Question

No 2. You are given the following information about four bonds traded in the market: Bond Coupon rate Price Time to maturity (years) 10% 99 10% 8% 97 7% 96 a) Find the spot rates in the given market. State all necessary assumptions. b) Estimate the expected spot rate for a two year investment that will be made in the end of the second year. c) Suppose bond N is currently traded at the market at a price of 95. Bond N has a coupon rate of 8% and 3 years to maturity. Are there any arbitrage opportunities? If yes, provide a detailed strategy to obtain an arbitrage profit. d) Find the yield to maturity of bond N directly and using approximation. What is the intuition behind the approximation formula? Are there any limitations of the approximation formula? e) Describe and explain different option-like characteristics of bonds. How do they affect bond prices?

Answer 1

A. use following formula to find spot rate.

Present value of bond = Coupon1/(1+r)ⁿ + Coupon2 / (1+r)ⁿ.......Coupon K / (1+r)^K + FV/(1+r)ⁿ

where n = no of year to maturity

r= spot rate

FV = face value of bond

or

Use Financial calculator

N = no of year to maturity PV= market value of Bond   PMT= Bond Coupon= Coupon rate* face value FV= face value  

I/Y=spot rate

here we assume face value of bond is 100

Bond N PV PMT FV I/Y.

A 1 -99 0.10*100 = 10 100 11.11%

B 2 -96 0.10*100 = 10 100 12.38%

C 3 -97 0.08*100= 8 100 9.19%

D 4 -97 0.07*100= 7 100 7.90%

B.

Spot rate of two year after two year (S) =    [( 1+ s_t4)⁴ / (1+ s_t2)²]^1/2 - 1

    = [ (1+0.079)⁴ / (1+0.1238)²]^1/2-1

= [ 1.355/1.2629]^0.5 - 1

^where    = 1.0358 - 1

s_{t4 =} Four year spot rate = 3.58%

s_t2 = two year spot rate hence estimated two year spot rate after two year is 3.58%

C.

first find the market value as per 3 year spot rate,

we assume face value = 100

Market value of Bond N =    Coupon1/(1+r)ⁿ + Coupon2 / (1+r)ⁿ + Coupon 3 / (1+r)ⁿ + FV/(1+r)ⁿ

  = 8/ (1+0.0919)¹ + 8/ (1+0.0919)² + 8/ (1+0.0919)³ + 100/(1+0.0919)   

= 7.33 + 6.71 + 6.145 + 76.82

= 96.99

where n = no of year to maturity Bond currently trade at 95 so there is arbitrage opportunity.

r= spot rate we will buy Bond N from current market @ 95 and Sell in spot market FV = face value of bond     @96.77 so there will arbitrage profit of 1.677

D.

  

   PV of bond N = Coupon1/(1+YTM)ⁿ + Coupon2 / (1+YTM)ⁿ + Coupon 3 / (1+YTM)ⁿ + FV/(1+YTM)ⁿ

or

where n = no of year to maturity use financial calculater

FV = face value of bond    N = no of year to maturity PV= market value of Bond YTM = yield to maturity      PMT= Bond Coupon= Coupon rate* face value FV= face value  

I/Y = YTM= ?

N=3 PV= -95 PMT= 0.08*100= 8 FV= 100 I/Y = 10.01% Hence YTM is 10.01%

yes there is one limitation of Yield to maturity (YTM) it assume same rate up to maturity and that is not possible in real market .