Finance-Theory · Unit 23 · Video 5 · Interactive Practice

Course Summary I: Six Principles and Present Value Pricing

IKey Formulas

Formula	Name	What you need
$V_0 = CF_0 + \frac{CF_1}{1+r} + \frac{CF_2}{(1+r)^2} + \cdots$	Present value	The timeline of cash flows and the opportunity cost of capital $r$
$\text{PV} = \frac{C}{r} - \frac{C}{r}\,\frac{1}{(1+r)^T}$	Annuity (perpetuity minus delayed perpetuity)	Payment $C$ for $T$ years, rate $r$
$P_0 = \frac{D_1}{r-g},\ \ r > g$	Gordon growth model	Next dividend $D_1$ , growth $g$ , required return $r$
$C_0 = \frac{1}{r}\left(\frac{r-d}{u-d}\,C_u + \frac{u-r}{u-d}\,C_d\right)$	One-period binomial price	$u$ , $d$ , payoffs $C_u$ , $C_d$ ; $r = 1 + \text{riskless rate}$

Key Insight: One method prices every security in the course: write down its cash flows, then value them with present value and no free lunch. Bonds have fixed payments, stocks random dividends, forwards start at zero value, options pay a kinked function of the price.

IIVisualization 1 — An Annuity as Two Perpetuities

A $250 annuity for $T$ years is a perpetuity bought today minus one that starts paying after date $T$ .

Years of payments (T) 5

IIIVisualization 2 — A Coupon Bond Is a Bundle of Discount Bonds

Each payment of the bond is priced as its own discount bond: $P = C\,P_{0,1} + C\,P_{0,2} + \cdots + (C+F)\,P_{0,T}$ .

Yield r (%) 7

IVVisualization 3 — Copying a Call with Stock and Borrowing

A call is copied by $\Delta$ shares plus riskless borrowing $B$ , so no free lunch fixes its price.

Strike K ($) 30

VQuiz Questions

Problem 1 · Gordon Growth

Given: A stock will pay a dividend of $2.70 one year from now; dividends then grow at $g = 3\%$ a year forever, and investors require $r = 11\%$ . Find the stock's price today.

✅ Correct!

P_0 = \frac{2.70}{0.11 - 0.03} = \frac{2.70}{0.08} = 33.75

, so $33.75.

❌ That ignores growth.

\frac{2.70}{0.11} \approx 24.55

is a level perpetuity; growing dividends are discounted at

r - g

❌ Wrong sign on $g$ .

\frac{2.70}{0.14} \approx 19.29

uses

r + g

. Growth raises the value: the denominator is

r - g

, not

r + g

❌ The $2.70 is already next year's dividend.

\frac{2.70 \times 1.03}{0.08} \approx 34.76

grows it one year too many; the formula takes

D_1

as given.

Show solution

The dividends form a growing perpetuity whose first payment, $2.70, arrives at date 1. With $r > g$ :

P_0 = \frac{D_1}{r - g} = \frac{2.70}{0.11 - 0.03} = \frac{2.70}{0.08} = 33.75

The price is $33.75. Turned around, the same model reads the required return off the price: $r = \frac{D_1}{P_0} + g = \frac{2.70}{33.75} + 0.03 = 0.08 + 0.03 = 0.11$ .

Problem 2 · The Same Bond One Year Later

Given: A 4-year bond with face value $1,000 pays a 6% coupon once a year, and its yield is $r = 7\%$ with annual compounding. Find its price one year from now, just after the first coupon is paid, if the yield is still 7%.

✅ Correct! Three payments remain:

\frac{60}{1.07} + \frac{60}{1.07^2} + \frac{1060}{1.07^3} \approx 973.76

. With the yield unchanged, the price drifts up toward par as maturity approaches.

❌ That is today's price, with four payments left. A year later only three remain, each one year closer, so the price changes even though the yield does not.

❌ Not at par. A bond sells at par only when its yield equals its coupon rate; here 7% > 6%, so it still sells below $1,000.

❌ That counts the coupon just paid. $973.76 + $60 = $1,033.76 is what the holder has in total; the coupon has already left the bond.

Show solution

After the first coupon, the bond pays $60 at dates 1 and 2 and $1,060 at date 3 (measured from the new today). Each payment is a discount bond:

P_1 = \frac{60}{1.07} + \frac{60}{1.07^2} + \frac{1060}{1.07^3} \approx 56.07 + 52.41 + 865.28 = 973.76

The price is $973.76. Check against today's price, $\frac{60}{1.07} + \frac{60}{1.07^2} + \frac{60}{1.07^3} + \frac{1060}{1.07^4} \approx 966.13$ : a holder who buys at $966.13 ends the year with $973.76 + 60 = 1033.76$ , and $\frac{1033.76}{966.13} \approx 1.07$ . The bond earns exactly its 7% yield, part as coupon and part as the pull toward par.

Problem 3 · A Forward from No Free Lunch

Given: A stock trades at $64 and pays no dividends; the riskless rate is $r = 7\%$ with annual compounding. Two parties agree a one-year forward contract on the stock.

Forward price that gives the contract zero value at initiation

The stock ends the year at $60. The long side's payoff

✅ Correct!

F_{0,1} = 64 \times 1.07 = 68.48

, and the long receives

S_1 - F_{0,1} = 60 - 68.48 = -8.48

, which is exactly the short's gain.

❌ That is the spot price. Buying the stock today ties up $64 for a year; the forward price must cover that financing cost, or one side gets a free lunch.

❌ Wrong direction.

\frac{64}{1.07}

discounts the price; the forward price carries the spot price forward to the settlement date.

❌ That is only the financing cost.

64 \times 0.07 = 4.48

is the interest; the forward price is the spot price plus that interest.

❌ That is the short side's payoff. The buyer receives

S_1 - F_{0,1}

, which is negative when the stock ends below the forward price.

❌ The payoff is measured against the forward price, not today's spot. The long agreed to pay $68.48, not $64.

❌ Zero value holds only at initiation. After that the contract gains or loses value; at settlement it pays

S_1 - F_{0,1}

❌ Check the forward price. Carry the spot price forward one year at the riskless rate.

❌ Check the payoff. The long's payoff is

S_1 - F_{0,1}

Show solution

Forward price. Borrow $64 at 7%, buy the stock, and hold it for a year: you will own the stock and owe $64 \times 1.07 = 68.48$ . Selling forward at any $F_{0,1} > 68.48$ locks in a riskless profit; at any $F_{0,1} < 68.48$ the reverse trade does. No free lunch forces

F_{0,1} = S_0(1 + r) = 64 \times 1.07 = 68.48

so the forward price is $68.48, and the contract costs nothing to enter.

Payoff. At settlement the buyer pays $68.48 for a stock worth $60:

S_1 - F_{0,1} = 60 - 68.48 = -8.48

The long loses $8.48 and the short gains $8.48: one side's gain is the other side's loss.

Problem 4 · A Binomial Put

Given: A stock trades at $30. Next period it moves up by a factor $u = 1.3$ or down by $d = 0.85$ , and the gross riskless return is $r = 1.04$ . Price a one-period European put with strike $30.

Weight on the down-state payoff, $\frac{u-r}{u-d}$

Put price today

✅ Correct!

P_0 = \frac{1}{1.04}\left(\frac{19}{45}\cdot 0 + \frac{26}{45}\cdot 4.5\right) = \frac{2.6}{1.04} = 2.5

, so $2.50.

❌ That is the up-state weight.

\frac{r-d}{u-d} = \frac{0.19}{0.45}

multiplies

C_u

; the down-state payoff takes

\frac{u-r}{u-d} = \frac{0.26}{0.45}

❌ The weights are not probabilities you choose. They come from copying the payoff with stock and borrowing, so they depend only on

u

d

and

r

❌ That is the down factor itself. The weight is

\frac{u-r}{u-d} = \frac{1.3 - 1.04}{1.3 - 0.85}

❌ Missing the discount.

\frac{26}{45} \times 4.5 = 2.6

is a next-period amount; divide by the gross return

r = 1.04

❌ That is the down-state payoff. The put pays

30 - 25.5 = 4.5

only if the stock falls; weight it and discount it.

❌ The weights are swapped.

\frac{19}{45} \times 4.5 \div 1.04 \approx 1.83

puts the up-state weight on the down-state payoff.

❌ Check the weight. Substitute

u = 1.3

d = 0.85

r = 1.04

into

\frac{u-r}{u-d}

❌ Check the price. Weight each state's payoff, add, and divide by

r

Show solution

Tree. The stock ends at $uS_0 = 1.3 \times 30 = 39$ or $dS_0 = 0.85 \times 30 = 25.5$ . The put pays $P_u = \text{Max}[0,\ 30 - 39] = 0$ and $P_d = \text{Max}[0,\ 30 - 25.5] = 4.5$ .

Weights. $\frac{r-d}{u-d} = \frac{1.04 - 0.85}{1.3 - 0.85} = \frac{0.19}{0.45} = \frac{19}{45}$ and $\frac{u-r}{u-d} = \frac{1.3 - 1.04}{0.45} = \frac{0.26}{0.45} = \frac{26}{45} \approx 0.578$ .

Price.

P_0 = \frac{1}{1.04}\left(\frac{19}{45}\cdot 0 + \frac{26}{45}\cdot 4.5\right) = \frac{2.6}{1.04} = 2.5

The put is worth $2.50. The same answer comes from copying it: $\Delta = \frac{P_u - P_d}{uS_0 - dS_0} = \frac{0 - 4.5}{13.5} = -\frac{1}{3}$ (short one third of a share) and $B = \frac{P_u - \Delta\, uS_0}{r} = \frac{0 + 13}{1.04} = 12.5$ lent riskless, so $P_0 = \Delta S_0 + B = -10 + 12.5 = 2.5$ . Any other price is an arbitrage.

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