Chapter 4

1. Chapter 4 Understanding Interest Rates

2. At the web site for the class I have an Excel file and a PDF file where you see the mechanics of the idea known as the time value of money. This time value of money concept is related to the notion of what we call the interest rate. Here we will incorporate the time value of money concept when we study debt instruments from a cash flow point of view. Different debt instruments have different cash flows. In the text the author wants to keep things relatively simple, but also express some important results that will use later. If you are into finance you will see we will compound interest only once a year. We will simplify other realities as well. We will consider the debt instruments called the simple loan, the fixed-payment loan, the coupon bond and the discount bond (and a special case!).

3. A simple loan essentially is a single payment idea mentioned in the other set of notes. (By the way, in the other notes I used the letter r for what I here call the letter i.) The main idea in a single payment was F = P(1 + i)n. Remember the term (1 + i)n was put into Excel and calculated for us. I called it the future value factor. For each $1 amount of P the value of F is the P times the future value factor. If we have the values for F, P, and n, then the value I in our work here is called the yield to maturity and this concept is what we mean by an interest rate. Say F = 12597, P = 10,000, and n = 3. On the Excel spreadsheet you can change the value of i at the top of the page, but I have it saved at i = 0.08 (8%). Now in the F/P column in row n we see the value 1.2597. Note this is the ratio of F to P in our problem and so the I value that connects the F and the P is 0.08 or 8%. Thus the yield to maturity here is 8%. Say in a new problem F = 1500, P = 900 and n = 5. The F to P ratio is 1500/900 = 1.6667. notice in the table when i = .08 in the F/P column in row n = 5 the value is 1.4693. Since 1.6667 does not equal 1.4693 we know the yield to maturity is not 8%.

4. In the Excel spreadsheet let’s change the i value to 0.09. The future value factor becomes 1.5386. Thus we know that since we need the value 1.6667 that we are getting closer. Let’s go to an interest rate of 0.10. We have 1.6105 and are getting closer. You can play around and see the yield to maturity or interest rate in this example is about 10.75%. Let’s modify this last example by making P = 1000. This means that from the last example we are saying the future value of 1500 is the same and the term is 5 years, but the present value is now 1000. The F to P ratio is now 1.5000 and the yield to maturity or interest rate would now be just a shade over 8.4%. So, with the last two examples in mind, if the future value and term are the same, then a higher present value means a lower yield to maturity. So, the debt instrument that pays out 1500 in the future has a lower yield the more one has to pay today to acquire the instrument.

5. A fixed payment loan, or fully amortized loan, is a situation where an amount is given today and to pay off the loan n equal payments (what I call A in the other notes and the book call FP) are made at regular intervals (yearly here) and after the last, or nth, payment is made the loan is paid off. For example, say your grand mother gives you $10 and wants you to pay her pack in two equal payments at the end years 1 and 2 and she wants 8% interest per year. The Excel column labeled A/P has the value .5608 in the row n =2. Thus A or P = 10(.5608) = 5.608. Let’s look at this mechanically. When you get the $10 at the end of the year the interest clock will charge you 8% of the $10 you borrowed and so at the end of the year you will owe 10.80. But you will make a payment of 5.608 and so will still owe 10.80 – 5.608 = 5.192 at the end of the first period and by interest clock you will be charged 8% more on this the second period and thus you will owe 5.192(1.08) = 5.60736 and so when you make the second payment of 5.608 you have paid back the loan. The A/P formula takes this into consideration.

6. Now, as another example, say you will get a loan for $100,000 and you will pay it off in equal payments of 8880 at the end of each of 30 years. The A to P ratio here is 8880/100000 = .0888. When you use the Excel spreadsheet with the interest rate of 8% (you use 0.08) you see in the row n = 30 the A/P ratio is .0888. Thus, the yield to maturity here is 8%. Let’s change the problem and say you can only get a 30 year low with yearly payments of 8880 and the most you will be lent is $95,000. The A/P ratio now is .0935 and so we have to change the i value at the top of the Excel page to get the new yield to maturity. The new yield to maturity is about 8.6%. Here we see that if the future value(s) are not changed and the term is not changed, then a lowering of the present value means a higher interest rate. This is similar to the previous example in that the present value and the yield to maturity move in opposite directions. This is significant because in bond markets the present value is usually the price of the bond. So, if the price changes the yield to maturity changes.

7. A coupon bond will require us to define some terms up front. At the end of n periods, the maturity time of the bond, the face value or par value of the bond will be paid to the owner of the bond. The owner of the bond will also be paid the coupon rate at the end of each of the n periods. The coupon rate is a percentage of the face value. Say the price today of a bond is 889.20 and the buyer will be paid 10% of the par value of 1000 for each of 8 years and at the end of 8 years will be paid the par value. So, at the end of each of 8 years the owner gets 100 and at the end of the eighth year gets 1000. The idea to find the yield to maturity is to have 889.20 = 100(P/A) + 1000(P/F). So, we have to pick an i value so that the P/A and P/F plug into the formula and equals 889.20.

8. At 8% we have 100(P/A) + 1000(P/F) = 100(5.7466) + 1000(.5403) = 1114.96. So, this is not the correct value. At 12.25 % (after some trial and error) 100(P/A) + 1000(P/F) = 100(4.9245) + 1000(.3967) = 889.15. Thus the yield to maturity on this coupon bond is about 12.25%. If we change the example and the price or present value rises to 1000, then 1000 = 100(P/A) + 1000(P/F) would be need to find the yield to maturity. You will see that the rate is 10%. Again we see the bond price or present value is inversely related to the yield to maturity. A special case of the coupon bond is the consol or perpetuity. There is no maturity and the owner is paid the same amount every year. As an example, if a person pays $2000 for the right to be paid $100 a year forever, then the yield to maturity is a simple calculation – 100/2000 = 0.05 or 5%.

9. If the same $100 in perpetuity can be had for a price of only $1000, then the yield to maturity become 100/1000 = .10 or 10%. Bond prices (present value) and interest rates (the yield to maturity) move in opposite directions!!! A discount bond is similar to a simple loan. There is no coupon, but the holder does get paid the face value of the bond in n periods. The single payment concept is at work here. If a discount bond has a face value of 1000 in 5 years and the price today is 680.6, then we see the P/F ratio is .6806 and this occurs at an interest rate of 8%. But, if the price today is 620.9 then the P/F ratio is .6209 and you will see this happens when you put the i value in the Excel to 10% or 0.10. So, yet again we see the bond price and the interest rate are inversely related.

10. Can you foresee the future? Well, the sun will rise each day (even when it is cloudy), and, and, and,… other things will surely happen. But, if you deal with a coupon bond for a period of 3 years, for example, then your outcome may or not be to your liking because maybe you can not foresee everything in the future. Say there is a coupon bond that pays 100 each year for 3 years and the face value of 1000 is paid back at the end of the third year. The individual that pays a price or present value of 1000 would have a yield to maturity of 10%. A time line view of the bond is +1000 So, the idea is that if an individual puts down 1000 for a bond there is a promise to get paid 100 at the end of each of 3 years and at the end of the third year the face value of 1000 is also paid. 100 100 100 1000

11. If at the end of the year, when the first 100 is collected, the interest rate is still 10%, then this bond still will earn the individual 10%. In fact if the current owner had to sell out (needs the funds) the bond could be sold for 1000 because this bond is no better or no worse than other investments (one can earn 10% out there!). At the end of the first year the bond really becomes a two year bond (what is left until maturity). +1000 But, sometimes at the end of the first year the interest rate is no longer 10% (we will try to explain why in later chapters). Say the interest rate becomes 20% (exaggerate for teaching purposes). 100 100 100 1000 1000

12. When the interest rate is 20% at the end of the first year our picture below is no longer adequate. Since this bond is already set up to pay 100 at the end of the second and third years and also have 1000 paid back at the end of the third year, the bond can no longer be sold for 1000 at the end of the third year. People with 1000 would expect more then 100 each year and 1000 at the end if current rates are 20%. At the end of the first year the bond is now worth (at 20% for 2 years) 100(P/A) + 1000(P/F) = 100(1.5278) + 1000(.6944) = 847.18. So, if the owner had to sell the bond could be sold for 847.18. +1000 100 100 100 1000 1000 847.18

13. So, when the interest rate changes the bond has a change in its return. The rate of return is defined as the payments to the owner plus the change in value, expressed as a percentage of the purchase price. In our work we will only look at this from year to year, or one year at a time. So if a 3 year 10% coupon bond with 1000 face value is purchased for 1000 and then at the end of the first year the interest rate in the world is 20% the bond becomes worth 847.18 at that point. If sold the bond would have a return of [100 – (847.18 – 1000)] / 1000 = -52.82 / 1000 = - 0.05282 The return would be negative! What have we demonstrated? I have no idea, but this sure was fun!!!! Really though, I can summarize next.

14. The coupon bond as originally presented has a yield to maturity of 10%. If interest rates never change then the return on the bond will be 10%. But, when interest rates change the return on the bond changes. If interest rates rise from the time of the original purchase then the return on the bond become negative. Similarly if the interest rate falls the return on the bond becomes positive. The return from one year to the next on a bond is R and if the coupon is C and the price of the bond in year t is pt and the price of the bond in year t+1 is pt+1, then the return on the bond in year t+1 at time t+1 is R = (C + pt+1 – pt) / pt = [C / pt] + [(pt+1 – pt) / pt)] = coupon yield + rate of capital gain. Page 79 has a nice summary of ideas and you should write these ideas down 50 times, or how ever long it takes you to remember them. Here is what I take from this. When you have 1000 bucks to invest the longer time frame you lock in that investment the more you could get burned by interest changes.

15. If you have 1000 and you buy an investment for 30 days when the 30 days are over you can reinvest in the same place or elsewhere. But if you lock in for 10 years and rates change in 30 days or 60 days or 5 years you are still locked in for 10 years! Because day to today interest rates do not move up and down as much as they do over a longer period of time, prices and returns for longer-run bonds are more volatile than those for short-term bonds. So, folks are concerned and consider interest rate risk when considering bonds of various durations. With this in mind, today I would expect longer-term bonds to be cheaper per 1000 of par value to compensate for the greater risk.

16. The last detail we want to consider here is the idea about a nominal interest rate and a real interest rate. Say you have $100 and you put it in your freezer for a year. At the end of the year you have $100 in cold, hard cash! And you earned 0 percent interest! Now, if the prices of things you buy went up 10% during the year, then your $100 at the end of the year buys less than what it would buy at the beginning of the year. If at the beginning of the year cans of coke cost 0.50 and at the end of the year they cost 0.55 then in terms of the real thing you could have had 200 cokes at the beginning of the year but only 181.18 cokes at the end of the year. Does a can of coke go bad in a year if unopened? If not you would have been better off buying the 200 cans of coke and storing them over the year. If you can earn 5% for the year at a bank then your $100 can become $105 at the end of the year. But if prices go up 10% you still lose purchasing power. 200 cokes are given up and you only get back 190.91 cokes.

17. In the bank example the 5% given by the bank is called the nominal interest rate because at the end of the year you will get back 5% more dollars. But with inflation the real purchasing power of the dollars is eroded. So, inflation means real earnings are less than nominal earnings. We typically say the real rate of interest = nominal rate of interest minus the inflation rate. Ex ante and ex post are phrases that basically stand for, respectively, before the fact and after the fact. So, before you enter a deal if you really want to earn 5% and you expect inflation of 3% then the nominal rate should be 8%. After the deal if you are getting 8% in nominal terms and the inflation rate is 4% you only earn 4% in real terms.

18. The author suggests when the real rate of return is low there are greater incentives to borrow and fewer incentives to lend. The basic logic is borrows do not pay back much in terms of purchasing power and so they are eager to borrow (relatively). Lenders say why give up stuff today if I do not get back much more in the future!!!!!