Business Finance Michael Dimond

Annuity: Limited Number of Payments If you require a 10% annual return, what would you pay for… …$100 to be delivered in 1 year? ($90.9091) …$100 to be delivered in 2 years? ($82.6446) …$100 to be delivered in 3 years? ($75.1315) …all of the above (i.e. $100 to be paid at the end of each of the next three years)? By adding together the present values, you find the value of all the cash flows in the stream. i = 10% 1 2 3 ? 100 100 100 100 ÷ (1+0.10)1 90.9091 82.6446 + 75.1315 248.6852 100 ÷ (1+0.10)2 What if you add another payment to the end of the annuity? 100 ÷ (1+0.10)3

Annuity: Limited Number of Payments 1 2 3 4 ? 100 100 100 100 100 ÷ (1+0.10)1 90.9091 82.6446 75.1315 + ? ? 100 ÷ (1+0.10)2 What if you add another payment to the end of the annuity? 100 ÷ (1+0.10)3

Perpetuity: The annuity which doesn't end What happens to PV as n increases? If all other TVM factors are unchanged, PV gets smaller as n increases 100 ÷ (1+10%)1 = 90.9091 100 ÷ (1+10%)5 = 62.0921 100 ÷ (1+10%)20 = 14.8644 100 ÷ (1+10%)40 = 2.2095 100 ÷ (1+10%)60 = 0.3284 100 ÷ (1+10%)100 = 0.0073 What the value finally does If n gets large enough, the PV of a single CF becomes almost zero: 100 ÷ (1+10%)1000 = 0.0000000000000000000000000000000000000004 This means any single additional cash flow does not significantly increase the sum of the present values, even though all of the remaining CFs have value. With a little math, the discounting of a perpetuity simplifies to: PVperp = CF/r

Valuing a perpetuity Consider a $100 annual perpetuity (“$100 per year forever”). What if you require a 10% annual return? Rather than trying to discount an infinite number of cash flows, we use the perpetuity formula. The value of a perpetuity: PVperp = CF/r 100 ÷ 0.10 = 1000 :. You would be willing to pay $1,000 right now to receive $100 per year “forever.” What would happen if your required rate of return was higher (12%)? What would happen if your required rate of return was lower (8%)? 1 2 3 4 PV? i = 10% APR 100 The timeline for a perpetuity has an arrow at the right end to indicate there is no end to the timeline

Growing perpetuities Consider a $100 annual perpetuity which grows 8% each year. What if you require a 10% annual return? As long as the percent growth rate is constant, this formula will give the present value: PVperp = CF1/(r – g) PVperp = 100 ÷ (0.10 – 0.08) = 100 ÷ 0.02 = 5,000 Expected growth has value How much is the growth in this perpetuity worth? There is a rule: r > g Notice this formula still works for a non-growing perpetuity. When growth = 0%, PVperp = CF/(r – 0) = CF/r 1 2 3 4 PV? i = 10% APR g = 8% 108 100 116.64 125.95

Growing perpetuities The general formula for valuing any perpetuity: PVperp = CF1/(r-g) What a share of stock does: A stock which pays a dividend is a perpetuity. There is no anticipated end to the timeline, and there is an expected cash flow which behaves in a predictable fashion. For example, IBM stock has paid a dividend regularly since 1962. Looking at the quarterly amounts, the pattern is easy to see: 6-May-10 0.65 Dividend 6-Aug-10 0.65 Dividend 8-Nov-10 0.65 Dividend 8-Feb-11 0.65 Dividend 6-May-11 0.75 Dividend 8-Aug-11 0.75 Dividend 8-Nov-11 0.75 Dividend 8-Feb-12 0.75 Dividend 8-May-12 0.85 Dividend 8-Aug-12 0.85 Dividend You could probably predict the next several dividends without much doubt.

Stock: the Dividend Growth Model Stock acts like a perpetuity, so we can adapt the value of a perpetuity to value a share of stock: P0 = D1/(r-g) Notice the price (P0) is at time zero (right now) and the expected dividend (D1) is the cash flow which determines the current price. In many cases, the most recent dividend is given instead of the expected dividend. If this happens, you need to determine the expected dividend: D1 = D0 x (1+g)

Dividend Growth Model examples You require a 12% return on investment. If XYZ Company stock just paid a $1.00 dividend and dividends are expected to grow 4% per year forever, how much would you pay for a share of this stock? D0 = 1.00 :. D1 = 1.00 x (1 + 4%) = 1.04 1.04 ÷ (0.12 – 0.04) = 1.04 ÷ 0.08 = 13.00 :. You would be willing to pay $13.00 per share for XYZ Company If IBM stock has an expected annual dividend of $3.79 (four quarters of dividends), a growth rate of 14.9% and you require 16.8% return, what price would you pay for IBM stock? 3.79 ÷ (0.168 – 0.149) = 3.79 ÷ 0.0190 = 199.4737 :. You would be willing to pay $199.47 per share for IBM

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Dividend Growth Model: Determining % Return Equity investors have a required rate of return. We can derive this from the dividend discount model: P0 = D1/(r-g) :. (r-g) = D1/P0 :. r-g = (D1/P0) :. r = (D1/P0) + g How do you test the answer you get? What if you wanted to solve for the expected growth rate?

About Preferred Stock Preferred stock is an ownership stake (equity) which comes with a contracted payout. The dividend is frequently a percentage of the par value of the stock. For example, 5% preferred stock with a $10.00 par value would have an annual dividend of $0.50. Because it is a percent of the par value, the dividend does not grow. The dividend is a perpetuity, so we use the perpetuity formula to value preferred stock. XYZ Company has preferred stock with a $3.00 dividend and investors require a 9% return for this preferred equity. What is the market price? D0 = 3.00 :. D1 = 3.00 3.00 ÷ 0.09 = 33.3333 :. The market price is $33.33 per share for XYZ Company Preferred Stock.

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