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Bond prices in the absence of arbitrage Consider a market with two risk-free zero-coupon bonds, A...

Bond prices in the absence of arbitrage

Consider a market with two risk-free zero-coupon bonds, A and B. Their respective maturities are 1 and 2 years, and their market prices are 97.0874 and 95.1814 (expressed as percentage of the face value).

(a) Calculate the discount rates rt for t = 1 and 2 years.

(b) Suppose that a two-year bond C, with a coupon rate of 2.75%, also trades in the market. What should be its price if there is no arbitrage?

(c) What is the yield to maturity of bond C?

(d) If the market price of bond C is 100, is there an arbitrage opportunity? If yes, propose an arbitrage strategy and calculate the arbitrage profit.

(e) Calculate the implied forward rate between years one and two, f1;2 .

(f) According to the Pure-Expectations Theory, what are the investors views about the one-year rate r1 one year from now?

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Answer #1

Price of a bond is the present value of future cash flows. Spot rate of zero coupon bonds can be calculated using this formula.

To calculate the price of coupon bonds cash flow should be cash flow of the bond is calculated and then cash flow at particular is discounted using the spot rate.

For Bond A: Face Value Term Price $1,000 1 Year 970.87 10 Let discount rate for 1 year is ri, 12 13 14 15 16 17 -Face Value/ (1+r1) (Face Value / Price)-1 3.00%-07/09)-1 Hence discount rate for 1 year is 3.00% 19 For Bond B: $1,000 Face Value Term Price 2 Year $951.81 Let discount rate for 2 year is r2, 28 -Face Value/(1+2) ((Face Value/Price)(1/2))-1 31 2.50%-((D22/D24)^(1/D23))-1 Hence discount rate for 2 year is 2.50% 35 37 Price of bond will be the present value of cash flows of the bond Par value (F Coupon rate $1,000 2.75% 39 41 42 43 Time to maturity 2 Years Interest is paid once a year i.e. annual Annual coupon (C) Annual Period (n) $27.50 D39 D40 47 Discount rate for 1 and 2 years will be the same as calculated above Cash Flow of Bonds can be written as follows: Period Cash Flow of Bonds Discount rate Discount Factor Present value of Cash flow Total Present value of cash flows $1,004.69 E54+F54 49 51 52 53 27.50 $1,027.50 3.00% 2.50% 0.97 0.95-1/(1+F52)AF50) $26.70 $97799 -F51*F53 57 Hence price of the bond is $1,004.6959 61 Yield to maturity of the bond is the rate at which NPV of cash flows to investor is zero $1,000 2.75% $1,005 Face value Coupon rate Current Price Years to Maturity Annual Coupon Cash flow to investor will be as follows: Year Cash flow 63 2 years 27.5-D63 D64 69 0 2 ($1,004.69)$27.50 $1,027.50 71 72 Yield to maturity is the rate at which if future NPV to Investor will be zero 74 75 76 Rate(nper,pmt,PV, [fv],type) function of excel can be used to find the yield to maturity as follows: NPER PMT ($1,004.69 $1,000 78 Yield to maturity 2.51% RATE(D75,D76,D77,D78) 81 Thus yield to maturity is 2.51% 83 If the market price of the bond is 100% i.e Market price of coupon bond 86 87 1000 Price of Bond (Without arbitrage) $1,004.69 D57 91 92 93 Thus Bond market price is cheaper, therefore coupon bond should be bought and zero coupon bonds with face value 27.50 and 1027.50 and maturity 1 and 2 year respectively should be sold 95 Profit Amount received from Sale - Amount Paid to buy coupon bond $4.69 D89-D87 97 Hence profit is $4.69

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