Finance-Theory · Unit 2 · Video 8 · Interactive Practice

Convert Before You Add: A Worked Net Present Value and Its Three Assumptions

IKey Formulas

Formula	Name	What it does
$\left(\frac{\text{\textdollar}_t}{\text{\textdollar}_0}\right) \times \text{CF}_t$	Converting one cashflow	Turns a date- $t$ amount into today's dollars at the market's exchange rate for date $t$
$\text{NPV}_0 = \text{CF}_0 + \left(\frac{\text{\textdollar}_1}{\text{\textdollar}_0}\right) \times \text{CF}_1 + \left(\frac{\text{\textdollar}_2}{\text{\textdollar}_0}\right) \times \text{CF}_2$	Net present value	Convert every cashflow, then add; $\text{CF}_0$ is already in today's dollars and takes no rate
$-10 + 5 \times 0.90 + 7 \times 0.80 = 0.10$	The worked example	In millions of dollars: the project is worth $100,000 today, while the naive sum $-10 + 5 + 7 = 2$ means nothing
$\text{NPV}_0 > 0 \Rightarrow \text{take}, \quad \text{NPV}_0 < 0 \Rightarrow \text{reject}$	Decision rule	Projects competing for the same money are ranked by $\text{NPV}_0$ and funded down the list

Key Insight: Dollars at different dates are different currencies, so every cashflow is converted into today's dollars before anything is added. Once $\text{NPV}_0$ is on the table, the management decision reads itself off its sign: the valuation is the hard part, the decision the easy one. Every such calculation rests on three assumptions: the cashflows are known (magnitudes, signs and timing), the exchange rates are known (read off the market), and there are no frictions in the conversions.

IIConvert Before You Add

Three amounts at three dates are three currencies; only after each is converted does the sum mean anything.

Project Step 1

IIIThe Number Decides

The sign of $\text{NPV}_0$ , not the sign of the raw total, decides: take if positive, reject if negative.

Start from

💡 Challenge: find cashflows whose naive total is positive but whose $\text{NPV}_0$ is negative, so that the two exchange rates alone flip the verdict.

IVRanking Projects That Compete for the Same Money

Four projects each cost $10 million today: ranked by $\text{NPV}_0$ , equal naive totals land far apart.

VQuiz Questions

Problem 1 · Convert, Then Add

Given: the exchange rates $\left(\frac{\text{\textdollar}_1}{\text{\textdollar}_0}\right) = 0.90$ and $\left(\frac{\text{\textdollar}_2}{\text{\textdollar}_0}\right) = 0.80$ ; a project costs $6 million today and pays $3 million in Year 1 and $4 million in Year 2 — find $\text{NPV}_0$ in millions of dollars, and the decision.

What is $\text{NPV}_0$ ?

Take the project or reject it?

✅ Correct! Converted into today's dollars the two payments are worth

2.7 + 3.2 = 5.9

million, $100,000 short of the $6 million cost, so

\text{NPV}_0 = -0.10

and the project is rejected.

❌ That is the naive sum.

-6 + 3 + 4 = 1

adds date-0, date-1 and date-2 dollars as they stand: three currencies. Convert each payment at its own date's rate first.

❌ The rates multiply.

\left(\frac{\text{\textdollar}_1}{\text{\textdollar}_0}\right) = 0.90

is the price today of one Year-1 dollar, so 3 Year-1 dollars are worth

3 \times 0.90 = 2.7

today, not

3 / 0.90

❌ Each rate belongs to its own date. The Year-1 payment converts at

0.90

and the Year-2 payment at

0.80

; this answer applies them the other way round.

❌ Check the sign of the converted sum. A negative

\text{NPV}_0

means reject. It is the naive sum,

+1

, that makes this project look worth taking.

Show solution

Convert each cashflow into today's dollars at its own date's rate; the date-0 cost is already in today's dollars and takes no rate:

\text{NPV}_0 = -6 + 3 \times 0.90 + 4 \times 0.80 = -6 + 2.7 + 3.2 = -0.1

The amounts are in millions of dollars, so the project is worth −$100,000 today. Since $\text{NPV}_0 < 0$ , reject it. The naive sum $-6 + 3 + 4 = 1$ would have said take, but it adds three different currencies.

Problem 2 · A Cost That Comes Last

Given: the same rates, $0.90$ for Year 1 and $0.80$ for Year 2; a project pays you $3 million today and $4 million in Year 1, then requires an investment of $8 million in Year 2 — find $\text{NPV}_0$ in millions of dollars.

✅ Correct! The Year-2 cost is in Year-2 dollars like any other Year-2 amount:

8 \times 0.80 = 6.4

of today's dollars, entered with a minus sign. The project is worth $200,000 today, so you take it even though its raw total is negative.

❌ The cost was left unconverted. A Year-2 amount converts at the Year-2 rate whether it is coming to you or leaving you: it enters as

-8 \times 0.80 = -6.4

, not

-8

❌ That is the naive sum.

3 + 4 - 8 = -1

adds three currencies. Convert before you add.

❌ The investment is money going out. A cost is a negative cashflow whenever it falls, so it enters as

-6.4

, not

+6.4

Show solution

Money going out is a negative cashflow at whatever date it falls, and it converts at that date's rate like any other payment:

\text{NPV}_0 = 3 + 4 \times 0.90 - 8 \times 0.80 = 3 + 3.6 - 6.4 = 0.2

In millions of dollars that is $200,000 today; $\text{NPV}_0 > 0$ , so take the project. The raw total $3 + 4 - 8 = -1$ makes it look like a loser, but converting shrinks the Year-1 inflow by only $0.4$ while it shrinks the Year-2 cost by $1.6$ : the cost comes last and takes the deepest discount.

Problem 3 · Selling the Project for a Later Payment

Given: the same rates; you own a project that costs $4 million today and pays $2 million in Year 1 and $3 million in Year 2. A buyer offers to take it over, paying the cost and collecting the payments, in exchange for one payment to you at date 2, two years from today — find the date-2 payment that is worth exactly what the project is worth to you today.

✅ Correct! The project is worth

0.2

million today, and one date-2 dollar is worth only

0.80

today, so it takes

0.2 / 0.80 = 0.25

million date-2 dollars to match it.

❌ That converts in the wrong direction. Multiplying by

0.80

turns date-2 dollars into today's dollars. Here you start from today's value and need date-2 dollars, so divide:

0.2 / 0.80

❌ $200,000 is the project's value in today's dollars. A payment at date 2 is made in date-2 dollars, a different currency, and each of those is worth only

0.80

today.

❌ That uses the Year-1 rate. The payment lands at date 2, so it converts at

0.80

, not

0.90

. Mark the payment on a timeline before choosing the rate.

Show solution

Step 1: value the project today.

\text{NPV}_0 = -4 + 2 \times 0.90 + 3 \times 0.80 = -4 + 1.8 + 2.4 = 0.2

That is $200,000 of today's dollars.

Step 2: run the conversion in the other direction. A payment of $X$ date-2 dollars is worth $0.80\,X$ today, and that must equal the project's value:

0.80\,X = 0.2 \quad\Longrightarrow\quad X = \frac{0.2}{0.80} = 0.25

So the buyer must pay $250,000 at date 2. Had the payment landed at date 1 instead, it would be $0.2 / 0.90 \approx 0.222$ million: the date the payment lands on decides which exchange rate applies.

Problem 4 · Which Assumption Breaks?

Given: three situations, each breaking one of the three assumptions behind every calculation of this kind. (1) A dealer who turns Year-2 dollars into today's dollars keeps a fee on every conversion. (2) The $7 million is certain to arrive, but nobody can say whether in Year 2 or in Year 3. (3) No market quotes a price today for a dollar paid in Year 2 — find the assumption each one breaks.

Situation 1 · the dealer's fee

Situation 2 · the uncertain date

Situation 3 · no quoted price

✅ Correct! The fee is a friction, the unknown date breaks "cashflows are known" through their timing, and the missing quote breaks "exchange rates are known".

❌ Not for the dealer's fee. Nothing about the cashflows is in doubt: their magnitudes, signs and timing are all known. The fee sits in the conversion itself.

❌ Not for the dealer's fee. The rate is known: it is posted on the dealer's board. The trouble is that the rate you actually receive, after the fee, is not the posted one.

❌ Not for the uncertain date. Nothing here says a rate is missing. What is missing is the payment's date, and timing is one of the three respects in which the cashflows must be known.

❌ Not for the uncertain date. Nobody is charging for a conversion here. The payment's timing is unknown, and timing is one of the three respects in which the cashflows must be known.

❌ Not for the missing quote. The cashflows can be fully specified. What is missing is the price today of a Year-2 dollar, the exchange rate that would convert them.

❌ Not for the missing quote. A friction is a cost of converting at a known rate; here there is no rate to convert at, because no market sets it.

Show solution

Every net present value in this video rests on three assumptions:

Cashflows are known, in three respects: their magnitudes, their signs and their timing. Situation 2 leaves the timing open, and a Year-3 dollar converts at a different rate from a Year-2 dollar, so $\text{NPV}_0$ cannot be computed.
Exchange rates are known, read off the marketplace. Situation 3 removes the market that would supply $\left(\frac{\text{\textdollar}_2}{\text{\textdollar}_0}\right)$ .
No frictions in the conversions: turning a Year-2 dollar into today's dollars costs nothing. Situation 1's fee is exactly the airport exchange counter: the posted rate and the rate you actually receive are two different numbers.

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