2. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main1 Lecture 3 UNDERSTANDING INTEREST RATES (1)

3. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main2 What means “interest rate” ? Economists use the term “interest rate” usually in the sense of “yield to maturity” of a credit market instrument

4. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main3 Credit market instruments (1) Simple loan: the borrower receives an amount of funds (principal) that is to be repaid to the lender at the maturity date, plus an additional payment: interests Fixed-payment loan (annuity): the amount of funds, including interests, is to be repaid periodically in equal installments.

5. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main4 Credit market instruments (2) A coupon bond: the borrower makes a periodical “coupon payment” on his/her interests, and redeems the principal in full at maturity (at face or par value). A discount bond or zero(-coupon) bond: it is bought at a price below its face value (at a discount) and paid off at face value when maturing.

6. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main5 The concept of “present value” In order to render different credit market instruments commensurable, the concept of “present value” is useful. It “discounts” all payments connected to a loan made in different periods to a single point in time, for for instance “today” (present time).

7. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main6 Present value of a simple loan The interest payment divided by the amount of the loan is a sensible way of measuring the cost of borrowing funds. It is the simple “interest rate” p.a.. Example A loan of €1,000 redeemable in one year at €1100 (which includes €100 interests): i = €100 / € 1,000 = 0.10 = 10% p.a.

8. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main7 Present value for multiple periods Example: If a loan of €1,000 is made at 10% interests p.a., the following time profile of the loan is generated: €1,210€1,331€1,464 €1,000 1342 time €1,100 Or more formally: €1,000 * (1 + i) t

9. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main8 Present (discounted) value Similarly we can turn a future payment into today’s value (“discounting the future”). Today (t=0)Future (t=4) PV = R 0 = R n / (1 + i) n Rn Rn € 1,000 € 683 = € 1,000 / 1,464 € 1,464 € 1,000

10. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main9 Present value of a payment stream The following relationship holds: or more generally for T periods:

11. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main10 Present value of a fixed-payment loan If a loan of € 1,000 at 10% p.a. interest is to be paid back in four equal install- ments, the following time profile for the payments is obtained: €1,000 1342 time €315,47

12. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main11 Present value of an annuity It implies by definition that the present value at 10% p.a. of this annuity is exactly Often the present (or final) value and the annuities are known, and the implicit yield to maturity is to be calculated.

13. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main12 Present value of a coupon bond The typical payment stream of a coupon bond is (at a coupon rate of 10% p.a. and four periods to maturity): €1,000 1342 time €100 €100+1,000

14. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main13 The yield to maturity of a coupon bond The yield to maturity of the payment stream of a bond priced at €1.000 is the value of i in the following equation: In this case, the face value is identical to the price of the bond, i.e. the yield to maturity must be 10 % p.a.

15. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main14 The yield of a bond at varying prices (1) Yield to maturity on a 10% coupon rate bond maturing in 10 years (face value = €1,000) Price of Bond (€)Yield to Maturity (%) 1,2007.13 1,1008.48 1,00010.00 90011.75 80013.81

16. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main15 The yield of a bond at varying prices (2) The table illustrates the following: 1.When the bond price = face value: the yield to maturity = the coupon rate. 2.When the bond price the coupon rate 3.=>The bond price and the yield to maturity are negatively related.

17. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main16 Negative relationship between P and i This finding is not really surprising if one looks at the formula to be solved for i: The relationship is particularly simple for a bond without a maturity date (perpetuity or consol). It is: P = R / i, which implies i = R / P

18. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main17 Yield to maturity of a discount bond It is similar to that of a simple loan. For a bond at a face value of € 1.000 maturing in one year the relationship is:

19. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main18 The distinction between interest and return The rate of return measures how well a person does by holding a bond. The rate of return does not necessarily equal the interest rate of the bond. The return on a bond held from t to t+1 is

20. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main19 The rate of return The return consists of the current yield (or coupon payment), plus the capital gain (or loss) resulting from fluctuations of the bond price. The rate of return is the sum of the two components over the purchase price of the bond. It is interesting to explore what happens to the rate of return if prices change.

21. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main20 One-year Returns on Different-Maturity 10% Coupon Rate Bonds Purchased in t at € 1,000, When Interest Rates Rise from 10% to 20% Years to maturity P t+1 (€)g (%)r (%) 30503-49.7-39.7 20516-48.4-38.4 10597-40.3-30.3 5741-25.9-15.9 2917-8.3+1.7 11,0000.0+10.0

22. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main21 Key findings for an increase of interest rates Only if the holding period equals the time to maturity, then r = i c. Capital losses occur if the holding period is smaller than the time to maturity. The more distant a maturity, the greater the percentage price change and the lower the rate of return. The rate of return can turn negative.

23. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main22 Interest-rate risks If the holding period is extended as a result of the price change, there is “only” a “paper loss”. It is still a loss! Prices and returns for long-term bonds are more volatile than those for shorter- term bonds. It entails an interest-rate risk, which is a major concern for portfolio managers.

24. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main23 Real and nominal interest rates If the nominal interest rate is adjusted for inflation on the cost of borrowing, it is called the “real interest rate”, i r. The “Fisher equation” states that the nominal interest rate i equals the real interest rate i r plus the expected rate of inflation π e. i = i r + π e Irving Fisher 1867-1947

25. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main24 The working of the “Fisher equation” Suppose you have made a loan at 5% interests expecting an inflation rate of 2% over the course of a year. Your real rate of interest is then 3%. Assume, interest rates rise to 8%, but inflationary expectations become 10%. Your real rate of interest is then -2%. The lower the real interest rate, the greater the incentives to borrow, and smaller to lend.

26. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main25 Real returns A similar distinction can be made between nominal and real returns. This distinction is important because the real interest rate, the real costs of borrowing, is a better indicator of the incentives to borrow and lend. Inflationary expectations can be “stripped off” by indexing the bonds to inflation.

27. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main26 Nominal interests and price developments

28. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main27 Implicit real interest rate

29. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main28 Indexed bonds With indexed bonds, investors are guaranteed a fixed real rate of interest. The recent issuances of indexed government bonds in countries like the Canada, France, New Zealand, Sweden, and in the United States, encourage to analyze indexed bonds. With the start of European Monetary Union (EMU) on January 1 st 1999, the German currency regulation (Währungsgesetz) was eliminated. It prohibited the use of indexation. The German government is still opposed.

30. Paul Bernd Spahn, Goethe-Universität Frankfurt/Main29 Taxing returns The notion of “return” is very complex. It comprises –Real interests –Inflationary components –Capital gains (and losses) Income taxes have difficulties to differentiate between these elements Taxing inflationary gains is a winning proposition for the government. Indexed bonds are likely to spur a discussion on whether to tax or not the “index change”.