1. Part II Fundamentals of Interest Rates

2. Chapter Three Understanding Interest Rates

3. 3 Chapter Preview We examine the terminology and calculation of various rates, and we show the importance of these rates in our lives and the general economy. Topics include:  Measuring Interest Rates  The Distinction Between Real and Nominal Interest Rates  The Distinction Between Interest Rates and Returns

4. 4 3.1 Measuring Interest Rates Since interest rates are among the most closely watched variables in the economy, it is imperative that what exactly is meant by the phrase interest rates is understood. In this chapter, we will see that a concept known as yield to maturity (YTM, 到期收益率 ) is the most accurate measure of interest rates.

5. 5 3.1.1 Present Value Introduction Different debt instruments have very different streams of cash payments to the holder (known as cash flows), with very different timing. All else being equal, debt instruments are evaluated against one another based on the amount of each cash flow and the timing of each cash flow. This evaluation, where the analysis of the amount and timing of a debt instrument’s cash flows lead to its yield to maturity or interest rate, is called present value analysis.

6. 6 3.1.1.1 Interest Rates as Exchange Rates Present value is the concept based on the commonsense notion that a dollar of cash flow paid to you one year from now is less valuable to you than a dollar paid to you today. Interest rate is the “exchange rate” between present value and future value.

7. 7 3.1.1.2 Concept of Present Value

8. 8 3.1.1.3 Four Types of Credit Instruments Simple Loan Fixed Payment Loan Coupon Bond Discount Bond

9. 9 3.1.2 Simple loan ( 简单贷款 ) Simple loan is a loan, in which the lender provides the borrower with an amount of funds, which must be repaid to the lender at the maturity date along with an additional payment for the interest.

10. 10 3.1.2.1 Simple Loan: Yield to Maturity （到期收益率） Yield to maturity = interest rate that equates today's value with present value of all future payments Simple Loan (one year loan for principal $100, on maturity creditor will be paid $110 (P+I) ：

11. 11 3.1.3 Fixed-payment loan ( 分期固定支付贷款 ) Fixed-payment loans are also called fully amortized loans, in which the lender provides the borrower with an amount of funds, which must be repaid by making the same payment every period (such as a month), consisting of part of the principal and interest for a set number of years.

12. 12 3.1.3.1 : Fixed Payment Loans: Yield to Maturity Fixed Payment Loan (i = 12%) Note: LV = Loan Value; FP= Fixed Payment

13. 13 3.1.3.2 A Mortgage Payment Table (Figure 3-1)

14. 14 3.1.4 Coupon Bond ( 息票债券 ) A coupon bond pays the owner of the bond a fixed interest payment (coupon payment) every year until the maturity date, when a specified final amount (face value or par value) is repaid.

15. 15 3.1.4.1 Coupon Bonds: Yield to Maturity Coupon Bond (Coupon rate = 10% = Coupon/Face value) Consol ( 无期限债券 ): Fixed coupon payments of $C forever

16. 16 3.1.4.2 A Bond Table (Figure 3-2) Coupon rate = 10% = C/F

17. 17 3.1.5 A Discount Bond A discount bond, also called a zero-coupon bond, is bought at a price below its face value (at discount), and the face value is repaid at the maturity date.

18. 18 3.1.5.1 Discounted Bonds ( 贴现债券 ): Yield to Maturity One-Year Discount Bond (P = $900, F = $1000)

19. 19 3.1.5.2 Relationship Between Price and Yield to Maturity Three interesting facts in Table 3-1 1. When bond is at par, yield equals coupon rate 2. Price and yield are negatively related 3. Yield greater than coupon rate when bond price is below par value

20. 20 3.1.6 Current Yield ( 本期收益率 ) Current yield is the approximation to describe interest rate on long-term bonds, including a perpetuity ( 永续年金 ), which is also called as a consol( 无到期日债券 ), a perpetual bond with no maturity date and no repayment of principal that makes fixed coupon payments of $c forever. C=yearly payment; P= Price of the bond

21. 21 3.1.6.1 Two Characteristics of Current Yield CY is better approximation to yield to maturity, nearer price is to par and longer is maturity of bond; Change in current yield always signals change in same direction as yield to maturity

22. 22 Bond Page of the Newspaper

23. 23 3.2 Nominal and Real Interest Rates ( 名义 利率与实际利率 ) So far our discussion has not taken account of the effects of inflation on the cost of borrowing. The interest rate which makes no allowance for inflation is called Nominal Interest Rate. The interest rate which is adjusted by eliminating expected changes in the price level (inflation), is called Real Interest Rate.

24. 24 3.2.1 Fisher Equation Fisher equation states that the nominal interest rate equals the real interest rate plus the expected rate of inflation. Real interest rate:

25. 25 3.2.2 Distinction Between Real and Nominal Interest Rates If i = 5% and π e = 0% then If i = 10% and π e = 20% then

26. 26 3.2.3 Conclusion of Fisher Equation Real interest rate more accurately reflects true cost of borrowing; When real rate is low, greater incentives to borrow and less to lend

27. 27 3.2.4 U.S. Real and Nominal Interest Rates Figure 3-3: Real and Nominal Interest Rates (Three-Month Treasury Bill), 1953–2001 Sample of current rates and indexes http://www.martincapital.com/charts.htm

28. 28 3.3 Interest Rate and Return Because of the possible fluctuation on the price of a bond, the return of a bond is not the same as its interest rate in most of the cases.

29. 29 3.3.1 Distinction Between Interest Rates and Returns Rate of Return = Current yield + Capital gain

30. 30 3.3.2 Key Facts in the Relationship Between Rates and Returns The above data can be calculated by Equation 3, or by financial calculator: e.g. the 1 st one, N=30-1, PMT=100, FV=1000, I/Y=20, CPT(PV) =502.53(4)

31. 31 3.3.3 Maturity and the Volatility of Bond Returns ( 期限与债券回报率的波动 ) Conclusion from Table 3-2 (P58) analysis ： When studying the affect from a change in interest rate, we talk about the expected price or future value of bonds; Prices and returns more volatile for long-term bonds because have higher interest-rate risk; No interest-rate risk for any bond whose maturity equals holding period. On maturity, the bond is repaid in cash and therefore, the change in interest rate does not affect its value.

32. 32 3.3.4 Key findings from Table 2 Only bond whose return = yield is one with maturity = holding period ( 期限等于持有期， P+I will be repaid in cash in one year); For bonds with maturity > holding period, i  P , implying capital loss; Longer is maturity, greater is price change associated with interest rate change; Longer is maturity, more return changes with change in interest rate; Bond with high initial interest rate can still have negative return if i 

33. 33 3.3.4 Reinvestment Risk ( 再投资风险 ) If an investor’s holding period is longer than the term to maturity of the bond, the investor is exposed to a type of interest-rate risk called reinvestment risk. There exist re-inverment risk if the proceeds from the short-term bond need to be reinvested ； i at which reinvest is uncertain ； Gain from i , lose when i  Example on P60 --------------Chapter end------------------------

34. 34 3.4 Duration( 久期 ) and Interest-Rate Risk When interest rates change, a bond with a longer term to maturity has a larger change in its price and hence more interest rate risk than a bond with a shorter term to maturity. To have more precisely information on the actual capital gain or loss that occur when interest rate changes by a certain amount, we need to calculate Duration. Duration is the average lifetime of a debt security’s stream of payments.

35. 35 3.4.1 Interest Rate Risk on Different Bonds Two bonds with same term to maturity can have different interest rate risk (P61). The following example.

36. 36 3.4.1.1 Example (1) Bond 1: 10-year zero-coupon bond, interest rate rise from 10% to 20%, the effect on rate of capital gain (g)? year one: N=10, I/Y=10, PMT=0, FV=1000 PV=385.54 year two: N=9, I/Y=20, PMT=0, FV=1000 PV=193.81 g=(193.81-385.54)/385.54=-49.7% Rate of return=-49.7+0(coupen received ） =-49.7%

37. 37 3.4.1.2 Example (2) Bond 2: 10-year bond, initial current yield 10%, initial price $1000, interest rate rise from 10% to 20%, the effect on rate of capital gain (g)? year one: PV=1000(price first year) year two: N=9, I/Y=20, PMT=100, FV=1000 PV=596.90 (price second year) g=(596.90-1000)/1000=-40.3% Rate of return=-40.3+10% （ coupen received ） =-30.3% Table 2 on P58(line 3)

38. 38 3.4.2 Calculating Duration Frederick Macaulay invented the concept of duration, to calculate the duration or effective maturity on any debt security. Macaulay realizsed that he could measure the effective maturity of a coupon bond by recognizing that a coupon bond is equivalent to a set of zero- coupon dicount bonds, as shown in the Timeline on P62. The duration calcuation is shown on P63.

39. 39 3.4.2.1 Calculating Duration (P63) i = 20%, 10-Year 10% Coupon Bond

40. 40 3.4.2.2 Particulars on Calculation Column (3): the PV of each of the zero-coupon bonds when the interest rate is 10%; Column (4): divide each of these PV by $1000, the total PV of the set of zero-coupen bonds (also the real PV of this 10% interest rate coupon bond), to get the percentage of the total value of all the bonds that each bond represents, or so called the weight; Column (5): time each of these weights by the relative number of years ((1)*(4)) and we get the weighted maturities (the years for each of the effective payments). Adding up allthe weighted maturities, we obtain the duration of the 10% 10-year coupon bond. Duration is a weighted average of the maturities of the each payments.

41. 41 3.4.2.3 Formula for Duration DUR=Duration ； t=years until cash payment is made ； CPt =cash payment (P+I) at time t; i=interest rate; n=years to maturity of the security. Practice calculating duration on basis of data in Table 3 on P63 with financial calculator, assuming interest rate rises from 10% to 20% (compare your result with Table 4 on P65).

42. 42 3.4.3 Key facts about duration All else equal, when the maturity of a bond lengthens, the duration rises as well; All else equal, when interest rates rise, the duration of a coupon bond fall; The higher is the coupon rate on the bond, the shorter is the duration of the bond; Duration is also useful in measuring the duration of a portfolio: the duration of a portfolio of securities is the weighted-average of the durations of the individual securities, with the weights equaling the proportion of the portfolio invested in each. Example 9 on P64.

43. 43 3.4.4 Duration and Interest-Rate Risk Duration is useful to measure interest-rate risk. It provides a good approximation( 近似值 ), particularly when interest rate changes are small, for how much the security price changes for a given change in interest rates.

44. 44 3.4.4.1 Formula for Measuring Interest- Rate Risk by duration Where % Δ P=(P t+1 –P t )/ P t =percentage change in the price of the security from t to t+1 = rate of capital gain; DUR = duration i=interest rate

45. 45 3.4.4.2 Example 1 (P66): Assuming that i  10% to 11%, Calculate the price change with the 10% coupon bond, on Table 3 (P63): The result tells that the rise in interest rate from 10% to 11%, will cause bond price decrease by 6.15%.

46. 46 3.4.4.3 Example 2: Assuming that i  10% to 11% Calculate the price change with the 20% coupon bond, on Table 4 (P65): The result tells that the rise in interest rate from 10% to 11%, will cause bond price decrease by 5.20%.

47. 47 3.4.4.4 Example 3 : A pension manager ’ s decision making (P67) The pension manager want to decide: whether to invest in a ten-year coupon bond with a coupon rate of 20%; or a ten-year coupon bond with a coupon rate of 10%.

48. 48 3.4.4.5 Conclusion of duration and Interest-Rate Risk The greater is the duration of a security, the greater is the percentage change in the market value of the security for a given change in interest rates; Therefore, the greater is the duration of a security, the greater is its interest-rate risk.

49. 49 Chapter Summary Measuring Interest Rates: We examined several techniques for measuring the interest rate required on debt instruments. The Distinction Between Real and Nominal Interest Rates: We examined the meaning of interest in the context of price inflation.

50. 50 Chapter Summary (cont.) The Distinction Between Interest Rates and Returns: We examined what each means and how they should be viewed for asset valuation.

51. 51 Quantitive Problems P69