Finance-Theory · Unit 2 · Video 6 · Interactive Practice

Yen, Pounds, and Next Year's Dollars: Where Net Present Value Comes From

IKey Formulas

Formula	Name	What it says
$\text{¥}150 + (\text{£}300)\times(153\ \text{¥}/\text{£}) = \text{¥}46{,}050$	Convert, then add	Both amounts go into one numeraire before the sum; with pounds as numeraire the same money reads £300.98, to the penny
$\left(\dfrac{\text{\textdollar}_t}{\text{\textdollar}_0}\right)$	Exchange rate for date $t$	A dollar at date $t$ , valued today: how many of today's dollars one date- $t$ dollar is worth
$V_0(\text{CF}_0, \text{CF}_1, \ldots) = \text{CF}_0 + \left(\tfrac{\text{\textdollar}_1}{\text{\textdollar}_0}\right)\times\text{CF}_1 + \left(\tfrac{\text{\textdollar}_2}{\text{\textdollar}_0}\right)\times\text{CF}_2 + \cdots$	Net present value operator	Every term in date-0 dollars; $\text{CF}_0$ carries no rate, and $\text{CF}_0 \lt 0$ is the initial investment

Key Insight: Dollars at different dates are different currencies, just as yen and pounds are. Choose a numeraire date (usually date 0, today's dollars), convert each cashflow at its own date's exchange rate, and only then add. Once the rates are given the sum is ordinary arithmetic; the whole difficulty sits in the rates.

IIConvert First — Yen, Pounds and a Numeraire

¥150 plus £300 has no unit until one currency is chosen as numeraire, and either choice holds the same money.

Numeraire

IIIEvery Date Its Own Currency — Building $V_0$

Each dated cashflow converts at its own exchange rate into date-0 dollars; only then do the terms add to $V_0$ .

💡 These three exchange rates are illustrative; like every rate in the video, they arrive ready-made, and where such numbers actually come from is the next question.

IVQuiz Questions

Problem 1 · Convert, Then Add

Given: you hold ¥600 and £200, and the market rate is $153\ \text{¥}/\text{£}$ — find the value of the holding with yen as the numeraire.

✅ Correct! Only the pounds need converting: £200 × 153 ¥/£ = ¥30,600, and with the ¥600 already in hand the total is ¥31,200, one number in one unit.

❌ That adds yen to pounds as they stand. 600 + 200 = 800 is a number with no unit to attach to it. Convert the pounds into yen first.

❌ The rate went onto the wrong amount. That multiplies the ¥600 by 153 and adds the £200 unconverted. The rate 153 ¥/£ turns pounds into yen: the ¥600 is already in the numeraire, and it is the £200 that gets multiplied.

❌ The pounds were divided by the rate. At 153 yen per pound, each pound is worth 153 yen, so £200 is 200 × 153 = ¥30,600, not 200 ÷ 153.

Show solution

With yen as the numeraire the ¥600 is already in the base currency, so only the pounds are converted, at the market rate:

\text{¥}600 + (\text{£}200)\times(153\ \text{¥}/\text{£}) = \text{¥}600 + \text{¥}30{,}600 = \text{¥}31{,}200

A London desk reads the same holding in pounds: $\text{¥}31{,}200 \div (153\ \text{¥}/\text{£}) \approx \text{£}203.92$ , to the penny (the quotient runs $203.9216\ldots$ ). A different number in a different unit, and the same money. The bare 800 is neither: it has no unit at all.

Problem 2 · The Term With No Rate

Given: a project costs 50 today and pays 30 at date 1 and 40 at date 2, so $\text{CF}_0 = -50$ , $\text{CF}_1 = 30$ , $\text{CF}_2 = 40$ ; the exchange rates are $\left(\tfrac{\text{\textdollar}_1}{\text{\textdollar}_0}\right) = 0.95$ and $\left(\tfrac{\text{\textdollar}_2}{\text{\textdollar}_0}\right) = 0.90$ — find $V_0$ in date-0 dollars.

✅ Correct!

\text{CF}_0

enters unchanged;

0.95\times 30 = 28.50

and

0.90\times 40 = 36.00

are in date-0 dollars, so

V_0 = -50 + 28.50 + 36.00 = 14.50

❌ That adds the cashflows as they stand.

-50 + 30 + 40 = 20

adds three currencies together; the date-1 and date-2 amounts must be converted before they join the sum.

❌ $\text{CF}_0$ was given a rate it does not carry. It sits at the numeraire date, already in date-0 dollars, so it enters as

-50

, not as

0.95\times(-50) = -47.50

❌ The rates were divided into the cashflows.

\left(\tfrac{\text{\textdollar}_1}{\text{\textdollar}_0}\right) = 0.95

is how many of today's dollars one date-1 dollar is worth, so it multiplies:

0.95\times 30 = 28.50

, not

30 \div 0.95

Show solution

Write the net present value operator for dates 0 to 2, then substitute:

V_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

V_0 = -50 + 0.95\times 30 + 0.90\times 40 = -50 + 28.50 + 36.00 = 14.50

The first term carries no exchange rate because it is already denominated in the base currency. Each later term is converted before it joins the sum, and that is the only reason the addition means anything.

Problem 3 · A Cost at the End

Given: a project costs 40 today, pays 70 at date 1 and 30 at date 2, and must be dismantled at date 3 at a cost of 60, so $\text{CF}_0 = -40$ , $\text{CF}_1 = 70$ , $\text{CF}_2 = 30$ , $\text{CF}_3 = -60$ ; the exchange rates for dates 1, 2 and 3 are $0.95$ , $0.90$ and $0.85$ — find the date-3 term in date-0 dollars, then $V_0$ .

The date-3 term, in date-0 dollars

The value $V_0$

✅ Correct! The dismantling cost converts at its own rate,

0.85\times(-60) = -51.00

, and

V_0 = -40 + 66.50 + 27.00 - 51.00 = 2.50

in date-0 dollars.

❌ The minus sign was lost. A future cost is a negative cashflow, and it converts like every other:

0.85\times(-60) = -51.00

❌ That divides by the rates. Each rate is how many of today's dollars one dollar of its date is worth, so it multiplies its cashflow: the date-3 term is

0.85\times(-60) = -51.00

, not

-60 \div 0.85

❌ That leaves the date-3 cashflow in date-3 dollars. Converting it into date-0 dollars means multiplying it by its own rate,

0.85

❌ That adds four currencies as they stand.

-40 + 70 + 30 - 60 = 0

is a number in no single unit; convert every later cashflow at its own rate first.

❌ The dismantling cost entered with a plus sign. Money going out is negative at whatever date it falls, so the date-3 term is

-51.00

and it reduces the sum.

Show solution

Convert term by term; the date-0 cashflow needs no rate:

\begin{aligned} \text{CF}_0 &= -40.00 \\ 0.95\times 70 &= 66.50 \\ 0.90\times 30 &= 27.00 \\ 0.85\times(-60) &= -51.00 \end{aligned}

V_0 = -40.00 + 66.50 + 27.00 - 51.00 = 2.50

The cost at date 3 is a negative cashflow and converts at the date-3 rate exactly as a receipt would. Added as they stand, the four cashflows give $0$ , a number in no single unit.

Problem 4 · The Borrower's Side

Given: a borrower receives 100 today and repays 36 at each of dates 1, 2 and 3; the lender's cashflows are the same amounts with the opposite signs. The exchange rates for dates 1, 2 and 3 are $0.95$ , $0.90$ and $0.85$ — find $V_0$ for each side.

Borrower: $\text{CF}_0 = 100$ , then $-36$ at dates 1 to 3

Lender: $\text{CF}_0 = -100$ , then $36$ at dates 1 to 3

✅ Correct! The three repayments are worth

34.20 + 32.40 + 30.60 = 97.20

in date-0 dollars, so the borrower's

V_0 = 100 - 97.20 = 2.80

and the lender's is

-2.80

: flip the sign of every cashflow and every term flips with it.

❌ That adds the cashflows as they stand.

100 - 3\times 36 = -8

for the borrower,

+8

for the lender. The repayments sit at three different dates, so each is converted before it is added.

❌ That is the other party's value. The lender's cashflows are the borrower's with every sign flipped, so the two values are negatives of each other; check which side receives the 100 at date 0.

❌ The rates were divided into the repayments. Each rate multiplies its cashflow: the date-1 repayment is worth

0.95\times 36 = 34.20

in date-0 dollars.

❌ $97.20$ is the value of the repayments alone. The lender's date-0 cashflow,

-100

, belongs in the sum too, unchanged, since it is already in date-0 dollars.

Show solution

The repayments are the same amount at three dates, but each date is its own currency, so each converts at its own rate:

0.95\times 36 + 0.90\times 36 + 0.85\times 36 = 34.20 + 32.40 + 30.60 = 97.20

The borrower's date-0 cashflow, $+100$ , needs no conversion, and the lender's is $-100$ :

V_0^{\text{borrower}} = 100 - 97.20 = 2.80, \qquad V_0^{\text{lender}} = -100 + 97.20 = -2.80

Nothing in the operator restricts which cashflows are positive or where the negative ones sit. Here the money comes in first and goes out later, and the operator nets the two inside one sum exactly as it does for a project that invests first.

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Yen, Pounds, and Next Year's Dollars: Where Net Present Value Comes From

IKey Formulas

IIConvert First — Yen, Pounds and a Numeraire

IIIEvery Date Its Own Currency — Building V0V_0V0​

IVQuiz Questions

IIIEvery Date Its Own Currency — Building $V_0$