The Cash Flow Additivity Principle is a fundamental concept in finance that states that the value of an investment portfolio is the sum of the present values of its individual cash flows. In other words, the principle asserts that the value of an investment is the sum of the present values of its expected future cash flows.

Suppose we are considering two series of cash flows (A and B). The annual interest rate is 5%. We want to know the future value of combined cash flows at t = 3.

• We can calculate the future value of each series and add them up. The future value of series A is 100 x 1.05² + 100 x 1.05 + 100 = 315.25 and the future value of series B is 150 x 1.05² + 150 x 1.05 + 150 = 472.875. The future value of A + B is 788.125.

• Alternatively, we can add the cash flows of each series first and then find the future value of the combined cash flows: 250 x 1.05² + 250 x 1.05 + 250 = 315.25 = 788.125.

We can use this principle to solve many uneven cash flow problems if we add dollars indexed at the same point in time. It allows for confirmation that asset prices are the same for economically equivalent assets, even if the assets have different cash flow streams. Dollar amounts indexed at the same point in time are additive.

Consider a cash flow series, A, with $100 indexed at t = 1, 2, 3 and 5, and $0 at t = 4. This series is an almost-even cash flow, flawed only by the missing $100 at t = 4. How do we find the present value of this series?

• We can create an annuity B with $100 indexed at t = 1, 2, 3, 4, 5. It's easy to find the present value of this series.
• Then we isolate an easily evaluated cash flow B - A; it has a single cash flow of $100 at t = 4. It's also easy to find the present value of this single cash flow.
• We then subtract the present value of B - A from the present value of B.

Implied Forward Rates

Implied forward rates refer to the interest rates that are implied by the current market prices of financial instruments with different maturities.

Suppose the yields-to-maturity on a 3-year and 4-year zero coupon bonds are 3.5% and 4% on a semi-annual basis. The implied forward rate IFR_6,2, can be calculated: (1 + 0.0175)⁶ x (1 + IFR_6,2)² = (1 + 0.02)⁸ => IFR_6,2 = 0.0275, so the forward rate is 5.5% (0.0275 x 2). If it is greater or less than 5.5%, the cash flow additivity principle would be violated and investors would have an arbitrage opportunity.

Implied Forward Exchange Rates

The forward exchange rate is the exchange rate at which a bank agrees to exchange one currency for another at a future date when it enters into a forward contract with an investor. If the one-year risk-free interest rates are 4.2% in Canada and 4.5% in the United States and the spot CAD/USD exchange rate is 1.35, what is the one-year forward CAD/USD exchange rate that prevents investors from earning arbitrage profits?

An investment of US$10,000 would be worth US$10,000 x e^0.045 = US$10,460.87

Alternatively, you can convert US$10,000 to CA$13,500 at the spot rate and investment the CA$ at 4.2%, and get CA$13,500 x e^0.042 = CA$14,079.08

The implied one-year forward exchange rate must be CA$14,079.08/US$10,460.87 = 1.3459

Option Pricing

Similarly, an investor can use options to create a hedged portfolio that is risk free, irregardless of the change in the price of the underlying asset.