Tutorial-Week1 (Exercises and Solutions)
Investment and Portfolio Theory 1 (6012B0233Y)
Universiteit van Amsterdam
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IPT1 (2017) ‐ Week 1 Page 1 Week 1: Tutorial CHAPTER 14 (Q14) A GE bond with an annual coupon rate of 4% and face value of $1000 sells for $980. What is the bond's current yield? CY = 48/980=4% (Q14) Which security has a higher effective annual yield? a. A 3‐month T‐bill selling at $97,645 with par value $100,000 b. A BMW coupon bond selling at par and paying a 10% coupon semi‐annually 3‐month T‐Bill: 3‐month interest rate: 97,645 ∗ 1 Effective Annual Yield: 100,000 ⇔ 1 1 1 , 1 , 0 . 2% 1 % BMW coupon Bond: Sells at par there will be no income from this bond due to capital gains/losses but only from coupons. 5% payment each 6 months annualize this with the effective annual yield formula: 1 0 . 1 1 1 . % (Q14) Consider a Boeing bond with a 10% coupon and with yield to maturity of 8%. If the bond's yield to maturity remains constant, then in 1 year, will the bond price be higher, lower, or unchanged? Why? Coupon rate > Yield to Maturity (note, the yield to maturity consists of the time value of money plus any potentially any credit risk; the annual coupon of the bond is higher than the fair compensation the investor should be receiving given the time value of money and the riskiness of the bond, so the price of the bond will be above the face value). Premium bond. Premium bonds approach their face value as they get closer to maturity. Thus the price of the bond will be lower 1 year later. (For more explanation , see "Bond prices over time" on the slides) (Q14) Consider an 8% coupon McDonalds bond with face value of $1,000 selling for $953 with 3 years until maturity making annual coupon payments. The interest rates in the first, second and 8%, 10%, 12%. Calculate the yield to maturity and third years will be, with certainty, realized compound yield of the bond. Dr. T. Jochem (U of Amsterdam) Not to be distributed outside of class room without prior consent. © IPT1 (2017) ‐ Week 1 Yield to Maturity: $953 Page 2 $80 1 $80 1 Realized Compound Yield: $953 ∗ 1 80 ∗ 1 ∗ 1 $1,268 ⇔ $ , $ . . 1 $1,080 ⇔ 9% 1 80 ∗ 1 1,080 ⇔ $953 ∗ 1 9% (Note: coupons and principals are paid at the end of each period. So the first coupon is paid as the first year ends and the second year is about to start. Thus, the reinvestment of the first coupon payment occurs at rate 10% for the second year and 12% for the third year.) (Q14) Assume you have a 1‐year investment horizon and are trying to choose among three bonds. All have the same degree of default risk and mature in 10 years. The first is a zero‐coupon bond that pays $1,000 at maturity. The second has an 8% coupon rate and pays the $80 coupon once per year. The third has a 10% coupon rate and pays the $100 coupon once per year. a. If all three bonds are now priced to yield 8% to maturity, what are their prices? Bond 1: P = PV factor (8%, 10) = 1,000/1^10 = $463 Bond 2: P = $1000 (since 8% coupon rate and 8% yield to maturity) P $80 ∗ Annuity factor 8%, 10 $1000 ∗ PV factor 8%, 10 1 1 1 $80 ∗ 1 $1000 $1000 0 1 1 Bond 3: P = $1,134 P $100 ∗ Annuity factor 8%, 10 $1000 ∗ PV factor 8%, 10 1 1 1 $100 ∗ 1 $1000 $1,134 0 1 1 Illustrated in Excel: Dr. T. Jochem (U of Amsterdam) Not to be distributed outside of class room without prior consent. © IPT1 (2017) ‐ Week 1 ii. iii. Page 4 yield to maturity realized compound yield Current yield: $70 $960 Yield to Maturity: $960 ⋯ 7% 3% semi‐annual YTM * in financial calculator: N = 10; PV = (‐)960; FV = 1000; PMT = 35; press CPT, I/Y) * in Excel, you can use … =IRR({‐960,35,35,35,35,35,35,35,35,35,1035}) = 3% =YIELD(settlement date, maturity date, coupon rate, price, redemption value in % of face value, frequency)1 =YIELD(“1/1/2000”, “1/1/2005”, 0, 96, 100, 2)=7% (bond‐eq. YTM) annual bond equivalent YTM: 3%*2=7% effective annual YTM: (1+3%)^2 ‐ 1 = 8% Realized Compound Yield: Step 1: Future value of reinvested coupons: 35 ∗ 1 35 ∗ 1 ⋯ 35 ∗ 1 $226 (PV=0; PMT=35; N=6; I/Y=3%; CPT FV=226] Step 2: Sales price at the end of year 3 is $1,000 (which is the face value, since in year 3 the YtM of 7% equals the coupon payment of 7%) Step 3: Compute realized compound yield: 960 ∗ 1 1000 226 ⇔ 8% annually. (alternatively: 960*(1+i)^6 = 1,226 i=4% semi‐annual realized compound yield then annualizing with EAR (1+4%)^2‐1 = 8%.) Shortcomings with… current yield: ignores capital gains/losses and coupon reinvestments YTM: assumes that bond is held until maturity and that the reinvestment rate equals YTM. Only if the YTM of 8% stays the same the following years and the coupon is re‐ invested at that rate, then the realized compound return ends up to be 8% too. 1 Note that in computers using the European decimal system, the decimal point is a comma and parameters are separated by semicolon rather than a comma! (See also the file on Excel Skills that is uploaded to Blackboard.) Dr. T. Jochem (U of Amsterdam) Not to be distributed outside of class room without prior consent. © IPT1 (2017) ‐ Week 1 Page 5 Realized compound yield: can only be computed ex‐post; (ex‐ante, needs to assume some yield at end of holding period and forecast some reinvestment rates) (Q14) A convertible bond has the following features: Bond characteristics Coupon 5% Maturity June 15, 2007 Market price of bond $775 Market price of underlying common stock $28 Annual dividend $1 Conversion ratio 20 shares Is the conversion option in the money? current bond price ‐ conversion value = current bond price ‐ (current value of stocks if converted) = 775 ‐ (20*28) = $191 not, not in the money; CHAPTER 15 (Q15) The following is a list of prices for zero‐coupon bonds of various maturities and with par values of $1,000. Calculate the yields to maturity of each bond and the implied sequence of forward rates. Maturity (Years) Price of Bond 1 $943 2 $898 3 $847 4 $792 The formula for computing forward rates is Maturity (Years) 1 2 3 4 1. Price of Bond YtM of Bond Forward rate $943 $898 $847 $792 6%* 5% 5% 6% f2=1^2/1 ‐1=5% f3=1^3/1^2‐1=6% f4=1^4/1^3‐1=7% Dr. T. Jochem (U of Amsterdam) Not to be distributed outside of class room without prior consent. © IPT1 (2017) ‐ Week 1 Page 7 The 3‐year zero‐coupon bond will sell next year for $1,000/(1*1)=$783 The 3‐year zero‐coupon bond today costs $1,000/1^3=$711 783.21/711 ‐ 1 = 10% This makes sense, since there should be no arbitrage between both bonds. Part (d) Current price: $120/1 + $120/(1*1)+$1,120/(1*1*1) = $1,003 Price next year: $120/1+$1,120/(1*1) = $984 Expected rate of return: ($120+($984‐$1003))/$1003=10% Makes again sense, there should be no arbitrage. (Q15) Suppose that the prices of zero‐coupon bond with various maturities are given in the following table. The face value of each bond is $1,000. Maturity (Years) Price of Bond 1 $925 2 $853 3 $782 4 $715 5 $650 a. Calculate the forward rate of interest for each year. Maturity Price of Bond YTM Forward Rate (Years) 1 $925 8% 2 $853 8% 8% 3 $782 8% 9% 4 $715 8% 9% 5 $650 9% 10% The formula for computing forward rates is 1. f2 = 1^2/1 = 8% f3 = 1^3/1^2 = 9% … b. How could you construct a 1‐year forward loan beginning in year 3 where you act as a lender and you’d like to lend $1,000? Confirm that the rate on that loan equals the forward rate. For each 3‐year zero bond issued today, use the proceeds to buy today 4‐year zero bonds issued today. You will be able to buy for each 3‐year zero bond, 782/ 715 =1 4‐year zero bonds. Dr. T. Jochem (U of Amsterdam) Not to be distributed outside of class room without prior consent. © IPT1 (2017) ‐ Week 1 Cash flows: Time 0 Cash Flow $0 1 2 3 $0 $0 ‐$1,000 4 $1,095 Page 8 Issue/sell 3‐year zero coupon bonds; buy 1 times as many 4‐year zero coupon bonds. (Inflows and outflows canceling out each other.) ‐ ‐ The 3‐year zero coupon bonds at time 0 matures; issuer pays out $1,000 face value per bond. The 4‐year zero coupon bonds purchased at time 0 matures; receive $1,000 face value per bond (and there are 1 times as many 4‐year bonds as we purchased 3‐year bonds) This is a synthetic one‐year loan starting at year 3. The rate on that loan is 9% (=1095/1000‐1) which equals the forward rate for year 4. Note that at the $1000 outflow in year 3 has to be financed (after all, you are giving a loan in year 3). If the one‐year annual interest rate in year 3 is lower than the forward rate of 9% that you have “locked in”, then you made a profit from this forward loan because you receive 9% interests even though the market interest rate is at that time lower. If, however, the one‐year annual interest rate turns out to be higher than the 9% that is predicted by the market right now, then one could have lend out the money in t=3 at a higher interest rate than the 9%. Given that the 9% is however “locked in”, one is losing out in this case from a potentially higher interest rate. In summary, constructing such a forward 1‐year loan locks in the 9% forward rate and foremost looks attractive if one expects future interest rates to turn out lower than the currently predicted forward rate of 9%. c. How could you construct a 2‐year maturity forward loan commencing in 3 years where act as a borrower and you would like to borrow $1000? For each 3‐year zero coupon bond you buy today, you issue $782/$650= 1 5‐year zero coupon bonds. Cash flows in time zero exactly cancel out. Cash flows: Time 0 Cash Flow $0 1 2 3 $0 $0 $1000 4 ‐ Dr. T. Jochem (U of Amsterdam) For each 3‐year zero coupon bond you buy today, you issue $782/$650.00=1 5‐year zero coupon bonds. Cash flows in time zero exactly cancel out. ‐ ‐ The 3‐year zero coupon bond purchased at time 0 matures; you receive face value $1,000. ‐ Not to be distributed outside of class room without prior consent. ©
Tutorial-Week1 (Exercises and Solutions)
Vak: Investment and Portfolio Theory 1 (6012B0233Y)
Universiteit: Universiteit van Amsterdam
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