Finance-Theory · Unit 3 · Video 1 · Interactive Practice

Can the Currency Flip an NPV's Sign? And What the Fannie–Freddie Rescue Actually Protected

IKey Relations

Relation	Name	What it says
$V_0^{\yen} = S \cdot V_0^{\text{\textdollar}}$	Fixed exchange rate	One rate $S$ at every date multiplies every cash flow, so it factors out of the sum; $S > 0$ , so the sign cannot change
$d_t^{\yen} = d_t^{\text{\textdollar}}\,\dfrac{S_0}{S_t}$	Yen discount factor	The price today of one yen at date $t$ , found by routing it through dollars; it equals the dollar factor only when $S_t = S_0$
$V_0^{\yen} = S_0 \cdot V_0^{\text{\textdollar}}$	Known rate path	With each currency discounted at its own factors, $S_t$ cancels term by term and today's rate scales the whole value
$\text{claims} = \text{equity} + \text{paper}$	Two sets of claimants	Owners hold the equity; counterparties hold the paper — the IOUs and other obligations. A rescue is judged claim by claim

Key Insight: A currency is a scale factor, never a sign change. A sign flip can only come from mixing one currency's cash flows with another currency's discount factors — and under a fixed rate that mistake leaves no trace, because $S_0 / S_t = 1$ .

IIPricing a Yen at a Future Date

A yen at date $t$ has its own price today: find it by routing one yen through dollars.

Step 1 Yen per dollar at date t 95

💡 No other price survives the market: value a date- $t$ yen any other way and you could borrow in one currency, lend in the other, and pocket a riskless profit.

IIIOne Project, Two Sets of Factors

The project is worth $0.10 million today — can the same cash flows count as a loss in yen?

Yen per dollar at date 2 95

💡 All of this holds under certainty. Once the future is uncertain, the exchange rate becomes a second source of risk and has to be dealt with separately.

IVThe Rescue, Claim by Claim

The shares collapsed and the rest of the market rose: which claim did the takeover make good?

Step 0

VQuiz Questions

Problem 1 · A Rate That Never Moves

Given: under certainty a project is worth $0.40 million today, and the exchange rate is fixed at 120 yen per dollar at every date — find the project's value today in yen.

✅ Correct! One rate multiplies every cash flow, so it factors out of the sum:

120 \times 0.40 = 48

million yen.

❌ Check the sign. An exchange rate is always positive, so multiplying by

S

rescales a value but can never reverse it. A project worth $0.40 million is worth

+48

million yen.

❌ The rate multiplies.

S = 120

is yen per dollar, so 0.40 million dollars is 0.40 million

\times\, 120

yen. Dividing would answer a different question: how many dollars one yen buys.

❌ The cash flows are not needed here. Every one of them is multiplied by the same

S

, and because the rate never moves the same discount factors serve both currencies, so

S

comes straight out of the sum.

Show solution

Write the dollar value with its discount factors, then convert every cash flow at the one rate $S$ :

V_0^{\text{\textdollar}} = C_0 + d_1 C_1 + d_2 C_2 + \cdots

V_0^{\yen} = S\,C_0 + d_1 S\,C_1 + d_2 S\,C_2 + \cdots = S\left(C_0 + d_1 C_1 + d_2 C_2 + \cdots\right) = S \cdot V_0^{\text{\textdollar}}

Because the rate never moves, the same discount factors $d_t$ serve both currencies, which is what lets $S$ come out of the bracket. With $S = 120$ and $V_0^{\text{\textdollar}} = 0.40$ million dollars:

V_0^{\yen} = 120 \times 0.40 = 48 \text{ million yen}

And since an exchange rate is always positive:

V_0^{\text{\textdollar}} > 0 \Rightarrow V_0^{\yen} > 0, \qquad V_0^{\text{\textdollar}} < 0 \Rightarrow V_0^{\yen} < 0

A fixed rate cannot flip the sign — it only changes the units the value is quoted in.

Problem 2 · The Factor That Belongs to the Yen

Given: the dollar discount factor for date 2 is $d_2^{\text{\textdollar}} = 0.75$ , and the rate is known to follow the path $S_0 = 120$ and $S_2 = 144$ yen per dollar — find the yen discount factor $d_2^{\yen}$ .

✅ Correct!

0.75 \times \frac{120}{144} = 0.625

. The yen is known to be weaker at date 2, so a date-2 yen is worth less today than the dollar factor alone would suggest.

❌ That is the error the video marks. The factors match only when the rate has not moved. Here

S_2 \neq S_0

, so a date-2 yen converts at a different rate from a yen today and needs its own price.

❌ The ratio is the other way up. Routing a date-2 yen through dollars divides by

S_2

at date 2 and multiplies by

S_0

today, giving

S_0 / S_2

, not

S_2 / S_0

❌ That is the rate ratio on its own.

S_0 / S_2 = 120/144

only handles the currency; the waiting still has to be paid for by the dollar discount factor

0.75

Show solution

Price one date-2 yen by taking it through dollars:

\yen_2 \;\xrightarrow{\ \div\, S_2\ }\; \text{\textdollar}_2 \;\xrightarrow{\ \times\, d_2^{\text{\textdollar}}\ }\; \text{\textdollar}_0 \;\xrightarrow{\ \times\, S_0\ }\; \yen_0

At date 2, one yen is $1/S_2$ dollars. Those dollars are worth $d_2^{\text{\textdollar}} \times 1/S_2$ dollars today, and today's dollars convert at $S_0$ :

d_2^{\yen} = S_0 \cdot d_2^{\text{\textdollar}} \cdot \frac{1}{S_2} = d_2^{\text{\textdollar}}\,\frac{S_0}{S_2} = 0.75 \times \frac{120}{144} = 0.75 \times \frac{5}{6} = 0.625

Sanity check on the direction: $S_2 = 144 > 120 = S_0$ means more yen per dollar at date 2, so the yen is known to be weaker then. A date-2 yen must therefore be worth less in today's yen than $0.75$ : indeed $0.625 < 0.75$ .

Had the rate been fixed, $S_0 / S_2 = 1$ and the two factors would coincide — which is exactly why the mistake of reusing the dollar factor is invisible under a fixed rate.

Problem 3 · The Same Project, Counted in Yen

Given: a project pays $-10$ , $5$ and $8$ million dollars at dates 0, 1 and 2, with dollar discount factors $d_1^{\text{\textdollar}} = 0.90$ and $d_2^{\text{\textdollar}} = 0.80$ . The rate is known to follow the path 150 yen per dollar at dates 0 and 1, and 125 yen per dollar at date 2, so in millions of yen the cash flows are $-1500$ , $750$ and $1000$ .

What is the date-2 yen discount factor?

What is the project's value today, in millions of yen?

✅ Correct!

-1500 + 675 + 960 = 135

million yen, which is exactly

150 \times 0.90

: today's rate times the dollar value.

❌ Check the factor. Take one date-2 yen through dollars: divide by

S_2 = 125

, discount at

0.80

, convert back at

S_0 = 150

. The rate ratio that survives is

S_0 / S_2

❌ Check the sum. Discount each yen cash flow at its own yen factor:

-1500 + 750(0.90) + 1000\,d_2^{\yen}

. Applying the dollar factor

0.80

to the date-2 yen cash flow instead is the mistake that drags the total below zero.

Show solution

Step 1 — the value in dollars.

V_0^{\text{\textdollar}} = -10 + 5(0.90) + 8(0.80) = -10 + 4.5 + 6.4 = 0.90 \text{ million dollars}

Step 2 — the yen discount factors. The rate has not moved by date 1, so that factor is unchanged; by date 2 it has:

d_1^{\yen} = 0.90\,\frac{150}{150} = 0.90, \qquad d_2^{\yen} = 0.80\,\frac{150}{125} = 0.80 \times 1.2 = 0.96

Step 3 — the value in yen.

V_0^{\yen} = -1500 + 750(0.90) + 1000(0.96) = -1500 + 675 + 960 = 135 \text{ million yen}

And $135 = 150 \times 0.90 = S_0 \cdot V_0^{\text{\textdollar}}$ , as it must be: in each term $S_t$ cancels between the cash flow and the discount factor, leaving $S_0$ multiplying the whole dollar value.

The tempting mistake. Discounting the yen cash flows at the dollar factors gives

-1500 + 750(0.90) + 1000(0.80) = -1500 + 675 + 800 = -25

a project that looks like a loss of 25 million yen. That sign flip does not exist: it comes from pairing one currency's cash flows with another currency's discount factors. The other wrong answers come from converting at the wrong date's rate ( $125 \times 0.90 = 112.5$ ) or from never converting at all ( $0.90$ is still in dollars).

Problem 4 · Which Claim Was Protected

Given: in September 2008 a finance company holds $300 million of Fannie Mae commercial paper (a short-term IOU) and $50 million of Fannie Mae common shares. The federal government takes Fannie Mae over; its shares are nearly wiped out and the stock market as a whole rises.

What happens to the finance company's two holdings?

A manufacturer in the S&P 500 holds none of this paper. Why did its shares rise on the news?

✅ Correct! The rescue protected the paper, not the shares — and the broad market rose because the chain of failures running through the paper's holders had been cut.

❌ Separate the claims. A company has two sets of claimants: the owners, whose equity was worth very little after enormous losses, and the counterparties holding its IOUs and other obligations, which the government stood behind.

❌ Look for the channel that reaches a firm with no exposure. The manufacturer holds none of the paper and none of the equity, so the news has to reach it through what would have happened to the institutions that did hold the paper.

Show solution

Claim by claim. Fannie Mae's claimants fall into two layers. The owners hold the equity, which was worth very little once the losses had been absorbed; the counterparties hold the paper — commercial paper and other obligations — and it was the paper that the government stood behind. So the shareholders lost and the paper was made good: the rescue protected the paper, not the shares. The finance company keeps its $300 million; its $50 million of stock is gone.

Why everyone else rose. Many companies in the S&P 500 owned none of this paper and their shares rose all the same, so the explanation cannot be their own holdings. Had Fannie Mae and Freddie Mac been allowed to fail, their paper would have been worthless, the institutions holding it would have taken the losses, those businesses would have come under pressure, and the damage would have spread from firm to firm through the entire market. The takeover cut that chain, and the market read it as business conditions stabilized and knock-on failures ruled out.

The limit of the reading. It did not make everyone safe. Lehman Brothers — a large mortgage lender and a major investor in CDOs, securities built from pools of debt — remained under heavy pressure even after the rescue and filed for bankruptcy on 15 September 2008, eight days after the takeover was announced.

Solved: 0 / 4