FNCE90060 Chap.1 Present Value and Arbitrage
Present Value and Arbitrage
Week 1 establishes the common language used throughout Financial Management: cash flows have amounts, dates, signs and risk. Present value converts a later certain payment into an equivalent amount today, while future value moves cash forward. Net present value compares benefits and costs only after they have been expressed at one date.
The Law of One Price then turns equivalence into a pricing rule: positions with identical future cash flows should carry the same current value. Replication makes that claim testable by constructing the same payoff from traded pieces. A genuine arbitrage is stricter than an attractive expected return. It requires a long-short trade with no net investment, no loss state and at least one positive payoff.
The chapter therefore joins calculation with a disciplined trading explanation.
What this chapter covers
- 01
Cash flows and the valuation date
- 02
Present value and future value
- 03
Net present value as cash today
- 04
Replication and the Law of One Price
- 05
Risk-free arbitrage conditions
- 06
Limits created by risk and trading frictions
Worked example · free
Value a two-payment contract
- 1Discount the first receipt: $520 divided by 1.04 equals $500.00.
- 1Discount the second receipt: $780 divided by 1.04 squared equals $721.15.
- 1Add the same-date values to obtain $1,221.15.
- 1Check that positive discounting makes the answer lower than $1,300.
Key terms
- Cash Flow
- A payment or receipt identified by amount, date, direction and uncertainty.
- Present Value
- The cash amount today equivalent to a future payment at the relevant discount rate.
- Net Present Value
- The sum of discounted inflows and outflows measured at one valuation date.
- Replication
- Constructing a portfolio whose future cash flows match another asset exactly.
- Law Of One Price
- The principle that identical traded payoffs should have identical current values.
- Arbitrage
- A long-short strategy with no net investment, no possible loss and some positive payoff.
Present Value and Arbitrage FAQ
Why must cash flows share one date before they are added?
Money available at different dates can earn different amounts before the chosen comparison point. Discounting or compounding first makes each cash flow economically comparable, so the total represents one coherent value rather than a sum of unlike amounts.
Does a higher quoted return prove an arbitrage?
No. A rate spread may reflect currency risk, credit risk, access restrictions or mismatched timing. Arbitrage requires an exact tradable payoff match and a long-short construction whose net terminal cash flow cannot be negative.
Why does present value fall when the rate rises?
A higher return means less money is needed today to reproduce the same future payment. The discount factor becomes larger, so dividing the fixed future cash flow by that factor produces a lower current equivalent.
What should an arbitrage answer show?
It should name the cheap and expensive positions, state what is bought and sold, record the initial net cash flow and verify that later receipts cover every obligation in all relevant states.
How do I choose a valuation date?
Use the date requested by the problem, or choose one date that makes every alternative easy to compare. Mark the distance from each cash flow to that date, then discount backward or compound forward by exactly that many matching periods. The choice of date changes the reported number but not the underlying economic equivalence.
Why must the discount rate match risk?
A certain receipt and an uncertain receipt are not interchangeable merely because they have the same promised amount and date. The opportunity cost must reflect the cash flow's risk as well as its timing. Using a risk-free rate for a risky claim would overstate its present value and would not establish a valid replicating price.
What is a useful direction check for NPV?
Verify signs and timing before interpreting the total. Costs paid today are normally negative, later benefits are positive, and a positive discount rate reduces the current value of a fixed future receipt. If a quoted purchase price rises while the payoff and rate remain fixed, the buyer's NPV should fall rather than rise.
When can the Law of One Price fail as a practical trade?
The theoretical comparison requires identical attainable payoffs. Transaction costs, funding limits, short-sale restrictions, taxes or unmatched states can stop a price difference from becoming a locked-in trade. State the friction explicitly; do not relabel a risky relative-value position as arbitrage just because two assets appear similar.
Exam move
Draw timelines until the valuation date, sign convention and exponent can be read without prose. Recompute simple one-period values in both directions to prove equivalence. For arbitrage, practise writing four lines: identify identical payoffs, compare prices, buy the cheaper side and sell the dearer side, then show initial and terminal net cash flows. Do not use the word arbitrage for a risky expected-profit strategy.
Finish each valuation with a direction check: a positive rate makes a future certain payment worth less today, and delaying the same payment lowers its current value. Build a correction log that names whether each error came from the date, rate, sign, risk match or trade direction. Before repeating a problem, predict which line must change and which cash flows must remain economically equivalent.
Explain the final price as the cost of reproducing the payoff, because that sentence connects the arithmetic to the no-arbitrage principle and makes an unsupported intuition easy to spot.
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