We introduced in class the terminal value formula and explained that it makes more sense to assume a constant long-term growth and long-term discount rate for horizons above 5 -years. For your convenience, we restate the formula here, but we make it shorter by assuming that the long-term growth and discount are reached after 2 years. This is a valid assumption and should not throw you off.

V = CF1/(1+r_st) + CF2/(1+r_st)^2 + 1/(1+r_st)^2 * CF3/(r_LT- g_LT).

Which statement below is correct?

Question

We introduced in class the terminal value formula and explained that it makes more sense to assume a constant long-term growth and long-term discount rate for horizons above 5 -years. For your convenience, we restate the formula here, but we make it shorter by assuming that the long-term growth and discount are reached after 2 years. This is a valid assumption and should not throw you off.

V = CF1/(1+r_st) +  CF2/(1+r_st)^2 + 1/(1+r_st)^2  * CF3/(r_LT- g_LT).

Which statement below is correct?

BlackTom AI · Accepted Answer

Terminal value at time 2 (TV2) equals the present value at t=2 of all cash flows from t=3 onward under perpetual growth, so TV2 = CF3 / (r_LT - g_LT) when CF3 is the first cash flow one period after t=2. The overall value at t=0 is then CF1 and CF2 discounted by the short-term discount rate, plus TV2 discounted two periods to t=0.

1. Relevant knowledge points and concepts
- Gordon (perpetuity growth) formula: value at time t of cash flows that start one period later and grow at g is CF_{t+1} / (r - g).
- Discounting: cash flows or terminal values must be discounted back to the valuation date using the appropriate discount rate for those periods.
- The short-term discount rate r_st applies for years 1–2; the long-term rates r_LT and g_LT apply from year 3 onward.

2. Step-by-step reasoning
- Compute TV at t=2 for cash flows from year 3 onward: TV2 = CF3 / (r_LT - g_LT).
- Present-value of TV2 at t=0: PV(TV2) = TV2 / (1 + r_st)^2 because it is two years before t=0.
- Add year-by-year discounted cash flows for years 1 and 2 using r_st.

3. Algebraic substitution (intermediate steps)
- V = CF1/(1 + r_st) + CF2/(1 + r_st)^2 + TV2/(1 + r_st)^2
- Substitute TV2 = CF3/(r_LT - g_LT):
  V = CF1/(1 + r_st) + CF2/(1 + r_st)^2 + [CF3/(r_LT - g_LT)]/(1 + r_st)^2
- Rearranged: V = CF1/(1 + r_st) + CF2/(1 + r_st)^2 + 1/(1 + r_st)^2 * CF3/(r_LT - g_LT), which matches the given equation.

4. Why the correct answer is correct
- The equation exactly implements the Gordon growth model for the terminal value at t=2 and correctly discounts that terminal value back to t=0 by (1 + r_st)^2. CF3 appears inside the terminal-value numerator (representing the first perpetuity cash flow at t=3) and is not being double-discounted beyond the required discounting of the terminal value.

5. Why the other options are incorrect
- Option 2 (CF3 is discounted twice) misunderstands the structure: CF3 is used to compute value at t=2 via the denominator (r_LT - g_LT); that valuation is then discounted twice to t=0, which is correct — not an erroneous double-discount of the same timing.
- Option 3 (forgot to discount CF1 and CF2 by g_st) confuses growth and discounting: CF1 and CF2 are actual cash flows at those dates and should be discounted by the short-term discount rate r_st only; there is no g_st discounting required.

6. Conclusion
The algebraic derivation and timing interpretation confirm the formula is correct. Final answer: The equation is correct.

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