Finance-Theory · Unit 2 · Video 7 · Interactive Practice

Ninety-Seven Cents for Next Year's Dollar: Where Discount Factors Are Set

IKey Formulas

FormulaNameWhere it comes from
($1$0)=0.97\left(\frac{\text{\textdollar}_1}{\text{\textdollar}_0}\right) = 0.97One-year exchange rateThe clearing price of a certain $1 payable at date 1: a traded price, not an estimate
V0=CF0+($1$0)×CF1+($2$0)×CF2+⋯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 + \cdotsPresent value operatorCashflows from the business; one exchange rate per date from the market
($5$0)<($1$0)<1\left(\frac{\text{\textdollar}_5}{\text{\textdollar}_0}\right) < \left(\frac{\text{\textdollar}_1}{\text{\textdollar}_0}\right) < 1Discount factorsTypically below one, and lower the longer the wait
CF1−0.97×CF1=0.03×CF1\text{CF}_1 - 0.97\times\text{CF}_1 = 0.03\times\text{CF}_1The discount3% of face; with payment certain, the payment for waiting

Key Insight: The exchange rates in the value operator are neither derived nor assumed: each one is a price read off a market, one auction per date. They typically come back below one, and in a world where payment is certain the shortfall is not a charge for risk: it is payment for waiting.

IIThe Auction Sets the Rate

The exchange rate is not derived or assumed: it is the price a certain date-1 dollar fetches at auction.

IIIOne Auction per Date

One auction per date sets each factor; a longer wait has to be paid more, so in ordinary times it sells for less.

💡 Inflation is a genuine further reason for a future dollar to be worth less today, but a different one: impatience is about when you consume, inflation about what a dollar buys. The two usually travel together and are taken up separately.

IVQuiz Questions

Problem 1 · From Auction Price to Value

Given: a certain $1 payable at date 1 sold at auction for $0.97 — find the value today of a certain $2,500 payable at date 1.

✅ Correct! Every certain date-1 dollar converts at the auctioned rate, so 0.97×2500=24250.97 \times 2500 = 2425: the claim is worth $2,425 today.
❌ No conversion was made. $2,500 is an amount of date-1 dollars. Until it is converted at the exchange rate, it is not a value in today's dollars.
❌ Wrong direction. Dividing by 0.97 pushes the amount above $2,500, but a certain future dollar sells for less than a dollar today. Multiply by the rate instead.
❌ That is the discount, not the value. $75 is the 3% of face the buyer is paid for waiting; the value is what the claim costs today, $2,500 − $75.
Show solution

The auction fixed the exchange rate between date-1 dollars and today's dollars:

($1$0)=0.97\left(\frac{\text{\textdollar}_1}{\text{\textdollar}_0}\right) = 0.97

Under certainty every date-1 dollar is the same currency, so the $2,500 claim converts at that same rate:

V0=($1$0)×CF1=0.97×2500=2425V_0 = \left(\frac{\text{\textdollar}_1}{\text{\textdollar}_0}\right)\times\text{CF}_1 = 0.97 \times 2500 = 2425

The claim is worth $2,425 today. The gap, $2,500 − $2,425 = $75, is 3% of face, the same 3% as on the $1 certificate.

Problem 2 · Certain Payment, Still a Discount

Given: the certificate pays $1 at date 1 with certainty (no default, no doubt the money arrives), and it sold for $0.97 — find what the three-cent discount pays for.

✅ Correct! With default ruled out, the three cents are the price of postponing consumption for a year: payment for waiting.
❌ Risk has been assumed away. The payment is certain by construction, so the chance of default is zero and there is nothing of that kind to price, yet the discount is still there.
❌ The money runs the other way. The issuer receives $0.97 and repays $1.00, so the issuer pays the three cents and the buyer, who waits, receives them.
❌ Certainty removes risk, not impatience. People would rather consume now than later, so a dollar a year away is worth less today even when it is certain to arrive. The $0.97 is the rate itself, not a mistake about it.
Show solution

Rule out what the setup eliminates. Default risk: the payment is certain, so there is none. A fee: the issuer takes in $0.97 and pays out $1.00, so the three cents flow to the buyer. Mispricing: the clearing price is the exchange rate, so there is no other number it could be wrong about.

What is left is impatience. The buyer hands over $0.97 now and waits a year to be made whole; nobody does that for free, and the auction settles the price of doing it:

1.00−0.97=0.03(3% of face)1.00 - 0.97 = 0.03 \quad (3\% \text{ of face})

Seen from the other side, the same premium is the foregone chance to lend the dollar out today. Inflation is a genuine further reason for discounting, but a different one: it is about what a dollar buys, not when you consume.

Problem 3 · Two Auctions, One Operator

Given: the one-year auction cleared at $0.97 per certain dollar, and a certain claim to $500 payable at date 2 sells today for $465. An asset pays, with certainty, $200 today, $300 at date 1 and $500 at date 2.

What is the two-year exchange rate?

What is the asset worth today?

✅ Correct! The two-year rate is 465/500=0.93465/500 = 0.93, and converting each cashflow at its own rate gives $200 + $291 + $465 = $956.
❌ 0.97 is the one-year rate. Each date has its own auction and its own price; the two-year rate is read off the date-2 claim.
❌ That is face over price. The rate is today's dollars paid per date-2 dollar promised, $465 ÷ $500, and it comes out below one.
❌ 0.07 is the discount as a share of face ($35 on $500). The rate is what is paid per dollar promised, not what is knocked off it.
❌ That adds three different currencies. Convert each cashflow at its own date's rate first, then add.
❌ The one-year rate was applied to the date-2 cashflow. A two-year wait has its own, lower price: 0.93.
❌ The date-0 cashflow was converted too. $200 today is already in today's dollars and enters the sum unchanged.
Show solution

Step 1: read the two-year rate off its own market. A certain $500 at date 2 costs $465 today, so each date-2 dollar costs

($2$0)=465500=0.93\left(\frac{\text{\textdollar}_2}{\text{\textdollar}_0}\right) = \frac{465}{500} = 0.93

It sits below the one-year rate of 0.97, as expected: the wait is longer.

Step 2: convert each cashflow at its own rate, then add. The date-0 cashflow needs no conversion:

V0=CF0+($1$0)×CF1+($2$0)×CF2V_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 V0=200+0.97×300+0.93×500=200+291+465=956V_0 = 200 + 0.97\times 300 + 0.93\times 500 = 200 + 291 + 465 = 956

The asset is worth $956 today. Its date-2 term is exactly the $465 the date-2 claim sold for: converting at a market rate reproduces the market's own price.

Problem 4 · A Consistent Set of Auction Prices

Given: under certainty, the auction for a certain $1 payable at date 1 cleared at $0.97 — find the list that could be the auction prices of a certain $1 payable at dates 1, 2 and 3 in ordinary times.

✅ Correct! It starts at the auctioned 0.97, stays below one, and falls with every extra year of waiting.
❌ Rising prices pay a longer wait less. Push the wait out and the compensation has to grow, so each later date must sell for less than the one before.
❌ A flat list leaves the extra years unpaid. A date-3 dollar would cost the same as a date-1 dollar even though the buyer waits two years longer for it.
❌ The one-year price is already set. The auction cleared at $0.97, so any list must start at 0.97; a date-1 price of $1.00 would also leave the first year of waiting unpaid.
Show solution

Three conditions, each from the video:

  • The date-1 price is a fact of the market, ($1$0)=0.97\left(\frac{\text{\textdollar}_1}{\text{\textdollar}_0}\right) = 0.97. That rules out (c).
  • Each longer wait must be paid more, so the prices fall strictly with the date: ($3$0)<($2$0)<($1$0)\left(\frac{\text{\textdollar}_3}{\text{\textdollar}_0}\right) < \left(\frac{\text{\textdollar}_2}{\text{\textdollar}_0}\right) < \left(\frac{\text{\textdollar}_1}{\text{\textdollar}_0}\right). That rules out (a), which rises, and (b), which is flat.
  • In ordinary times every factor is below one, so each shrinks the cashflow it multiplies. List (d) meets this too.

Only (d) survives. The one-year price does not pin down 0.94 and 0.91: those are set in their own auctions. What the need to pay for waiting fixes is their order.

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