# Perpetuities and Annuities EC Borja Larrain

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1 Perpetuities and Annuities EC 1745 Borja Larrain

2 Today: 1. Perpetuities. 2. Annuities. 3. More examples. Readings: Chapter 3 Welch (DidyoureadChapters1and2?Don twait.) Assignment 1 due next week (09/29).

3 Questions/Goal of this class: NPVandexcelareapain: Dowealwayshaveto compute these long summations for projects with many periods? Some people seem to have a calculator in their heads: How do they compute the NPV of a project so fast? Examples: If a firm produces \$5 million/year forever, and the interest rate is a constant 5% forever, what is the value of the firm? (A: \$100 million) In this class we ll see some shortcuts that can make these calculations very easy. These formulas also illustrate the importance of assumptions about interest rates for evaluating projects and assets. Yes, 1% can be the difference between being a millionaire and bankruptcy.

4 1 Perpetuities A perpetuity is a financial instrument that pays C dollars per period, forever. If the interest rate is constant and the first payment from the perpetuity arrives in period 1, then the Present Value of the perpetuity is: PV = X t=1 C (1 + r) t = C (1 + r) + C (1 + r) = C r IMPORTANT: the first cash flow is tomorrow (t =1), not today (t =0).

5 If a firm produces \$5 million/year forever, beginning next year, and the interest rate is a constant 5% forever, what is the value of the firm? X t=1 5 (1 + 5%) t = =100 What is the value of a perpetuity if the first cash flow is today rather than tomorrow? X t=0 C (1 + r) t = C + X t=1 C (1 + r) t = C + C r So, if the firm starts producing 5 million right away its value is \$105.

6 What if they offer you today a perpetuity with a 5% rate, but that starts paying \$5 in two more years (t =2)? In one more year the perpetuity will be worth 5/0.05 = 100. And having 100 in one more year is worth today: r = =95.23

7 1.1 Growing Perpetuities A growing perpetuity pays C, thenc (1 + g), then C (1 + g) 2, then... For example, if C =\$100and g =0.10 = 10%, then you will receive the following payments: C 0 = \$0 C 1 = \$100 C 2 = \$100 (1 + 10%) = \$110 C 3 = \$100 (1 + 10%) 2 =\$ IMPORTANT: It s easy to get confused with the timing. In general: C t = C 1 (1 + g) t 1 The relevant discount rate for the cash flow at t is r 0,t.

8 Definition The Present Value of a growing perpetuity is: X t=1 C t (1 + r) t = X t=1 C (1 + g) t 1 (1 + r) t = C r g Memorize this formula! It s very helpful. Thegrowthincashflowsactsasareductioninthe interest rate. Or, to put it in a slightly more convoluted way, a reduction in the interest rate acts as an increase in cash flows...aha!, do you see how you can "improve" the outlook on a company or project with only a slight modification of the interest rate you are assuming?

9 What is the value of a promise to receive \$10 next year, growing by 2% (just the inflation rate) forever, if the interest rate is 6% per year? = \$250 What is the value of a firm that just paid \$10 this year, growing by 2% (just the inflation rate) forever, if the interest rate is 6% per year? 10(1 + 2%) =\$255 What is the value of a firm that will only grow at the inflation rate (π), and which will have \$E million in earnings next year? P = E r π In 10 years a firm will have cash flows of \$100 million. Thereafter, its cash flow will grow at the inflation

10 rate of 3%. if the interest rate is 8%, estimate the value of the firm if you sell the firm in the 10th year =\$2.06 billion This selling (or residual) value assumption is typical in many applications as a way to simplify NPV computations. The firm is assumed to become a perpetuity at some point in the future (say 10 or 20 years) and the first years are studied in more detail.

11 1.2 The Gordon Dividend Growth Model What is the share price of a firm that will pay dividends of \$1 next year, with dividends that grow at 4% annually forever, and with a cost of capital of 12%? P = D r g = =12.5 There are many "ifs" in applying the perpetuity formula to stocks: dividends forever and always growing? Still, these assumptions are not that bad as a first approximation: dividends are quite stable in time once a firm starts paying them (more on this at the end of this course).

12 Another way to use this formula: what is the cost of capital of a firm with a dividend yield of 5% and with dividends that grow at 3% forever? r = D P + g =5%+3%=8% But be careful with naive applications of this formula. Example: Glassman and Hassett s "Dow 36, 000" written at the end of the 1990s (peak of the internet bubble). When they wrote the book the Dow Jones was at 9, 000 and the dividend yield at 2%. These imply D =2% 9, 000 = 180. They assumed that the long-run growth rate of the economy was 2.5% and r =3%(yield on Treasury Bills at the time). With these assumptions: P =180/(3% 2.5%) = 36, 000.

13 However, the Dow Jones today is at ±11, 000. What went wrong? Hint: is the cost of capital they assumed a reasonable number for stocks? Probably not, the number is too low. Why? More on this later on the course. If r =6%, P =180/(6% 2.5%) = 5, 142 Calculations are very sensitive to the r you assume.

14 2 Annuities An annuity is a financial instrument that pays C dollars for T years. It has the following PV formula: PV = TX t=1 C (1 + r) t = C r " 1 # 1 (1 + r) T You should be able to remember this formula. One trick: think of the annuity as a perpetuity today minus a perpetuity given at time T : C r 1 (1+r) T ³ Cr.

15 Annuities Perpetuity A pays a for ever starting next period Annuity C pays for T periods starting next period. Perpetuity B Perpetuity B pays a for ever starting T periods from now A B = C = a 1 r r (1 1 + r ) T

16 Proof: Define 1/(1 + r) =q. The value of a an annuity is: P = Cq + Cq 2 + Cq Cq T Now, apply the "Gaussian" trick and multiply everything by q: Pq = Cq 2 + Cq 3 + Cq Cq T +1 Subtract the second line from the first: P Pq = Cq Cq T +1 Therefore, P = Cq(1 qt ) 1 q Once you substitute q for what it is you ll get our formula. For the special case where T (i.e., a perpetuity) it turns out that q T 0 because q<1, and therefore P = C/r.

17 2.1 Annuity Example: Mortgage Loan A 30-year mortgage is an annuity with 360 payments, starting one month from today. Because payments are monthly we need the monthly interest rate. The monthly rate is the quoted rate divided by 12 (In other words, like bank interest, your actual annual rate is higher than quoted, nice!) The monthly rate on a 9% mortgage is: r monthly =0.09/12 = 75 bp per month If you take a \$1.2 million fixed rate mortgage with 30 years to maturity, 360 equal monthly payments, and a quoted interest rate of 9%: what is your monthly

18 mortgage payment? C 1, 200, 000 = C (1.0075) 360 " C 1 1, 200, 000 = ( ) 360 C = \$9, #

19 2.2 Annuities vs. Perpetuities What fraction of a perpetuity s value comes from the first T years? The difference between a perpetuity and an annuity (in % terms) is: PV(perpetuity) PV(annuity) PV(perpetuity) = 1 (1 + r) T This number gets smaller as r or T get bigger. For example, if r =5%and T =30, this number is 23%, but if r =10%and T =30, this number is only 6%. So, for high interest rates, the cash flows after 30 years or so are basically irrelevant.

20 Put differently, is it reasonable to use a perpetuity as an approximation for an annuity? (the formula is easier, isn t it?) Yes, if interest rates are high or if the annuity is of sufficientlylongmaturity.

21 3 More examples/applications 3.1 Rental Equivalents Think of 3 alternatives: Machine A: costs \$20k up-front and lasts 18 years. It has annual maintenance costs of \$1k per year. Machine B: costs \$25k up-front and lasts 30 years. It has annual maintenance costs of \$900 per year. Outsource production for \$2,500 per year Theinterestrateis10%peryear. Machine A costs \$2,438.60: \$20 = C 0.1 " 1 1 ( ) 18 # C =2, 438.6

22 Plus maintenance, Option A costs \$3, Option B costs \$2, plus 900 maintenance = \$3,551. So, Option C is a bargain! Careful: this answer depends on the implicit assumption of "equal repetition".

23 3.2 How to cheat on loans The advertisement of an actual automobile loan agreement claimed: "12 months car loans. Only 9%" Butthisishowthiscardealercalculatedpayments on a \$10,000 car: a \$10k loan at 9% implies you owe \$10, equal payments imply 10, 900/12 = \$ per month. Is the PV of this loan \$10,000? Actually, no. What is the implied interest rate (or IRR) on this loan? 10, 000 = r monthly = 1.35% r monthly ³ 1+rmonthly 12

24 But this monthly rates does not imply 9% annually: (1 + r monthly ) 12 1 r annual =17.5% The dealer is assuming you borrow 10k for the whole year, but you are actually making monthly payments and thereby reducing what you owe during the year. It s cheating! A true loan of 9% a year implies a monthly payment of just \$ (homework: get this number! First get the true monthly rate if the annual rate is 9%). This is how ordinary loans and mortgages work.

25 3.3 Growing annuities A growing annuity pays C (1+g) t 1 per year starting in period 1 for T periods. The Present Value of a growing annuity is: PV = TX t=1 C (1 + g) t 1 (1 + r) t = C " 1 r g # (1 + g)t (1 + r) T Don t bother to memorize this formula! Example: An insurance company offers a retirement annuity that pays \$100 per year for 15 years, growing at an "inflation-compensator" rate of 3%, and sells for \$ What is the interest rate? = 100 r 3% r = 11.76% " 1 (1 + 3%)15 (1 + r) 15 #

26 READ CHAPTERS 3 & 4, NOW! (ASSIGNMENT 1 COVERS MATERIAL UP TO Ch. 4)

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