1. GEK2507 1 Frederick H. Willeboordse frederik@chaos.nus.edu.sg Compound & Prosper!

2. GEK2507 2 Bonds Lecture 9

3. GEK2507 3 Today’s Lecture Excel Inflation What are Bonds in finance How can we use Excel to calculate a reasonable bond price. A bond from 1863

4. GEK2507 4 Excel Basics – Date Math Calculating the number of days between two dates or doing math on dates is tricky. Hence Excel stores dates to ‘number of days since Jan 1, 1900’. This is done with the DATE function. How the date is displayed in a cell depends on how the cell is formatted. In the top picture it is formatted as a ‘date’ and in the bottom picture it is formatted as a number. Syntax:

5. GEK2507 5 Excel Basics – Date Math This allows us to easily calculate the number of days from one date to another.

6. GEK2507 6 Excel Basics – Series Fill 1) Enter the first value 2) Select a range to fill 3) Go to the Fill-Series Menu 4) Choose how you want to do the fill Result

7. GEK2507 7 Inflation is the phenomenon that goods become more expensive (and hence that thus their price ‘inflates’). As a consequence either one dollar can buy less goods, or one needs more dollars to pay for the same item. Hence inflation has an effect inverse to that of receiving interest. Inflation

8. GEK2507 8 Inverse of the future value formula. Same as present value formula! Inflation Effect of inflation on the value of $1000 in current dollars.

9. GEK2507 9 Note the subtle point - Though close, there is a difference between: 1000 * (1-r) and 1000/(1+r) Eg. 1000 * 0.97 = 970 and 1000/1.03 = 970.87 This may look like a small difference, but differences can add up! Note Inflation

10. GEK2507 10 From this calculation we see that in terms of today’s buying power our original 1000 will only be worth 744 dollars in ten years. But we had also seen in the lecture on values that our 1000 will grow to 2594 if invested at 10% a year. Oh that’s only 256 dollars less so in 10 years, after inflation, we still should have: 2594 – 256 = 2338 dollars Not too bad... WRONG! Inflation

11. GEK2507 11 If we know that we have 2,594 dollars in 10 years then we need to discount this back to today with the prevailing inflation rate in order to see how much that is in today’s dollars. Hence we’ll only have 1930 Inflation

12. GEK2507 12 For this case it is more useful to combine the two calculations: Indeed the same Inflation

13. GEK2507 13 Couldn’t we just say Effective Interest = Interest – Inflation? It’s different! Inflation

14. GEK2507 14 It’s different because interest and inflation are multiplicative factors. Or: 1/1.03 * 1.1 = 1.06796 unequal 1.07! It’s small but nevertheless important. Inflation

15. GEK2507 15 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

16. GEK2507 16 The are some terms one should be familiar with: Face Value: This is also called “Par Value”, “Maturity Value” or “Principal”. This is the amount that needs to be paid back at the end. Maturity Date: The date on which the principal needs to be returned. Coupon Rate: The percentage of the regular interest. Bonds

17. GEK2507 17 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

18. GEK2507 18 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

19. GEK2507 19 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

20. GEK2507 20 Time for Excel! Why the 1? Bonds Inverse of the future value formula. Same as present value formula!

21. GEK2507 21 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

22. GEK2507 22 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

23. GEK2507 23 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

24. GEK2507 24 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

25. GEK2507 25 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

26. GEK2507 26 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

27. GEK2507 27 With Excel we find that at the end we have: Interest on Bond Cash from previous year + current interest This is the cash we end up with Bonds

28. GEK2507 28 Now we can discount this back to today And we find that the current value is 5,197 dollars. Bonds

29. GEK2507 29 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: Present Value of Coupons Present Value of Face Value Bond Value With T = Time to Maturity, r = Current Interest Rate Bonds

30. GEK2507 30 Let’s try that out: This is straightforwardly calculated in Excel: (Excel is a great general purpose calculator) Bonds

31. GEK2507 31 Now we can discount this back to today And we find that the current value is 5,197 dollars. Hey! That is the same value we found before. Bonds

32. GEK2507 32 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 PV Bonds

33. GEK2507 33 Using the Excel built-in function And we find that the current value is …. 5,197 dollars. Yes, we did it again! Note the minus, an Excel convention PV() Bonds

34. GEK2507 34 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

35. GEK2507 35 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? Bonds

36. GEK2507 36 We can again use the Solver to do the job: The key thing to realize is that the Premium is 0 at the correct yield. Bonds

37. GEK2507 37 And (should I say naturally) there’s also an Excel function that can do the job: Number of years left Coupon Purchase Price Face Value RATE() Bonds

38. GEK2507 38 Bonds Discount Rate Using Excel to Understand and Calculated Yields and Values Key Points of the Day