FRM 2017 part i schweser book 4 part 2

【梦轩考资网www.mxkaozi.com】 QQ106454842 专业提供CFA FRM全程高清视频+讲义 Topic 59 Cross Reference to GARP Assigned Reading - Tuckman, Chapter Given the following bonds and forward rates: M aturity YTM Coupon Price year 4.5% 0% 95.694 years 7% 0% 87.344 years 9% 0% 77.218 • 1-year forward rate one year from today = 9.56% • 1-year forward rate two years from today = 10.77% • 2-year forward rate one year from today = 11.32% Which of the following statements about the forward rates, based on the bond prices, is true? A The 1-year forward rate one year from today is too low B The 2-year forward rate one year from today is too high C The 1-year forward rate two years from today is too low D The forward rates and bond prices provide no opportunities for arbitrage © 2017 Kaplan, Inc Page 147 【梦轩考资网www.mxkaozi.com】 QQ106454842 专业提供CFA FRM全程高清视频+讲义 Topic 59 Cross Reference to GARP Assigned Reading - Tuckman, Chapter C o n c ep t C h e c k e r A n s w er s C First compute the 2-year spot rate: N = 4; PV = -93.2775; PM T = 0; FV = 100; CPT I/Y = 1.755% z(0.5) = 1.755% x = 3.51% Next compute the forward rate in 1.5 years ending in year 1+ 0.0351) 1+ 0.0326) f (2.0)') x 1+ _A— ' f (2.0) = 4.26% C bond price = $3 ( 1+ B 0.0250) + $3 ( 1+ 0.0300) + $103 ( 1+ 0.0326) = $104.00 / (1.0836)^ _ = 20% (1.0875)3 C The easiest way to find the bond value is to first calculate the appropriate spot rates to discount each cash flow Sj = 5.5% 52 = [(1.055)(1.0763)]1/2 —1 = 6.56% 53 = [(1.055)(1.0763)(1.1218)]1/3 - = 8.39% S4 =[(1.055)(1.0763)(1.1218)(1.155)]1/4- l = 10.13% Then use the spot rates to discount each cash flow and take the sum of the discounted cash flows to find the value of the bond bond price $100 | 1.055 $100 | 1.06562 $100 1.08393 | $ 1,100 1.10134 $1,009.16 Note that you could also this in one step using the forward rates, but breaking the problem into two steps makes the math easier to on your calculator C Given the bond spot rates on the zero-coupon bonds, the appropriate forward rates should be: • 1-year forward rate one year from today = [(1 + 0.07)2 / (1 + 0.045)] —1 = 0.0956, or 9.56% • 1-year forward rate two years from today = [(1 + 0.09)3 / (1 + 0.07)2] —1 = 0.1311, or 13.11% • 2-year forward rate one year from today = [(1 + 0.09)3 / (1 + 0.045)] = 1.2393 1.23930 - = 0.1132= 11.32% The 1-year forward rate two years from today is too low Page 148 © 2017 Kaplan, Inc 【梦轩考资网www.mxkaozi.com】 QQ106454842 专业提供CFA FRM全程高清视频+讲义 T he following is a review o f the Valuation and Risk Models principles designed to address the learning objectives set forth by GARP® T his topic is also covered in: Re t ur ns, Spr e a d s , a n d Yi e l ds Topic 60 Ex a m F o c u s Bonds with coupons that are greater than market rates are said to trade at a premium, while bonds with coupon rates less than market rates are said to be trading at a discount For coupon bonds, yield to maturity (YTM) is not a good measure of actual returns to maturity When a bondholder receives coupon payments, the investor runs the risk that these cash flows will be reinvested at a rate of return that is lower than the original promised yield on the bond This is known as reinvestment risk For the exam, know how to calculate YTM given different compounding frequencies Re a l iz e d Re t ur n LO 60.1: Distinguish between gross and net realized returns, and calculate the realized return for a bond over a holding period including reinvestments The gross realized return for a bond is its end-of-period total value minus its beginningof-period value divided by its beginning-of-period value The end-of-period total value will include both ending bond price and any coupons paid during the period If we denote current bond price at time t as B V , coupons received during time period t as Ct, and initial bond price as BVt l, then the realized return for a bond from time period t- to t is computed as follows: BVt + C t - BVt_1 BVt-i Example: Calculating gross realized return What is the gross realized return for a bond that is currently selling for $112 if it was purchased exactly six-months ago for $ 105 and paid a $2 coupon today? Answer: Substituting the appropriate values into the realized return equation, we get: R t-i,t — $112 H- $2 —$105 $105 Rt_i)t = 8.57% © 2017 Kaplan, Inc Page 149 【梦轩考资网www.mxkaozi.com】 QQ106454842 专业提供CFA FRM全程高清视频+讲义 Topic 60 Cross Reference to GARP Assigned Reading - Tuckman, Chapter The net realized return for a bond is its gross realized return minus per period financing costs Cost of financing would arise from borrowing cash to purchase the bond Even though borrowing cash to pay for the entire price of the bond would technically reduce the initial cash outlay to zero, convention is to use the initial bond price as the beginning-ofperiod value Example: Calculating net realized return W hat is the net realized return for a bond that is currently selling for $112 and paid a $2 coupon today if its purchase price of $103 was entirely financed at an annual rate of 0.6% exactly six-months ago? Answer: Substituting the appropriate values into the realized return equation and then subtracting per period financing costs, we get: D _ $112 + $ - $ r ’ $105 0.6% R t_ l t = 8.57% - 0.3% = 8.27% In order to compute the realized return for a bond over multiple periods, we must keep track of the rates at which coupons received are reinvested When a bondholder receives coupon payments, the investor runs the risk that these cash flows will be reinvested at a rate that is lower than the expected rate For example, if interest rates go down across the board, the reinvestment rate will also be lower This is known as reinvestment risk Example: Calculating realized return with reinvested coupons W hat is the realized return for a bond that is currently selling for $112 if it was purchased exactly one year ago for $105, paid a $2 coupon today, and paid a $2 coupon six months ago? Assume the coupon received six months ago was reinvested at an annual rate of 1% Answer: $112 + $2 + $2x f i + %] -$ l R t-l,t — $105 R t-l,t — Page 150 $ 1 + $2 + 2.01 —$105 $105 10.49% © 2017 Kaplan, Inc 【梦轩考资网www.mxkaozi.com】 QQ106454842 专业提供CFA FRM全程高清视频+讲义 Topic 60 Cross Reference to GARP Assigned Reading - Tuckman, Chapter B o n d Sp r ea d LO 60.2: Define and interpret the spread o f a bond, and explain how a spread is derived from a bond price and a term structure o f rates The market price of a bond may differ from the computed price of a bond using spot rates or forward rates Any difference between bond market price and bond price according to the term structure of interest rates is known as the spread of a bond A bond’s spread is a relative measure of value which helps investors identify whether investments are trading cheap or rich relative to the yield curve (i.e., the term structure of rates) Recall the calculation of bond price using forward rates from the previous topic Assume a 2-year bond pays annual coupon payments, C, and a principal payment, P, at the end of year two The bond’s price will be computed by discounting all cash flows by corresponding 1-year forward rates as follows: b0nd price = [l + f (1.0)]+ [l + f(l.0 )]x [l + f(2.0)] If the market price of this bond trades at a premium or discount to this computed price, we can find the spread of the bond by adding a spread, s, to the forward rates as follows: C C+ P market bond price = T -—— - , + -r -—— - ;— -r -— - , [l + f (l 0) + s] [l + f (l 0) + s] X Jl + f (2.0) + s] By deriving this spread, we can identify how much the bond is trading cheap or rich in terms of the bond’s return For example, rather than saying the market price of the bond is trading 10 cents cheap, relative to the price determined by the term structure of rates, we can say that that the bond is trading 4.9 basis points cheap Note that this spread could be the result of either bond-specific factors or sector-specific factors Yi e l d t o Ma t u r it y LO 60.3: Define, interpret, and apply a bonds yield-to-m aturity (YTM) to bond pricing LO 60.4: Com pute a bond’s YTM given a bond structure and price The yield to maturity, or YTM, of a fixed-income security is equivalent to its internal rate of return The YTM is the discount rate that equates the present value of all cash flows associated with the instrument to its price 【梦轩考资网www.mxkaozi.com】QQ106454842 专业提供CFA FRM全程高清视频+讲义 © 2017 Kaplan, Inc Page 151 【梦轩考资网www.mxkaozi.com】 QQ106454842 专业提供CFA FRM全程高清视频+讲义 Topic 60 Cross Reference to GARP Assigned Reading - Tuckman, Chapter For a security that pays a series of known annual cash flows, the computation of yield uses the following relationship: (l + y )1 + C (1 + y )2 + (1 + y ) + + c N (l + y) N where: P = the price of the security = the annual cash flow in year k N = term to maturity in years y = the annual yield or YTM on the security Example: Yield to maturity Suppose a fixed-income instrument offers annual payments in the amount of $100 for ten years The current value for this instrument is $700 Compute the YTM on this security Answer: The YTM is the y that solves the following equation: $700 = $100 (1 + y)1 + $100 (1 + y)2 + $100 (1 + y) + $100 + (l + y)10 We can solve for YTM using a financial calculator: N = 10; PMT - 100; PV = -700; CPT =>- I/Y = 7.07% If cash flows occur more frequently than annually, the previous equation can be rewritten as: P— Cj Co Ca Cn r H y H t + ••• H (l + y)1 (1 + y)2 (1 + y)3 (1 + y)" where: n = N x m = the number of periods (years multiplied by payments per year) C^ = the periodic cash flow in time period k y = the periodic yield or periodic interest rate Example: Periodic yield and YTM Suppose now that the security in the previous example pays the $100 semiannually for five years Compute the periodic yield and the YTM on this security Page 152 ©2017 Kaplan, Inc 【梦轩考资网www.mxkaozi.com】 QQ106454842 专业提供CFA FRM全程高清视频+讲义 Topic 60 Cross Reference to GARP Assigned Reading - Tuckman, Chapter Answer: The periodic yield is they that solves the following equation: $700 = $100 $100 (1 + y)1 + (1 + y)2 $100 $100 (1 + y)3 (1 + y)10 Using a financial calculator: N = 10; PMT = 100; PV = -700; CPT => I/Y = 7.07% Why is this the same value as in the previous example? Remember that this yield corresponds to a 6-month period To compute the annual YTM, we must multiply the periodic yield by the number of periods per year, m = This produces a YTM of 14.14% The yield to maturity can be viewed as the realized return on the bond assuming all cash flows are reinvested at the YTM Example: Realized return Suppose a bond pays $50 every six months for five years and a final payment of $1,000 at maturity in five years If the price is $900, calculate the realized return on the security Assume all cash flows are reinvested at the YTM Answer: The semiannual rate is the y that solves the following equation: $50 $50 $50 $ + $1,000 $900 = r + — T + — T + - + (l + y)1 (1 + y)2 (1 + y)3 (1 + y)10 Using a financial calculator, we arrive at a semiannual discount rate of 6.3835% and a YTM of 12.77%: N = 10; PMT - 50; PV = -900; FV = 1,000; CPT +> I/Y = 6.3835; YTM - 6.3835 x = 12.77% The yield to maturity calculated above (2 x the semiannual discount rate) is referred to as a bond equivalent yield (BEY), and we will also refer to it as a semiannual YTM or semiannual-pay YTM If you are given yields that are identified as BEY, you will know that you must divide by two to get the semiannual discount rate W ith bonds that make annual coupon payments, we can calculate an annual-pay yield to maturity, which is simply the internal rate of return for the expected annual cash flows For zero-coupon Treasury bonds, the convention is to quote the yields as BEYs (semiannualpay YTMs) © 2017 Kaplan, Inc Page 153 【梦轩考资网www.mxkaozi.com】 QQ106454842 专业提供CFA FRM全程高清视频+讲义 Topic 60 Cross Reference to GARP Assigned Reading - Tuckman, Chapter Example: Calculating YTM for zero-coupon bonds A 3-year Treasury STRIP is priced at $768 Calculate the semiannual-pay YTM and annual-pay YTM Answer: The direct calculation method, based on the geometric mean, is: l 1,000 10 semiannual-pay YTM or BEY — - x = 5 % l 768 J l annual-pay YTM= Th e L im it a t io n s o f 1,000 768 Tr - = 5.42% a d it io n a l Yi e l d M ea su r es Reinvestment risk is a major threat to the bond’s computed YTM, as it is assumed in such calculations that the coupon cash flows can be reinvested at a rate of return that’s equal to the computed yield (i.e., if the computed yield is 8%, it is assumed the investor will be able to reinvest all coupons at 8%) Reinvestment risk applies not only to coupons but also to the repayment of principal Thus, it is present with bonds that can be prematurely retired, as well as with amortizing bonds where both principal and interest are received periodically over the life of the bond Reinvestment risk becomes more of a problem with longer term bonds and with bonds that carry larger coupons Reinvestment risk, therefore, is high for long-maturity, high-coupon bonds and is low for short-maturity, low-coupon bonds The realized yield on a bond is the actual compound return that was earned on the initial investment It is usually computed at the end of the investment horizon For a bond to have a realized yield equal to its YTM, all cash flows prior to maturity must be reinvested at the YTM, and the bond must be held until maturity If the “average” reinvestment rate is below the YTM, the realized yield will be below the YTM For this reason, it is often stated that: The yield to maturity assumes cash flows will be reinvested at the YTM and assumes that the bond will be held until maturity 【梦轩考资网www.mxkaozi.com】QQ106454842 专业提供CFA FRM全程高清视频+讲义 Page 154 ©2017 Kaplan, Inc 【梦轩考资网www.mxkaozi.com】 QQ106454842 专业提供CFA FRM全程高清视频+讲义 Topic 60 Cross Reference to GARP Assigned Reading - Tuckman, Chapter LO 60.5: Calculate the price o f an annuity and a perpetuity Ca Pr l c u l a t in g t h e An n u it ic e o f a n y We can easily calculate the price of cash flows (annuities) if given the YTM and cash flows Example: Present value of an annuity Suppose a fixed-income instrument offers annual payments in the amount of $100 for 10 years The YTM for this instrument is 10% Compute the price (PV) of this security Answer: The price is the PV that solves the following equation: PV = $100 (1 + 10)1 + $100 (1 + 10)2 + $100 (1 + 10)3 + + $100 (1 + 0.10)10 Using a financial calculator the price equals $614.46: N = 10; PMT = 100; I/Y= 10; CPT => PV = $614.46 Ca l c u l a t in g t h e Pr ic e o f a Pe r p e t u it y The perpetuity formula is straightforward and does not require an iterative process: PV of a perpetuity = C y where: C = the cash flow that will occur every period into perpetuity y = yield to maturity Example: Price of perpetuity Suppose we have a security paying $1,000 annually into perpetuity The interest rate is 10% Calculate the price of the perpetuity Answer: We don’t need a financial calculator to this calculation The price of the perpetuity is simply $10,000: p v = $L000 0.10 $ 10,000 © 2017 Kaplan, Inc Page 155 【梦轩考资网www.mxkaozi.com】 QQ106454842 专业提供CFA FRM全程高清视频+讲义 Topic 60 Cross Reference to GARP Assigned Reading - Tuckman, Chapter Sp o Ra t t es a n d YTM LO 60.6: Explain the relationship between spot rates and YTM In the previous topic, we discussed the calculation of spot rates and examined how to value a bond given a spot rate curve Pricing a bond using YTM is similar to using spot rates in that YTM is a blend of the given spot rates Consider the following example Example: Spot rates and YTM A bond with a $100 par value pays a 5% coupon annually for years The spot rates corresponding to the payment dates are as follows: Year 1: 4.0% Year 2: 4.5% Year 3: 5.0% Year 4: 5.5% Assume the price of the bond is $98.47 Show the calculation of the price of the bond using spot rates and determine the YTM for the bond Answer: The formula for the price of the bond using the spot rates is as follows: p = ^ + ^ + ^ + (1.04) (1.045)2 (1.05)3 105 (1.055)' $98.47 = 4.81 + 4.58 + 4.32 + 84.76 Now compute the YTM: $98.47 = (1 + YTM) + (1 + YTM)2 + (1 + YTM)3 + 105 (1 + YTM)4 FV = $100; PV = -$98.47; PMT = 5; N = 4; CPT —►I / Y = 5.44% YTM = 5.44% We see from this example that the YTM is closest to the 4-year spot rate This is because the largest cash flow occurs at year as the bond matures If the spot curve is upward sloping, as in this example, the YTM will be less than the 4-year spot (i.e., the last spot rate) If the spot curve is flat, the YTM will be equal to the 4-year spot, and if the spot curve is downward sloping, the YTM will be greater than the 4-year spot Page 156 ©2017 Kaplan, Inc ... associated with the instrument to its price 【梦轩考资网www.mxkaozi.com】QQ10 645 4 8 42 专业提供CFA FRM? ??程高清视频+讲义 © 20 17 Kaplan, Inc Page 151 【梦轩考资网www.mxkaozi.com】 QQ10 645 4 8 42 专业提供CFA FRM? ??程高清视频+讲义 Topic 60... that the bond will be held until maturity 【梦轩考资网www.mxkaozi.com】QQ10 645 4 8 42 专业提供CFA FRM? ??程高清视频+讲义 Page 1 54 ? ?20 17 Kaplan, Inc 【梦轩考资网www.mxkaozi.com】 QQ10 645 4 8 42 专业提供CFA FRM? ??程高清视频+讲义 Topic 60 Cross... realized return for a bond is its end-of-period total value minus its beginningof-period value divided by its beginning-of-period value The end-of-period total value will include both ending