = [ 100 * (1+r) ] / (1+y)^N

PVN =  100 * (1+r) / (1+y)^N
```
Combining the final discounted cash flow with the discounted cash flows for the annual interest payments, we can write the present value of discounted cash flows formula as:
```  PV = [  CF1  ]  / (1+y)^1 + [  CF2  ] / (1+y)^2 + ... + [    CFN    ] / (1+y)^N

PV = [ 100*r ]  / (1+y)^1 + [ 100*r ] / (1+y)^2 + ... + [ 100*(1+r) ] / (1+y)^N
```
Since a bond's price is the present value of its future cash flows discounted at a rate equal to the bond's yield, and since bonds are priced as a percent of par value (so a bond's price at maturity is 100), we have derived the formula for the price of a bond with with coupon rate r, yield y, and term to maturity N. So we can rewrite the present value formula as a bond price formula by using P for Price instead of PV for Present Value:
```  P = 100*r / (1+y)^1 + 100*r / (1+y)^2 + ... + 100*(1+r) / (1+y)^N
```
As an example calculation, let's calculate the price of a 3-year bond (N=3) with a yield, y, of 1.5% (the yield of a 3-year Treasury as I write this), and a coupon rate, r, of 1.5%. Writing 1.5% in decimal form as 0.015, the bond price formula is:
```  P = 100 * r   / (1  + y )^1 + 100 * r   / ( 1 + y )^2 + 100*( 1 + r ) / ( 1 + y )^3

P = 100*0.015 / (1+0.015)^1 + 100*0.015 / (1+0.015)^2 + 100*(1+0.015) / (1+0.015)^3
```
You can either copy and paste the right-hand side of the equation into a Google search box or into a spreadsheet (preceded by = to indicate that it's a formula), or use a calculator (careful with the parentheses) to determine that in this case P = 100.

This result is not surprising if you recall from Part 2 that with some reasoning about bond price and yield, we determined that yield equals coupon rate for a bond priced at par (100). In other words, bond price is 100 if yield equals coupon rate.

We've learned in this series that bond price and yield move in opposite directions, and now we can verify this for a 3-year bond with the bond price formula. Let's assume that the 3-year Treasury yield jumps from 1.5% to 2.0% today. Coupon rate is fixed, so r would remain 1.5% (0.015), but the yield, y, increases to 2.0% (0.020), and our bond price formula becomes:
```  P = 100 * r   / (1 + y  )^1 + 100 * r   / (1 + y )^2 + 100*( 1 + r  ) / (1 + y  )^3

P = 100*0.015 / (1+0.020)^1 + 100*0.015 / (1+0.020)^2 + 100*(1+0.015) / (1+0.020)^3
```
Performing this calculation with one of the methods mentioned, the result is P = 98.56, which verifies that the bond price decreases when the yield increases.

When the yield increases by 0.5 percentage points from 1.5% to 2.0%, the price decreases by 1.44% (98.56 / 100 - 1 = -1.44%). Note that the price percentage decrease is a little less than three times the percentage point increase (1.44 / 0.5 = 2.88), which is a little less than the bond maturity of three years. This is not a coincidence, but is related to the bond concept of duration, which I intend to discuss in a future post in this series.

Although I think understanding the bond price formula is a great way to deepen one's understanding of the relationship between bond price and yield, the formula is cumbersome to use, especially for a bond with a long term to maturity; e.g., for a 20-year bond we'd have 20 discounted cash flow terms in the formula. Fortunately there are spreadsheet formulas and online calculators that can be used to easily calculate bond price, which I plan to discuss in the next post in this series.