Finance-Theory · Unit 2 · Video 9 · Interactive Practice

The Whole Table of Exchange Rates, Replaced by One Number

IKey Formulas

Formula	Name	What it gives
$(1+r)^T$	Growth factor	What $1 today is worth at Year $T$ : the future value of a dollar, a power of one factor
$\left(\frac{\text{\textdollar}_T}{\text{\textdollar}_0}\right) = \frac{1}{(1+r)^T}$	Discount factor	Today's value of $1 paid at Year $T$ : the reciprocal of the growth factor, below 1 when $r > 0$
$\dfrac{1}{(1+r)^{\,j-i}}$	Rate between two dates	Value at Year $i$ of $1 paid at Year $j$ : the exponent counts the years between payment and valuation
$V_0 = \text{CF}_0 + \frac{\text{CF}_1}{(1+r)} + \frac{\text{CF}_2}{(1+r)^2} + \cdots$	Present value operator	Every cashflow divided by its own power of $(1+r)$ , then added

Key Insight: Once the opportunity cost of capital $r$ is known, every exchange rate between every pair of dates is a power of the single factor $(1+r)$ , and the value operator becomes the explicit sum $V_0 = \sum_t \text{CF}_t/(1+r)^t$ .

IIOne Number Fills the Table

Once $r$ is known, the exchange rate between any two dates is a power of $(1+r)$ , set by the years between them.

Opportunity cost of capital r 8% Last date T 3

💡 In practice the market quotes many different rates rather than one; treating it as handing you a single $r$ is a simplification the course relaxes later.

IIICompounding: Year by Year Equals One Power

Growing $100 one year at a time at rate $r$ lands exactly on the one-step power $100\,(1+r)^t$ .

Rate r 7% Year t 3

💡 Writing two years of growth as $(1+r)^2$ rather than a separate two-year rate $1+z$ takes the year as the unit of account: at 7%, $1 + z = 1.07^2 = 1.1449$ , so $z = 14.49\%$ rather than $14\%$ .

IVThe Present Value Operator

Each cashflow is divided by its own power of $(1+r)$ , and the converted terms add up to $V_0$ .

Opportunity cost of capital r 10% Cashflows, $ millions

💡 An asset is a sequence of cashflows, so this one expression values any asset: under certainty, and with perfect markets among the assumptions still in force.

VQuiz Questions

Problem 1 · Discount a Single Payment

Given: $500 will be paid to you at Year 4, and the opportunity cost of capital is $r = 6\%$ . Find its value today, to the nearest cent.

✅ Correct! The payment is four years away, so it is divided by

1.06^4 = 1.26247696

, leaving $396.05 in today's dollars.

❌ Wrong direction. $631.24 is $500 multiplied by

1.06^4

: what $500 held today would grow to by Year 4. A dollar arriving later is worth less today, so divide by the growth factor.

❌ Close, but there is no compounding in that. $403.23 divides by

1 + 4 \times 0.06 = 1.24

, as if the four years of growth were four separate 6% pieces. Each year's growth applies to the previous year's balance, so the divisor is

1.06^4 \approx 1.2625

❌ Only one year of discounting. $471.70 is $500 divided by

1.06

, which is the payment's value at Year 3. The exponent counts the years between the payment and today: four.

❌ Not quite. Divide the payment by

(1+r)

raised to the number of years until it arrives.

Show solution

A payment $T$ years away is multiplied by the discount factor $\frac{1}{(1+r)^T}$ . Here $T = 4$ and $r = 0.06$ :

1.06^4 = 1.26247696

V_0 = \frac{500}{1.26247696} = 396.0468\ldots \approx 396.05

The payment is worth $396.05 today. Check in the other direction: $396.05 grown at 6% for four years comes back to $500.00 to the cent.

Problem 2 · The Timeline Trap

Given: you are owed $240 at Year 5, and the opportunity cost of capital is $r = 5\%$ . Find its value at Year 2, not today, to the nearest cent.

✅ Correct! The payment lands

5 - 2 = 3

years after the valuation date, so it is divided by

1.05^3 = 1.157625

❌ That is its value today. Dividing by

1.05^5 \approx 1.2763

carries the payment all the way back to Year 0; the question stops at Year 2, three years before the payment.

❌ The exponent is not the valuation date. Dividing by

1.05^2

discounts for 2 years, but the payment is

5 - 2 = 3

years away from Year 2.

❌ Wrong direction. The payment arrives after Year 2, so bringing it back to Year 2 means dividing by

1.05^3

; multiplying carries money forward in time.

❌ Not quite. Draw the timeline first: the payment sits at Year 5, the valuation at Year 2.

Show solution

Draw the timeline: the $240 sits at Year 5 and you are valuing it at Year 2. The gap is $5 - 2 = 3$ years, so the exponent is 3, not 5 and not 2.

1.05^3 = 1.157625

\text{value at Year 2} = \frac{240}{(1.05)^{3}} = \frac{240}{1.157625} = 207.3210\ldots \approx 207.32

The claim is worth $207.32 at Year 2. The exponent is the distance along the timeline between the payment and the valuation date, not the distance from today.

Problem 3 · Put the Operator to Work

Given: a project costs $1,000 today and pays $400 at Year 1, $500 at Year 2 and $300 at Year 3; the opportunity cost of capital is $r = 10\%$ . Find $V_0$ in dollars, and the decision it implies.

✅ Correct! The three inflows are worth $1,002.25 in today's dollars against a $1,000 cost, so

V_0 \approx 2.25 > 0

: take it, by a thin margin.

❌ That adds dollars of four different dates as if they were one currency. A raw sum is the operator with

r = 0

; at 10% each later cashflow must first be divided by its own power of

1.1

❌ Close, but those divisors are $1.1$ , $1.2$ , $1.3$ . That is

1 + rt

, simple interest. The divisors are powers of one factor:

1.1

1.21

1.331

❌ The exponents are off by one. Dividing the Year-1 inflow by

1.1^2

(and so on) discounts every inflow one year too far; the Year-

t

cashflow is divided by

(1.1)^t

❌ Not quite. Convert each cashflow to today's dollars with its own power of

1.1

, then add.

Show solution

Write the operator out term by term, with $\text{CF}_0 = -1000$ entering unconverted:

V_0 = -1000 + \frac{400}{1.1} + \frac{500}{(1.1)^2} + \frac{300}{(1.1)^3} = -1000 + \frac{400}{1.1} + \frac{500}{1.21} + \frac{300}{1.331}

V_0 \approx -1000 + 363.6364 + 413.2231 + 225.3944 = 2.2539

So $V_0 \approx 2.25$ dollars. It is positive, so the project is taken: in today's dollars it is worth $2.25 more than it costs. Adding the raw amounts would have said $200, which is the same operator with $r = 0$ .

Problem 4 · One Price Fixes the Rest

Given: the market prices a dollar paid at Year 1 at $0.97 today, and a single opportunity cost of capital governs every date.

What opportunity cost of capital does that price imply?

What does a dollar paid at Year 3 sell for today?

✅ Correct! One price pins the rate,

1 + r = 1/0.97

, and every other date follows as a power of that factor: a Year-3 dollar is worth

0.97^3 \approx 0.9127

today.

❌ The 3-cent discount is not the rate. The rate is defined through the growth factor,

1 + r = 1/0.97 \approx 1.0309

, so

r \approx 3.09\%

, a little more than 3%.

❌ That is the growth factor written as a percent.

1/0.97 \approx 1.0309

1 + r

; the rate is what remains after subtracting 1, about 3.09%.

❌ Under a single rate, one price is enough. The Year-1 discount factor alone gives

1 + r = 1/0.97

, and every other exchange rate is a power of that one factor.

❌ That knocks 3 cents off per year. The Year-3 factor is the Year-1 factor applied three times,

0.97^3

, so the discount compounds rather than adding up.

❌ Close, but dividing by $1 + 3r$ ignores compounding. Three years at one rate means dividing by

(1+r)^3 = (1/0.97)^3

, which gives

0.97^3

❌ A dollar paid later must be worth less when $r > 0$ . $0.97 is the Year-1 price; the Year-3 price is

0.97^3

❌ Not quite. Recover

1 + r

from the one price you have, then raise the discount factor to the number of years.

Show solution

The rate. With one rate, the Year-1 price alone pins it, because the discount factor is the reciprocal of the growth factor:

1 + r = \frac{1}{0.97} \approx 1.030928 \quad\Rightarrow\quad r \approx 3.09\%

The Year-3 price. Every other date is a power of the same factor, and a Year-3 dollar is three years away:

\frac{1}{(1+r)^3} = 0.97^3 = 0.912673 \approx 0.9127

A Year-3 dollar sells for about $0.9127 today. The discount compounds, running 0.97, then 0.9409, then 0.9127, rather than 0.97, 0.94, 0.91.

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