Bonds

Lecture 8

This lecture is part of Chapter 4:Investing in the Company

Today’s Lecture

What are Bonds in finance

How can we use Excel to calculate a reasonable bond price.

A bond from 1863

Bonds

In finance, a bond is a loan contract where an investor lends a government or a company a certain amount of money and then receives interest at regular intervals for some period of time.

The contract has a certain length and the originally stated loan amount is paid back in full at the end of the contract.

The US Treasury

Bonds

The are some terms one should be familiar with:

• Face Value: This is also called “Par Value”, “MaturityValue” or “Principal”. This is the amount that

needs tobe paid back at the end.

• Maturity Date: The date on which the principal needs tobe returned.

• Coupon Rate: The percentage of the regular interest.

Bonds

Bonds also have certificates like stocks. And also just like stocks, bonds can be traded. Hence the prices of bonds can fluctuate quite a bit. In fact because there’s a fixed regular interest on the bond and because market interest rates can change, the calculation of a reasonable price for a bond is not trivial. More on this later…

Even when the bonds are issued, one does not necessarily have to pay exactly the principal amount. There can be a premium or a discount.

Bonds

An important concept in finance is that of a/the discount rate. In the case of bonds this equals the interest rate that the Federal Reserve charges other banks on short term loans.The discount rate is an important reference rate for other interest rates.

Why is it called the discount rate?

It has to do with the present value of an investment that will pay off in the future.

Bonds

If e.g. we know that we will get 1000 dollars for a bond next year, and if the current interest rate is 5%, how much would the bond be worth now.

If the interest rate is 5%, the investment will be worth 105% in one year. Hence: 1000 dollars (the end value of our investment) = 105% of current value.

In other words current value = 1000/105% = 952 dollars.

How about if we get 1000 dollars in two years?

Bonds

Time for Excel!A B C D E F G H I

23 My first bond calculations45 Face Value 1,000.006 Current interest rate 5%78 Maturing in Current 9 years Value1011 1 952.38 =D5/(1+D6)12 2 907.03 =C11/(1+$D$6)13 3 863.84 =C12/(1+$D$6)14 4 822.7015 5 783.5316 6 746.2217 7 710.6814 8 676.8415 9 644.6116 10 613.91 Why the 1?

Bonds

Hence, for a bond which will give us 1000 dollars in 10 years time, 614 dollars is a reasonable price.

Technically speaking, this is a zero coupon bond since we are not getting any interest paid each year.

Let’s do the calculation again for the case that we get a yearly interest payment.

Bonds

The face value of the bond is 1000 dollars and at the end of the year we get 50 dollars interest (the interest rate is 5%).

This is a bit trivial so don’t think about it too much! In one year our investment will be worth 105% but we will also have 1050 dollars.

As before: current value = (end value)/105% = 1050/105%.

This is of course just 1000 dollars!

The same would be true for ten years.

Bonds

Hence:

10 year 1000 dollar bond with 5% interest paid yearly: Current Value 1000.

10 year 1000 dollar bond with no interest paid yearly: Current Value 614.

Perhaps this makes it a bit clearer why it’s called the ‘discount rate’.And now … bang … the big point:

Bonds

Interest rates change!!!!!

When a bond is issued, it’s interest rate is set to the prevailing rate of that moment.

It should come as no surprise that the value of a bond will also be influenced by the difference between it’s originally set (and let me stress fixed) coupon rate and the currently prevailing rate.

Bonds

Let us first illustrate this with an extreme example:

During freak-year 1, the interest rate is 50%. A 1000 dollar 10-year bond is issued and it will pay 500 dollars a year in interest.

A year later we’re in ‘Japan during the nineties’ mode and interest has dropped to 1%.

Since a new bond will only give us 1% a year (1 dollar vs the 500 dollars of the freak-bond), the freak-bond must be worth a lot. Roughly: 9 (the remaining years) * 500 + principal =5500 dollars.

Bonds

Let us look at this a bit more carefully:

We have a bond that will give us 1000 dollars in 9 years time that will also pay us 500 dollars of interest a year. Since the interest received can be reinvested at the end of the nine years we should have a bit more than the 5500 dollars.

Naturally bonds can be traded.

The CBOT in action.

Bonds

With Excel we find that at the end we have:A B C D E F G H I

23 My first bond calculations45 Face Value 1,000.006 Coupon Rate 50% Current Interest Rate 1%789 Year Total1011 1 500.0012 2 1,005.00 =C11*(1+$H$6) + $D$5*$D$613 3 1,515.0514 4 2,030.2015 5 2,550.5016 6 3,076.0117 7 3,606.7714 8 4,142.8415 9 4,684.2616 inc principal 5,684.26

Interest on Bond

Cash from previous year + current interest

This is the cash we end up with

Bonds

Now we can discount this back to todayA B C D E F G H I

23 My first bond calculations45 Face Value 5,684.006 Current interest rate 1%78 Maturing in Current 9 years Value1011 1 5,627.72 =D5/(1+D6)12 2 5,572.00 =C11/(1+$D$6)13 3 5,516.83 =C12/(1+$D$6)14 4 5,462.2115 5 5,408.1316 6 5,354.5917 7 5,301.5714 8 5,249.0815 9 5,197.1116

And we find that the current value is 5,197 dollars.

Bonds

Excellent! Again, a conceptually far from trivial problem could be solved with Excel by “UNDERSTANDING” it.

Of course we can also put this in equations:

T

T

tt r

FaceValue

r

CouponP

)1()1(1

Present Value of Coupons

Present Value of Face Value

Bond Value

With T = Maturity Date and r = Current Interest Rate

Bonds

Let’s try that out:

T

T

tt r

FaceValue

r

CouponP

)1()1(1

9

9

1 )01.01(

1000

)01.01(

500

tt

P

This is straightforwardly calculated in Excel: (Excel is a great general purpose calculator)

Bonds

Now we can discount this back to todayA B C D E F G H I

23 Year Value of Current Value of Total4 Coupon Face Value Value56 1 495.057 2 490.158 3 485.30 =500/POWER(1.01,B8)9 4 480.4910 5 475.7311 6 471.0212 7 466.3613 8 461.7414 9 457.17 =1000/POWER(1.01,9) =C16+E161516 Total 4,283.01 914.34 5,197.3517

And we find that the current value is 5,197 dollars.Hey! That is the same value we found before.

Bonds

Good to know that the Math works!

Excel has a large number of built-in functions. Hence if one tries to do something that appears to be quite standard, it’s always a good idea to try and see whether there’s a corresponding Excel function.

And indeed: the function is called PV

Presumably standing for Present Value

Bonds

Using the Excel built-in functionA B C D E F G H I

234 5197 =PV(0.01,9,-500,-1000)56

And we find that the current value is …. 5,197 dollars.

Yes, we did it again!

Note the minus, an Excel convention

PV()

Bonds

Understand the problem and use simple Excel functions

Use Math

Use the ‘black-box’ Excel built-in function

Hence we have found three ways to solve our problem:

Which one is the best?

Bonds

Thus far we have determined the current value of the bond based on the present and past interest rates.

However, on the bond market one does not directly trade in yields but in bonds. This basically reverses the problem as:

If I buy a bond at certain price, what is its yield?

BondsWe can again use the Solver to do the job:

A B C D E F G H I J K L23 Calcultate the total receipts Discount it back to the current value Purchase Price45 Face Value 1,000.00 Total Receipts 5,818.04 50006 Coupon Rate 50%7 Effective Yield 1.70%8 Maturing in Current 9 Year Total years Value Premium1011 1 500.00 1 5,720.90 0.00 =G19-J512 2 1,008.49 2 5,625.3913 3 1,525.61 3 5,531.4814 4 2,051.51 4 5,439.1315 5 2,586.35 5 5,348.3216 6 3,130.26 6 5,259.0317 7 3,683.41 7 5,171.2318 8 4,245.95 8 5,084.8919 9 4,818.04 9 5,000.0020 inc principal 5,818.04

The key thing to realize is that the Premium is 0 at the correct yield.

Bonds

And (should I say naturally) there’s also an Excel function that can do the job:

A B C D E F G234 Yield:56 1.70% =RATE(9,-500,5000,-1000)78910

Number of years left

Coupon

Purchase Price

Face Value

RATE()

Key Points of the Day

Bonds

Discount Rate

Using Excel to Understand and Calculated Yields and Values