Question

Consider the following risk-free securities available to buy or sell to all investors in the market: 6. Cash flow (t=2) Price (t=0) Cash flow (t=1) Cash flow (t=3) Security 76 80 120 68 80 216 80 80 80 D 102 a) Compute the YTM offered by securities A and C. b) Write-down the equation that you would need to solve to find the YTM of security D. c) Compute the no-arbitrage price of security B. What is its YTM? d) Compute the missing cash flow paid by security E at t = 3 so that its no-arbitrage price at t = 0 is $102. e) If security D was suddenly trading for $205, but all the other prices stayed the same, is there an arbitrage opportunity that could be exploited? If so, clearly explain how you would implement the strategy and the profit that you expect to make.

Can someone help me with this question. I was assuming a 1% ytm for C and D but not sure if I did it correctly.

Answer 1

Part (a)

For Security A: Price at t = 0 will be = 76 = (Cash flow at t = 1) / (1 + YTM) = 80 / (1 + YTM)

Hence, YTM = 80 / 76 - 1= 5.26%

For security C: 68 = 80 / (1 + YTM)³; Hence, YTM = (80 /68)^1/3 - 1 = 5.57%

Part (b)

Let y be the YTM of the security D then, we need to solve the equation:

216 = 80 / (1 + y) + 80 / (1 + y)² + 80 / (1 + y)³ = 80 / y x [1 - ( 1+y)^-3]; Or,

2011 – (1 + y)-3) = 216

Part (c)

Price of security D = Present value of all its future cash flows

Hence, 216 = 80 / (1 + s1) + 80 / (1 + S2)² + 80 / (1 + s3)³ = 80 / (1 + 5.26%) + 80 / (1 + s2)² + 80 / (1 + 5.57%)³ = 144.00 + 80 / (1 + s2)²

Hence, 80 / (1 + s2)² = 216 - 144 = 72.00

Hence, s2 = (80 / 72)^1/2 - 1 = 5.41%

hence, no arbitrage price for security B = 120 / (1 + s2)² = 120 / (1 + 5.41%)² = $ 108.00

Part (d)

E is same as C in terms of the cash flows. E = 102 / 68 x C = 1.5C

Hence, the missing cash flow of E at t = 3 will be = 1.5 x corresponding cash flow of C = 1.5 x 80 = 120

Security D can be expressed as a liner combination of A, B and C.

Create a security A + 2/3B + C

It will have a cash flows of 80 at t = 1; 2/3 x 120 = 80 at t = 2 and 80 at t = 3

Hence its cash flows are identical to that of D. Hence, price of D = Price of A + 2/3B + C = 76 + 2/3 x 108 + 68 = 216

If D starts trading at a lower price of 205, then clearly there is an arbitrage opportunity and can be exploited as shown below:

• Buy 1 unit of D
• and short the synthetic portfolio A + 2/3B + C i.e.
  • Short (sell) 1 unit of A
  • Short (Sell) 2/3 unit of B
  • Short (Sell) 1 unit of C

And the arbitrage profit = (A + 2/3B + C) - D = 216 - 205 = $ 11