0
Upcoming SlideShare
×

Thanks for flagging this SlideShare!

Oops! An error has occurred.

×
Saving this for later? Get the SlideShare app to save on your phone or tablet. Read anywhere, anytime – even offline.
Standard text messaging rates apply

# Lecture 4 2012 valuation and risk

1,430

Published on

0 Likes
Statistics
Notes
• Full Name
Comment goes here.

Are you sure you want to Yes No
• Be the first to comment

• Be the first to like this

Views
Total Views
1,430
On Slideshare
0
From Embeds
0
Number of Embeds
0
Actions
Shares
0
8
0
Likes
0
Embeds 0
No embeds

No notes for slide

### Transcript

• 1. 固定收益课题 4 ： 债券定价和风险 / Valuation and Risk 陈国辉博士 ／ 南洋理工大学商学院
• 2. 债券定价模式 <ul><li>C = coupon rate p.a. </li></ul><ul><li>m = number of coupon payment per annum </li></ul><ul><li>y = yield to maturity p.a. </li></ul><ul><li>n = number of period </li></ul><ul><li>P = price of bond </li></ul>
• 3. 价格和收益率的关系 (Price-Yield Relationship) Price Yield <ul><li>价格和收益率成反向关系 (Inverse relation) </li></ul><ul><li>高收益率使到价格下滑 (High yield causes price to decrease) </li></ul><ul><li>凸性 (convexity) </li></ul>
• 4. 到期收益率 ( Yield-to-Maturity,YTM) 的局限性 <ul><li>YTM 不是真正能实现的收益率 </li></ul><ul><li>YTM is not the realized rate of return </li></ul><ul><li>实现 YTM 的 2 个条件 (2 conditions for realizing YTM as return) ： </li></ul><ul><ul><li>持有债券到到期日 (Hold the security up to maturity) </li></ul></ul><ul><ul><li>利息的再投资率必须等于 YTM (Reinvestment rate has to equal to YTM) </li></ul></ul><ul><li>债券投资者更关心总收益 ( total return) ：资本收益 (capital gain or price return) + 票息收益 coupon return) + 再投资收益 ( reinvestment return) </li></ul>
• 5. 总收益 Total Return <ul><li>假设一投资者的投资期限是 2 年。再假设他买了 9 ％票息的 2 年期债券，价格为 98.26 (ytm = 10%) </li></ul><ul><li>请问此投资的总收益是多少 , 假设其再投资收益率是 10 ％？ </li></ul><ul><ul><li>Future value of his investment </li></ul></ul><ul><ul><ul><li>First coupon: \$9(1.1) </li></ul></ul></ul><ul><ul><ul><li>Second coupon \$9 </li></ul></ul></ul><ul><ul><ul><li>Maturity value = 100 </li></ul></ul></ul><ul><ul><ul><li>Total future value = 9(1.1) + 9 + 100 = 118.9 </li></ul></ul></ul><ul><ul><li>Today ’s investment = 98.26 </li></ul></ul><ul><ul><li>Total return: </li></ul></ul><ul><ul><ul><li>118.9/(1+R) 2 =98.26 </li></ul></ul></ul><ul><ul><ul><li>R = 10% </li></ul></ul></ul><ul><li>如果再投资收益率是 9 ％，那总收益是多少？ </li></ul><ul><li>如果再投资收益率是 11 ％，那总收益是多少？ </li></ul>Reinvestment Risk
• 6. 总收益 Total Return <ul><li>假设一投资者的投资期限是 1 年。再假设他买了 9 ％票息的 2 年期债券，价格为 98.26 (ytm = 10%) </li></ul><ul><li>请问他的总收益率是多少？ </li></ul><ul><ul><li>假设他卖出债券时的价格的 YTM 是 10%, ie., 109/(1+0.1) or 99.0909 </li></ul></ul><ul><ul><li>或假设以低于 99.0909 的价格卖出？ </li></ul></ul><ul><ul><li>或假设以高过 99.0909 的价格卖出？ </li></ul></ul>Price Risk
• 7. 平均到期期限 Average Maturity <ul><li>投资者在乎到期日吗？ </li></ul><ul><li>假设 3 个债券 </li></ul><ul><ul><li>Debt A: Pay \$10,000 in 5 years </li></ul></ul><ul><ul><li>Debt B: Pay \$1 in 0.25 year and \$9,999 in 5 years. </li></ul></ul><ul><ul><li>Debt C: Pay \$9,999 in 0.25 year and \$1 in 5 years. </li></ul></ul><ul><li>注意付息的时间和额度 </li></ul>
• 8. 久期 Duration <ul><li>为了更准确的计算平均到期期限，我们可以考虑用时间来权重每个现金流。得到的结果叫久期 ( duration) 或麦考利久期 Macaulay Duration 。 </li></ul><ul><ul><li>假设债券每年付 \$C ， YTM=y 。 先假设每年复利一次，那么久期： </li></ul></ul><ul><ul><li>因为其分母是债券的价值，所以我们也可以用以下公式表示： </li></ul></ul>
• 9. 久期：债券价格对利率变动的敏感度 <ul><ul><li>假设年票息 </li></ul></ul><ul><ul><li>dp / dy ： </li></ul></ul>
• 10. 久期：债券价格对利率变动的敏感度 <ul><li>dp/dy ＝实值久期 ( Dollar Duration) (Bloomberg calls it RISK.) </li></ul><ul><li>如果价格变化是以百分比来表示，那么叫修正久期 ( modified duration) ： </li></ul>
• 11. 国债的久期 在付息期间 （ w ＝从今天到下个付息日的部分时间）： 在付息日（ w ＝ 1 ）：
• 12. 久期 Duration <ul><li>假设以下 3 个债券： </li></ul>Bond Yr 1 Yr 2 Yr 3 Yr 4 Yr 5 Duration A 5 5 5 5 105 4.5 Yr B 20 20 20 20 120 3.8 Yr C 0 0 0 0 100 5.0 Yr
• 13. 久期的特性 （ 1 ） <ul><li>给予既定的到期期限，票息越高，久期越短 ( Holding maturity unchanged, increasing the coupon reduces duration) </li></ul><ul><ul><li>这是因为更高的现金流在短期内实现 </li></ul></ul><ul><ul><li>This follows because more cash flows are paid sooner. </li></ul></ul><ul><li>给予既定的到期期限和票息， 到期收益率越高，久期更短 ( Holding maturity and coupon unchanged, increasing yield leads to lower duration) </li></ul><ul><ul><li>这是因为高 YTM ，意味着高折现率，更远的现金流的现在值就越低了 </li></ul></ul><ul><ul><li>This follows because higher yield discount distant cash payment more heavily. </li></ul></ul><ul><li>给予既定的票息和 YTM ，更长的到期期限，更长的久期 ( Holding coupon and yield unchanged, increasing maturity leads to higher duration) </li></ul><ul><ul><li>看似很容易理解，但不是所有的债券都有此特性 </li></ul></ul><ul><ul><li>Seems clear but not true all the time. It only applies to premium bonds (see next page). </li></ul></ul>
• 14. Zero coupon bond Discount bond Par bond Premium bond Perpetual bond Maturity Duration
• 15. 久期的特性 （ 2 ） <ul><li>随着到期日的贴近，久期下滑 ( Duration declines as bond approaches maturity ) </li></ul><ul><ul><li>开始时慢速下滑 </li></ul></ul><ul><ul><li>期限低过 5 年后，下滑的速度加快。 </li></ul></ul><ul><ul><li>最后以 1 对 1 的比例下滑 </li></ul></ul><ul><li>成锯齿状变化 ( Sawtooth fluctuation ) </li></ul><ul><ul><li>形状来自于定期付票息： </li></ul></ul><ul><ul><ul><li>在付息日，久期上升 ( On coupon payment date, duration increases due to payment of the near-term coupon) </li></ul></ul></ul><ul><ul><ul><li>期限更长的债券，其上升的幅度更大 ( Long term bond has bigger jump (see table on next slide) ) </li></ul></ul></ul>
• 16. Duration (in year) of a 12%, 12% yield 20 years bond
• 17. 久期实用案例 1 <ul><li>债券免疫 Bond Immunization </li></ul><ul><ul><li>平衡再投资风险和价格风险 ( Balancing reinvestment risk and price risk is called immunization) </li></ul></ul><ul><ul><li>可通过匹配久期和投资期限来达到免疫 ( This can be done using duration matching: matching duration with the investor ’s horizon (See proof in class.)) </li></ul></ul>
• 18. 久期实用案例 2 <ul><li>久期衡量债券价格的波动率（即对利率的敏感度） ( Duration as a measure of volatility to estimate price change due to change in interest rate): </li></ul>
• 19. 案例 : 19.5 year 12% coupon bond priced at 12% yield. A 10 bp change in yield. <ul><li>Before coupon payment when duration = 7.48 </li></ul>
• 20. 久期实用案例 3 <ul><li>当久期在债券付息日升高时，是否意味债券的风险增加了？ When duration jumps on coupon payment day, does it mean that the bond suddenly begins to impose higher market risk on the investment? </li></ul><ul><ul><li>付息后，久期＝ 7.92 (duration = 7.92) </li></ul></ul><ul><ul><li>价格变化： </li></ul></ul><ul><ul><li>注意有现金流，而现金的久期等于零。投资经理必须决定如何投资收到的现金。这将影响其“组合”久期。 However, duration-matched manager has to decide how to reinvest the cash. His/her decision will change the “portfolio” duration. </li></ul></ul>
• 21. 久期实用案例 4 <ul><li>随着期限的变短，久期也在下降 ( Duration of the bond change as time passes) </li></ul><ul><ul><li>举例： </li></ul></ul><ul><ul><ul><li>假设初始投资时： 20-year bond ， 12 % coupon, 12% yield ， duration = 8 years = 投资期限 (investment horizon) </li></ul></ul></ul><ul><ul><ul><li>4 年之后，投资期限＝ 4 years ， 但是久期 ＝ 7.5 years </li></ul></ul></ul><ul><ul><ul><li>所以必须重新平衡组合 rebalance the “portfolio”. </li></ul></ul></ul><ul><ul><ul><li>在什么时候最适合做重新平衡 When is the best time to do rebalancing? </li></ul></ul></ul><ul><ul><ul><li>在付息日。因此国债在 Feb 15, May 15, Aug 15, 及 Nov 15 的交易量特高。 </li></ul></ul></ul>
• 22. 久期实用案例 5 <ul><li>债券市场用 DV01 来衡量久期 : </li></ul><ul><ul><li>That is, how much the price of bond will change for 1 bp change in yield. </li></ul></ul><ul><ul><li>DV01 = (Modified Duration x Full Price) x 0.0001 for \$100 face value bond. </li></ul></ul>
• 23. 久期实用案例 6 – 组合 <ul><li>一个债券组合的久期等于其加权平均久期。权重以债券市值计算。 ( A portfolio’s duration is simply the weighted average duration of the bonds in a portfolio. The weight is calculated by market value.) </li></ul>If yield curve shift up by 100 bp, what is the percentage value change of portfolio? Bond Market Value (mil \$) Weight Mod Duration Weight x Mod Dur A 10 0.1 4 0.4 B 40 0.4 7 2.8 C 30 0.3 6 1.8 D 20 0.2 2 0.4 Total 100 1 5.4
• 24. 凸性 Convexity <ul><li>凸性测量在利率和价格关系线上的弧线部分 Convexity measures the curvature (nonlinearity) in a bond-yield curve </li></ul><ul><li>所有没带期权的债券都存在正凸性 All non-callables exhibit some degree of positive convexity </li></ul><ul><li>正凸性意味在同量的利率下降和上升情况下， 债券价格的上升比下降来的大 Positive convexity indicates a bond ’s price rises more for a given decline in yield but fall less for a given increase in yield </li></ul><ul><li>凸性来自于债券的定价模式 Convexity exists because of bond price-yield relationship </li></ul><ul><li>凸性就像是在期权里的 Gamma Convexity is like Gamma in option </li></ul><ul><li>凸性越高越好 Convexity is desirable </li></ul>
• 25. 从 Dp/Dy 到 D 2 p/Dy 2 <ul><li>From duration, we know: </li></ul><ul><li>The second derivative: </li></ul>
• 26. 测量凸性 Measure Convexity 有些人用以下定义： 我们对凸性的定义 : <ul><li>注意： </li></ul><ul><li>我们测量凸性时收益是以小数表示。如果收益率是以％表示，那么当凸性＝ 100 时，凸性应写成 1 </li></ul>
• 27. 凸性的特性 <ul><li>久期越长，凸性更大 Higher the duration, higher the convexity (see next slide). </li></ul><ul><li>现金流越分散，凸性越大 Higher the dispersion of cash flows, higher the convexity. </li></ul><ul><ul><li>比较两种债券：固定票息和零息，哪种债券的凸性较大？ (no cash flow dispersion). </li></ul></ul><ul><ul><li>杠铃 ( Barbell) 组合的凸性比子弹的凸性来的高 Barbell has greater convexity than a bullet (draw a line between any two points on the duration-convexity curve to see the barbell has greater convexity.) </li></ul></ul>
• 28. 凸性的特性 Coupon bond ’s convexity is sum of a duration-matched zero’s convexity.
• 29. 凸性的特性 <ul><li>利率波动率增高时，凸性会越大 Higher the yield volatility, higher the value of convexity. </li></ul><ul><ul><li>波动率增高会对正凸性更有利 Higher volatility enhances the expected performance of positively convex positions. </li></ul></ul><ul><ul><li>那么，有谁会愿意买子弹式债券呢？ Why would anybody want to buy bullet then? No free lunch. </li></ul></ul>
• 30. 近似凸性 Approximate Convexity
• 31. 应用久期和凸性测量价格变化 应用泰勒系列关系式 Taylor series expansion ： http://en.wikipedia.org/wiki/Taylor_series
• 32. Duration Approximation Duration and Convexity Approximation Yield Price
• 33. 举例 <ul><li>假设修正久期＝ 4 年，凸性＝ 49 ，那么如果利率增加 100 基点，债券的价格会增加或降低多少？ Suppose Modified Duration is 4 years and convexity is 49, what is the % change of price for 100 bp increase in yield? </li></ul><ul><li>-4x0.01+0.5x49x(.01) 2 =-0.03755 = -3.755% </li></ul>
• 34. 近似久期 Approximate Duration <ul><li>举例 : 25-yr, 6% coupon, 9% yield, P 0 = 70.3570 </li></ul><ul><li>步骤： </li></ul><ul><ul><li>提高 YTM 10 bp (ie., Δ y=0.001), 用定价模式计算 P - = 69.6164 </li></ul></ul><ul><ul><li>降低 YTM 10 bp (ie., Δ y=0.001), 用定价模式计算 P + = 71.1105 </li></ul></ul><ul><ul><li>近似久期 Approximate duration = 10.62 </li></ul></ul>
• 35. 久期实用案例 7 －相对久期 Matching \$Duration <ul><li>假设你持有债券 X ，想置换债券 Y 。但是你期望保持其久期。 </li></ul><ul><li>Suppose you own Bond X and want to exchange for Bond Y. You want to ensure that after the swap, the \$Duration is maintained. </li></ul>Bond Price Par Amount (PA) Market Value (MV) Modified Duration (MD) Dollar Duration (\$D) X 80 10 mill 8 mill 5 8 mil x 5 / 100 = \$400,000 (for 1% change in yield) Y 90 ? ? 4 \$400,000 (imposed)
• 36. 久期实用案例 7 －相对久期 Matching \$Duration
• 37. 债券免疫 Bond Immunization <ul><li>等配资产和负债的价值 </li></ul><ul><li>Matching assets and liabilities value. </li></ul><ul><ul><li>要等配资产和负债的现金流是很难的 </li></ul></ul><ul><ul><li>Cannot exactly match the cash flows. </li></ul></ul><ul><ul><li>但是我们确保资产的价值比负债高。当需要现金时，我们售出的资产大过负债的现金需求 </li></ul></ul><ul><ul><li>But can ensure the value of asset is greater than the value of liabilities. When fund is needed, can sell assets to cover cash needs. </li></ul></ul>
• 38. 资金不足和资金过分充足 Under-funding and Over-funding <ul><li>考虑一退休基金有以下的现金需求： </li></ul><ul><li>Consider a pension fund has the following cash flow requirements: </li></ul>Year Pension Payment Present value (5% yield) 1 10,000 9,523.81 2 10,000 9,070.29 3 10,000 8,638.38 4 10,000 8,227.02 5 10,000 7,835.26 Total 50,000 43,294.77
• 39. <ul><li>假设基金买了一只 5 ％的平价债券（ \$43,294.77 ） </li></ul><ul><li>Suppose the fund invested in a bond with coupon 5%. Par amount = \$43,294.77. </li></ul><ul><li>当利率上升后，基金的资产 < 负债 </li></ul>Modified Duration Value today If yield is 5% If yield is 6% Assets 4.33 \$43,294.77 \$43,294.77 \$41,471.04 Liabilities 2.76 \$43,294.77 \$43,294.77 \$42,123.64 Difference 0 0 -\$652.6 (under-funded)
• 40. 债券免疫理论 Bond Immunization Theorem <ul><li>Recall this: </li></ul>Where P 1 = Value after yield change （利率变化后的价值） P 0 = Value before yield change （利率变化前的价值） MD = Modified duration （修正久期） Δ y = change in yield （利率变化） C = Convexity （凸性）
• 41. 债券免疫理论 Bond Immunization Theorem <ul><li>对负债来说 : </li></ul><ul><li>对资产来说 : </li></ul>
• 42. 债券免疫理论 Bond Immunization Theorem <ul><li>我们的目的是： </li></ul><ul><ul><li>A 1  L 1 </li></ul></ul><ul><li>假设 : </li></ul><ul><ul><li>A 0 = L 0 和 </li></ul></ul><ul><ul><li>MD A = MD L </li></ul></ul><ul><li>那么 : </li></ul><ul><li>假如我们能保证 C A  C L , 那么 A 1  L 1 </li></ul>
• 43. 债券免疫理论 Bond Immunization Theorem <ul><li>Back to the example. Suppose we invest in 2 bonds: </li></ul><ul><ul><li>2-year, value at par = 100, MD 2 = 1.85941 number of bonds = n 2 </li></ul></ul><ul><ul><li>5-year, value at par = 100, MD 5 = 4.329477, number of bonds = n 5 </li></ul></ul><ul><li>We can solve for the following equations: </li></ul>
• 44. 债券免疫理论 Bond Immunization Theorem <ul><li>n 2 = 274.3406 </li></ul><ul><li>n 5 = 158.6071 </li></ul><ul><li>你可以证明其总凸性大过负债的凸性 </li></ul><ul><li>You can verify that the total convexity is greater than the convexity of the liabilities (=12.08251) </li></ul><ul><li>你也应该发觉到，如果要得到更大的凸性，你应该选择现金流分散的债券或债券组合 </li></ul><ul><li>You should also notice that in general, to get higher convexity, you should choose a bond portfolio with higher cash flow dispersion. </li></ul>
• 45. 债券免疫理论应用时的问题 Bond Immunization in Practice <ul><li>实践中，可以供选择的债券很多 In practice, there are a large number of bonds to choose from, so there are many different combinations that may match the conditions: equal value, equal duration, and excess convexity. </li></ul><ul><li>要解决这个问题，我们需要多加其它的条件 To handle this problem, we need additional criteria </li></ul><ul><ul><li>收益最大化 Maximize yield (aggressive approach) </li></ul></ul>
• 46. Max imize Yield （收益率最大化） <ul><li>要求收益最大化，但必须免疫（条件） </li></ul><ul><li>We can pick bond (n 1 ,n 2 …n N ) to maximize y A : </li></ul><ul><li>Subject to </li></ul>
• 47. 债券免疫理论应用时的问题－重新组合 Bond Immunization in Practice － Rebalancing <ul><li>在一段时间内，负债的久期和资产的久期不会产生一样的变化 Change in duration of liabilities does not match change in duration of assets as time passes. </li></ul><ul><ul><li>举例 </li></ul></ul><ul><ul><ul><li>假设初开始投资期限为 8 年，我们购买了一只 20 年的债券（ 12 ％票息， 12 ％收益率），久期为 8 年 Assume initial investment of a 20-year bond (12% coupon, 12% yield) with duration to match 8 years horizon. </li></ul></ul></ul><ul><ul><ul><li>4 年后， 20 年债券的久期掉到 7.5 年（假设） 4 years later, horizon reduces to 4 years but 20-year bond’s duration only reduced to say 7.5 years duration. </li></ul></ul></ul><ul><ul><ul><li>因此我们必须重新组合 Therefore he has to rebalance the “portfolio”. </li></ul></ul></ul>
• 48. 为负债免疫的最佳方法是什么？ What is the best way to immunize a liability? <ul><li>假设其它条件不变，买零息债券最好 Holding everything constant, buy zeros of same maturity period. </li></ul><ul><li>但是，零息债券的价格一般偏高（低收益），因此你需要更多的钱来达到目标收益 However, zeros provide lower yield, hence higher price. That is, you need more money to satisfy the yield target. </li></ul>