Formula Sheet COMM1180 2021T3 - Week 5 PDF

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UNSW Business School COMM1180 Value Creation Term 3 2021

Week 5

Equity Valuation Formula Sheet General Valuation Principle (review)

Today’s price of any asset 𝑃0 is the sum of all discounted future cash flows 𝐶𝐹𝑡 : 𝑛

𝑛

𝑡=1

𝑡=1

𝑃0 = ∑ 𝑃𝑉(𝐶𝐹𝑡 ) = ∑

𝐶𝐹𝑡 (1 + 𝑟 )𝑡

Simple bond pricing formula

The fair value of a bond with exactly n 6-month periods remaining to maturity, face value 𝐹,

and a coupon rate c (i.e., a coupon amount of 𝐶 = 𝑐/2 × 𝐹) and a per period discount rate of 𝑟 is 1 − (1 + 𝑟)−𝑛 𝐹 𝑃0 = 𝐶 + ( 1 + 𝑟 )𝑛 𝑟

Growing annuities Assume that payments grow at constant rate g from one period to the next (typically 𝑔 < 𝑟), e.g., 𝐶2 = (1 + 𝑔)𝐶1 ⋯ 𝐶𝑛 = (1 + 𝑔)𝑛−1 𝐶1 . Ordinary annuity

If 𝐶 = 𝐶1 is the first payment at the end of the first period, then: 1+𝑔 𝑛 1−[ 1 + 𝑟] 𝑃𝑉 = 𝐶1 𝑟−𝑔 (1 + 𝑟 )𝑛 − (1 + 𝑔)𝑛 𝐹𝑉 = 𝐶1 𝑟−𝑔

Annuity due

If 𝐶 = 𝐶0 is the first payment at the beginning of the first period, then: 1+𝑔 𝑛 1−[ 1 + 𝑟 ] (1 + 𝑟) 𝑃𝑉 = 𝐶0 𝑟−𝑔 (1 + 𝑟)𝑛 − (1 + 𝑔)𝑛 𝐹𝑉 = 𝐶0 (1 + 𝑟) 𝑟−𝑔 Ordinary perpetuity If 𝐶 = 𝐶1 is the first payment at the end of the first period, then: 𝑃𝑉0 =

𝐶1 𝑟−𝑔

Perpetuity due If 𝐶 = 𝐶0 is the first payment at the beginning of the first period, then: 𝑃𝑉0 =

𝐶0 (1 + 𝑟) 𝑟−𝑔

Return decomposition

Given current price 𝑃0 , future price 𝑃1 , end of period dividend 𝐷1 , the expected return on equity 𝑟𝑒 can be decomposed into 2 parts, the forward dividend yield and expected capital gains: 𝑟𝑒 =

𝐷1 𝑃1 − 𝑃0 + 𝑃0 𝑃0

An alternative representation where g is the earnings (dividend) growth rate is: 𝑟𝑒 =

𝐷1

𝑃0

+𝑔

Re-invest to grow The growth rate of earnings in year t 𝑔𝑡 = Retention Rate%,𝑡−1 × Return on new investments%,𝑡 = 𝑅𝑅𝑡−1 × 𝑅𝑂𝐼𝑡

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Total Payout Model

Given total cash distributions of 𝑇𝑃1 at time 1, cost of equity 𝑟𝑒 , growth rate of total earnings 𝑔𝑇𝐸 and current number of shares outstanding #0 , total equity value is 𝑀𝑉0 (𝐸𝑞𝑢𝑖𝑡𝑦) = And today’s share price is

𝑇𝑃1 𝑟𝑒 − 𝑔𝑇𝐸

𝑇𝑃1 𝑀𝑉0 𝑟𝑒 − 𝑔𝑇𝐸 𝑃0 = = #0 #0

Customer Lifetime Value and Customer Equity

Assume a per period discount rate 𝑑 . The (expected) lifetime value of a customer that is acquired at time k at an acquisition cost CoA𝑘 , with a retention rate of 𝑅𝑘 per period is ∞

𝐶𝐿𝑉𝑘 = ∑(Revenue𝑘,𝑡 − Cost 𝑘,𝑡 ) [ 𝑡=𝑘

𝑅𝑘 𝑡−𝑘 ] − CoA𝑘 1+𝑑

Customer equity is the sum of all (expected) lifetime values 𝐶𝐿𝑉𝑘 of all customers, present and future: ∞

𝐶𝐸 = ∑ 𝑁𝑘 𝑘=0

𝐶𝐿𝑉𝑘 (1 + 𝑑)𝑘

Where 𝑁𝑘 is the number of customers in cohort 𝑘, i.e. who were acquired at time 𝑘.

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