The time to expiration is the time you decide whether or not to extend the franchise, and is at the end of year 2. P = $18,646 X = $20,000 t = 2. rRF = 0.06. = 0.2025 σ2 Although the problem stated to assume the variance of the rate of return was project’s also calculate it using the direct method. First calculate the rates of return 0.2025, we’ll using the decision tree. To do this, calculate the present values of the two branches as of the exercise date, year 2, and the rates of return assuming the initial value of the investment was P = $18,646. 0 1 2 3 4 40% Prob. | | | | | Good 43,388 = PV of two 25,000 cash flows Bad | | | | | 60% Prob. 8,678 = PV of two 5,000 cash flows Annual rate of return in good state = [43,388/18,646]1/2 -1 = 52.54% Annual rate of return in bad state = [8,678/18,646]1/2 1 = -31.78% – The expected value of these two returns is 52.54(0.40) 31.78(0.60) = 1.95% [Note: this – 10% as you might hope! The 2-year returns are 43,388/18,646 1 = 132.69% in isn’t – the good state and 8,678/18,646 1 = -53.46% in the bad state. The expected value of – these is 132.69% (0.40) - 53.46% (0.60) = 21.0%, which is exactly 10% compounded twice: 0.21 = (1.10)2 1.] – The variance is = (0.5254 - 0.0195)2(0.40) + (-0.3178 0.0195)2 (0.60) = 0.1706 σ2 –
The time to expiration is the time you decide whether or not to extend the franchise, and is at the end of year 2. P = $18,646 X = $20,000 t = 2. rRF = 0.06. σ2 = 0.2025 Although the problem stated to assume the variance of the project’s rate of return was 0.2025, we’ll also calculate it using the direct method. First calculate the rates of return using the decision tree. To do this, calculate the present values of the two branches as of the exercise date, year 2, and the rates of return assuming the initial value of the investment was P = $18,646. 0 1 2 3 4 40% Prob. | | | | | Good 43,388 = PV of two 25,000 cash flows Bad | | | | | 60% Prob. 8,678 = PV of two 5,000 cash flows Annual rate of return in good state = [43,388/18,646]1/2 -1 = 52.54% Annual rate of return in bad state = [8,678/18,646]1/2 – 1 = -31.78% The expected value of these two returns is 52.54(0.40) – 31.78(0.60) = 1.95% [Note: this isn’t 10% as you might hope! The 2-year returns are 43,388/18,646 – 1 = 132.69% in the good state and 8,678/18,646 – 1 = -53.46% in the bad state. The expected value of these is 132.69% (0.40) - 53.46% (0.60) = 21.0%, which is exactly 10% compounded twice: 0.21 = (1.10)2 – 1.] The variance is σ2 = (0.5254 - 0.0195)2(0.40) + (-0.3178 – 0.0195)2 (0.60) = 0.1706
FM14e_SM_Ch26.docx