MBA Finance Part-Time Present Value

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1 MBA Finance Part-Time Present Value Professor Hugues Pirotte Spéder Solvay Business School Université Libre de Bruxelles Fall

2 Present Value Objectives for this session : 1. Introduce present value calculation in a simple 1-period setting 2. Extend present value calculation to several periods 3. Analyse the impact of the compounding periods 4. Introduce shortcut formulas for PV calculations 2 Present value is the workhorse for measuring value creation. It captures two basic financial principles: A euro today is worth more than a euro tomorrow A safe euro is worth more than a risky one The first principle is based on the existence of financial markets. Borrowing and lending are used to move cash flows over time. Excess cash can be invested and earns interest: one euro invested today will be worth more than one euro in the future. Stated differently, the On this other hand, cash deficit can be met with borrowing. When borrowing, the future repayment will be greater than the amount borrowed as it will include both the principal and the interest. The second principle is based on the idea that individual are risk averse: they prefer a safe cash flow to a uncertain one with the same expected value. Stated differently, risk averse individual would pay more for the safe cash flow. 2

3 Present Value: teaching strategy 1-period: FV, PV, 1-year discount factor DF 1 NPV, IRR Several periods: start from Strips Zero-coupons & disc. Factors General PV formula From prices to interest rates: spot rates Shortcut formulas: perpetuities and annuities Compounding interval 3 In a one-period framework, both teaching strategies are equivalent. Starting from prices presents both advantages (+) and drawbacks (-): + prices are what we are looking for; starting from zc is natural + multiple discount rates are easy to introduce (term structure) + starting from price is more natural - 3

4 Interest rates and present value: 1 period Suppose that the 1-year interest rate r 1 = 5% 1 at time 0? 1.05 at time 1 1/1.05 = at time 0? 1 at time 1 1-year discount factor: DF 1 = 1 / (1+r 1 ) Suppose that the 1-year discount factor DF 1 = at time 0? 1 at time 1 1 at time 0? 1/0.95 = at time 1 The 1-year interest rate r 1 = 5.26% Future value of C 0 : FV 1 (C 0 ) = C 0 (1+r 1 ) = C 0 / DF 1 Present value of C 1 : PV(C 1 ) = C 1 / (1+r 1 ) = C 1 DF 1 Data: r 1? DF 1 = 1/(1+r 1 ) or Data: DF 1? r 1 = 1/DF We now introduce the 1-year interest rate. Starting from the 1-year discount factor (the market value of a 1-year zerocoupon with face value of 1), it is straightforward to calculate the 1-year interest rate. You pay 0.95 for a 1-year bond. You receive 1 one year later. The interest you earn over one year is 0.5. This correspond to an interest rate of 0.5/0.95 = 5.26% We are now in a position to calculate the present value of a cash flow one year from now. If we know the 1-year discount factor, we proceed as before by multiplying the future cash flow by the 1-year discount factor. If we know the 1-year interest rate, we get the PV as: PV = C 1 / (1+r 1 ) Note the relationship between present value and future value: PV = FV 1 * DF 1 4

5 Let s define the goal Time value of money: using PV Consider simple investment project: Interest rate r = 5%, DF 1 = We begin with the simplest possible setting: analyzing a 1-period project under certainty. The key point to realize is that, because of the interest rate, adding cash flows is not sufficient to reach a decision. To see this, imagine that the current 1-year interest rate is 10% and that you analyze a project with the following cash flows: C 0 = -100 C 1 = +105 The future cash inflow is greater than the initial cash outflow. But this is not a sufficient condition for this project to be profitable. With an interest rate of 10%, investing 100 in financial market would give a future cash inflow of 110. Comparing the cash flows associated with these two alternative strategies shows that you are better off lending at 10% rather than investing your money in the project. 5

6 Thus, we can compute the «Net Future value» NFV = = 20 = + C 1 - I (1+r) Decision rule: invest if NFV>0 Justification: takes into account the cost of capital cost of financing opportunity cost In this simple setting (1-period, certainty), several possible interpretations of the interest rate are possible. Cost of capital: if you don t have the require amount for the investment, you would have to borrow it. The repayment one year later would be 110 = I(1+r). The net future value tells you how much you will have in cash one year later if you go ahead with the project and borrow to fund the initial investment. Opportunity cost of capital: suppose that you have in cash the required amount for the project. You don t need to borrow but you still need to take interest rate in account because, if the projected is accepted, your money will not be available to earn interest. Committing the cash to the project has an opportunity cost: the interest foregone because cash in invested in the project rather than being lent. 6

7 or inversely the «Net Present Value» NPV = /1.05 = + 19 = - I + C 1 /(1+r) = - I + C 1 DF 1 = = +19 DF 1 = 1-year discount factor a market price C 1 DF 1 =PV(C 1 ) Decision rule: invest if NPV>0 NPV>0 NFV> Application: Valuing a 1-year bond Suppose that the current 1-year rate r = 5% How much would you be ready to pay for a bond with following characteristics? Solution: Maturity 1 year Face value 100 Coupon 8% The price to pay is the present value of the future cash flow P 0 = 108/1.05 = = Check: the expected return on this bond should be equal to the 1-year interest rate (5%) Expected return = [Coupon + (P 1 -P 0 )]/P 0 = [8+( )]/ =5% 7

8 Economic foundations of net present value Euros next year I. Fisher 1907, J. Hirshleifer Perfect capital markets Separate investment decisions from consumption decisions Slope = - (1 + r) = - (1 + 5%) -50 NPV Euros now 8 8

9 Entreprise Value Maximisation Euros next year Investment opportunities Investment NPV 0 Market value of company Euros today 9 9

10 Internal Rate of Return Alternative rule: compare the internal rate of return for the project to the opportunity cost of capital Definition of the Internal Rate of Return IRR : (1-period) IRR = Profit/Investment = (C 1 - I)/I In our example: IRR = ( )/100 = 25% The Rate of Return Rule: Invest if IRR > r In this simple setting, the NPV rule and the Rate of Return Rule lead to the same decision: NPV = -I+C 1 /(1+r) >0 C 1 >I(1+r) (C 1 -I)/I>r IRR>r 10 The definition of the IRR in this slide is only valid for 1-period projects. A more general definition is suggested by noting that, if discounted at the IRR, the NPV is always equal to 0. NPV=-I+C 1 /(1+IRR)= -I + C 1 /[1+(C 1 -I)/I]=0 The leads to a general definition. The internal rate of return is defined as the rate of discount which makes NPV=0 NPV IRR Discount rate 10

11 IRR: a general definition The Internal Rate of Return is the discount rate such that the NPV is equal to zero I + C 1 /(1+IRR) 0 In our example: /(1+IRR)=0 IRR=25% Net Present Value IRR 0.0% 2.5% 5.0% 7.5% 10.0% 12.5% 15.0% 17.5% 20.0% 22.5% 25.0% 27.5% 30.0% Discount Rate 11 Application: Yield to maturity on a 1-year bond The yield to maturity on a bond is the discount rate such that the current market value is equal to the present value of future cash flows. For a 1-year bond, the bond s yield y is obtained by solving the following equation P 0 = [Coupon + Face Value]/(1+y) The can be written as: -P 0 + [Coupon + Face Value]/(1+y) = 0 Comparing this later expression to the definition of the IRR show that the yield to maturity is the IRR for an investment the bond at the current market price. Example: Suppose that a 1-year 8% coupon bond with face value 100 sells for The yield to maturity on this bond is: y = (8+100)/ =6.40% 11

12 Present Value Calculation with Uncertainty Consider the following project: C 0 = -I = -100 Cash flow year 1: +50 with probability ½ +200 with probability ½ The expected future cash flow is C 1 = 0.5 * * 200 = 125 The discount rate to use is the expected return of a stock with similar risk r = Risk-free rate + Risk premium = 5% + 6% (this is an example) NPV = / (1.11) = Calculating present value when future cash flows are uncertain is achieved using the same formula as before with two main differences. The first one is that, as the future cash flow is uncertain, the amount to discount is the expected cash flow. The expected cash flow is obtained as the average of the possible realizations of the future weighted by their respective probabilities. The second difference is that the discount rate to use is no longer equal to the 1-year risk-free interest rate. If the future cash is uncertain, risk averse investors will require a higher expected return than when the cash flow is certain. This higher expected return of obtained by adding a risk premium to the risk-free interest rate to obtain the discount rate. The exact determination of this risk premium is a difficult topic that we will analyze in a later session. 12

13 A simple investment problem Consider the following project: Cash flow t = 0 C 0 = - I = Cash flow t = 5 C 5 = (risk-free) How to calculate the economic profit? Compare initial investment with the market value of the future cash flow. Market value of C 5 = Present value of C 5 = C 5 * Present value of \$1 in year 5 = C 5 * 5-year discount factor = C 5 * DF 5 Profit = Net Present Value = - I + C 5 * DF 5 13 When doing financial calculations, we have to juggle with market values and interest rates. The traditional presentation of present value starts with one single interest rate. It then proceeds to explain how to calculate prices (or present values) based on this interest rate. This is what Ross, Westerfield and Jaffee do in their Chapter 4. Brealey and Myers (Chap 3) recognize first that interest rate vary depending on the maturity of cash flow. But they then finesse the issue by assuming that the interest rate is the same regardless of the date of the cash flow. The analysis of the term structure (the relationship between interest rates and maturities) is deferred to an appendix (Appendix 5A in RWJ) or to a later chapter (Chap 23 in BM). In this presentation, we take a slightly different approach. We start from market prices of risk-free zero coupons. We show how these market prices can be used to calculate the net present value of a project involving certain cash flows. We then examine how interest rates can be determined based on these market prices. Of course, at the end, we will be able to take whatever starting point is more appropriate. But starting from prices will, hopefully, make clear that there is no such thing at The interest rate. Present value calculation taking into account the current shape of the term structure is not more difficult than assuming a flat term structure. It has the huge advantage to fit with current prices observed on the market. 13

14 Using prices of U.S. Treasury STRIPS Separate Trading of Registered Interest and Principal of Securities Prices of zero-coupons Example: Suppose you observe the following prices Maturity Price for \$100 face value The market price of \$1 in 5 years is DF 5 = 0.80 NPV = * 0.80 = = A zero-coupon is a pure discount bond. You pay a price today and you receive one single payment (the face value of the zero-coupon) at one future date (the maturity of the zero-coupon). In the early 1980s, several U.S. investment banks started stripping U.S. Treasury issues. They would buy a bond (consisting of a number of coupons and the principal) and sell each element separately as a zero-coupon using feline names as CATS, TIGRS, COUGARS and LIONS 1. The idea was taken over by the U.S. Treasury in 1985 who started offering its own zerocoupon instruments under the denomination STRIPS (nothing to do with strip tease, at least as far as I know..). Prices of U.S. Treasury strips are regularly published in the WSJ. Note that prices are quoted with colons that represent 32nds. For instance, 93:16 means /32 = Strips were later on introduced in various European markets. In the UK, Gilt strips exist (a gilt is a long term British government bond) and quotations can be found in the FT. France introduced stripped OATs (OAT means Obligation Assimilée du Trésor). In Belgium, strips on Belgian government bonds are known as obligations démambrées. 1 CATS: Certificates of Accrual on Treasury Securities (Salomon Brothers) TIGRS: Treasury Investment Growth Receipts (Merrill Lynch) COUGARS: Certificates on Government Receipts (A. G. Beckers Paribas) LIONS: Lehman Investment Opportunity Notes (Lehman Brothers) 14

15 Present Value: general formula Cash flows: C 1, C 2, C 3,,C t, C T Discount factors: DF 1, DF 2,,DF t,, DF T Present value: PV = C 1 DF 1 + C 2 DF C T DF T An example: Year Cash flow Discount factor Present value NPV = = We now have a general formula to compute present values. The formula is very simple. You simply multiply each cash flow by the corresponding discount factor and you take the sum. Keep in mind that the discount factors are market prices for zero-coupon. An analogy is helpful to understand the logic. Suppose that you are offered the following deal. You pay 100 EUR and you receive 30 USD (US dollar), 40 GBP (British pound) and 60 CHF (Swiss franc). Is this a good deal? Before accepting, you will convert all cash flows into one unit: the euro using current spot exchange rates. If the spot rates are: 1 USD = 1 EUR 1 GBP = 1.50 EUR 1 CHF = 0.70 EUR the net present value is NPV = * * * 0.70 This is a good deal. = = +32 Present value calculation is similar. Instead of cash flows being denominated in different currencies, they take place at different points in time. The NPV is obtained by valuing each cash flow at the appropriate exchange rate. 15

16 Several periods: future value and compounding Invests for 1,000 two years (r = 8%) with annual compounding After one year FV 1 = C 0 (1+r) = 1,080 After two years FV 2 = FV 1 (1+r) = C 0 (1+r) (1+r) = C 0 (1+r)² = 1, Decomposition of FV2 C 0 Principal amount 1,000 C 0 2 r Simple interest 160 C 0 r² Interest on interest 6.40 Investing for t years FV t = C 0 (1+r) t Example: Invest 1,000 for 10 years with annual compounding FV 10 = 1,000 (1.08) 10 = 2, Principal amount 1,000 Simple interest 800 Interest on interest Two important points to note: 1. Note that reinvestment takes place at the same rate r 2. Reinvestment is done once a year (annual compounding) The Power of Compounding The power of compounding increases with the investment horizon. interest on interest represents an increasing fraction of the future value as the investment horizon become longer. As an example, consider an investment of 1 at 8% for 50 years. The future value is FV 50 = (1.08) 50 = The decomposition of this future value is: Principal amount 1 Simple interest 4 Interest on interest Interest on interest is 89% of future value! For those of you having a mathematical background, note that the future value is a convex function of time to maturity: f(t) = C 0 (1+r) t f (t) = t C 0 (1+r) t-1 > 0 f (t) = t(t-1) C 0 (1+r) t-2 > 0 16

17 Present value and discounting How much would an investor pay today to receive C t in t years given market interest rate r t? We know that 1 0 => (1+r t ) t t Hence PV (1+r t ) t = C t => PV = C t /(1+r t ) t = C t DF t The process of calculating the present value of future cash flows is called discounting. The present value of a future cash flow is obtained by multiplying this cash flow by a discount factor (or present value factor) DF t The general formula for the t-year discount factor is: DF t 1 = (1 + r ) t t 17 This is the standard formula to calculate a discount factor. Note that the discount rate used r t is, in general, a function of the maturity. 17

18 Discount factors Interest rate per year # years 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

19 Spot interest rates Back to STRIPS. Suppose that the price of a 5-year zero-coupon with face value equal to 100 is 75. What is the underlying interest rate? The yield-to-maturity on a zero-coupon is the discount rate such that the market value is equal to the present value of future cash flows. We know that 75 = 100 * DF 5 and DF 5 = 1/(1+r 5 ) 5 The YTM r 5 is the solution of: = 5 (1 + r5) The solution is: r 5 5 = 1 = 5.92% 75 This is the 5-year spot interest rate 19 A spot transaction is a transaction where both delivery and payment take place immediately. In a forward transaction, the terms of the transaction are set now but the transaction itself is executed at some future date. The spot interest rate is the rate on an loan starting immediately. For example, you borrow 75 and you will repay 100 five year later. The spot interest rate is simply the annual interest rate charged on this transaction. The final payment (100) is the future value of the amount borrowed: 100 = 75 (1+r 5 ) 5? r 5 = 5.92% There are no intermediate interest payment. The previous formula can also be written as: 75 = 100 / (1+r 5 ) 5 If 75 is the current price of a 5-year zero-coupon with face value equal to 100, then this second presentation shows that the spot interest rate is the yield-to-maturity of the zero-coupon. 19

20 Term structure of interest rate Relationship between spot interest rate and maturity. Example: Maturity Price for 100 face value YTM (Spot rate) r 1 = 2.00% r 2 = 2.79% r 3 = 3.41% r 4 = 3.70% r 5 = 4.56% Term structure is: Upward sloping if r t > r t-1 for all t Flat if r t = r t-1 for all t Downward sloping (or inverted) if r t < r t-1 for all t 20 Starting from the prices of zero-coupons with different maturities, we calculate the yield-to-maturity of each of them to obtain the various spot interest rates. In general, spot rates vary with maturity. The term structure of interest rates is the relationship between spot interest rates (yields of zero-coupons) and maturity. The term structure of interest rates for the European Monetary Union is available of the web site of the European Commission: Click on Euro yield curve. 20

21 Euro Yield curve Sept 25,

22 Using one single discount rate When analyzing risk-free cash flows, it is important to capture the current term structure of interest rates: discount rates should vary with maturity. When dealing with risky cash flows, the term structure is often ignored. Present value are calculated using a single discount rate r, the same for all maturities. Remember: this discount rate represents the expected return. = Risk-free interest rate + Risk premium This simplifying assumption leads to a few useful formulas for: Perpetuities (constant or growing at a constant rate) Annuities (constant or growing at a constant rate) 22 22

23 Constant perpetuity C t =C for t =1, 2, 3,... C PV = r Proof: PV = C d + C d² + C d3 + PV(1+r) = C + C d + C d² + PV(1+r) PV = C PV = C/r Examples: Preferred stock (Stock paying a fixed dividend) Suppose r =10% Yearly dividend = 50 Market value P0? Note: expected price next year = Expected return = div P = = + ( P P 0 P 500 ) P = = = ( = ) 10 % 23 This is a simple and powerful formula. Always keep in mind that this formula gives you the present value of a perpetuity one year before the first cash flow. In our example, the first cash flow is receive at time t=1 and the formula gives the price at time 0. You can combine the formula with the discount factor to compute the present value of a deferred perpetuity, a perpetuity starting at some future date. As an example, suppose that you start a pension fund. You expect to pay 10m (millions) each year to your employees when they retire. Your first payment in due in 31 years (your employees are pretty young). What is the present value of your pension liability if the discount rate is 8%? First calculate the future value of the pension payments one year before the first payment (this is in year t = 30). This value is 10m / 0.08 = 125m Now discount it back to today by multiplying the future value by the 30- year discount factor DF 30 = 1/(1.08) 30 = The present value of your pension liability is: PV = 125m * = 12.42m 23

24 Growing perpetuity C t =C 1 (1+g) t-1 for t=1, 2, 3,... r>g Example: Stock valuation based on: Next dividend div1, long term growth of dividend g If r = 10%, div 1 = 50, g = 5% Expected return = PV C r g = 1 50 P = Note: expected price next year = 0 = 1, P1 = = 1, div + ( P1 P0 ) 50 + (1,050 1,000) = P 1,000 1 = 0 10% 24 This formula is widely used for valuing stocks. In that case, g is the growth rate of the dividend. Note that the growth rate should be less than the discount rate. This is because the growth rate is constant up to infinity. The condition is required for the geometric series to converge. For those of you willing to have a proof, here is a short one: PV = C 1 * d + C 1 (1+g) * d² + C 1 (1+g)² * d³ + where d = 1/(1+r) Multiply both side by (1+g). PV (1+g) = C 1 (1+g) * d + C 1 (1+g)² * d² + C 1 (1+g)³ * d³ + The right hand side is the value of a perpetuity with a discount rate x =(1+r)/(1+g) 1 = (r g)/(1+g) So: PV(1+g) = C 1 / x = C 1 (1+g) / (r-g) 24

25 Constant annuity A level stream of cash flows for a fixed numbers of periods C 1 = C 2 = = C T = C Examples: Equal-payment house mortgage Installment credit agreements PV = C * DF 1 + C * DF C * DF T + = C * [DF 1 + DF DF T ] = C * Annuity Factor Annuity Factor = present value of 1 paid at the end of each T periods

26 C t = C for t = 1, 2,,T Constant Annuity C PV = r 1 1 (1 + r) [ T Difference between two annuities: Starting at t = 1 PV=C/r Starting at t = T+1 PV = C/r [1/(1+r) T ] Example: 20-year mortgage Annual payment = 25,000 Borrowing rate = 10% PV =( 25,000/0.10)[1-1/(1.10) 20 ] = 25,000 * 10 *( ) = 25,000 * = 212,839 ] 26 The present value of a constant annuity can be expressed as the annual cash flow multiplied by an annuity factor. PV = C * Annuity Factor The T-year annuity factor is the sum of the first T discount factors: Annuity Factor = DF 1 + DF DF T This formulation is useful if you need to calculate an annuity factor when the discount rate is not a constant. When the discount rate is a constant, the annuity factor can be expressed as the difference between two perpetuity: one perpetuity beginning in year 1 minus a second deferred perpetuity beginning in year T+1. The value of the first perpetuity is C/r (250,000 in our example). The value of the second perpetuity is C/r in year T. It s value today is equal to C/r * DF T (in our example: 250,000 * ) As a consequence, the annuity factor can be expressed as: Annuity factor = (1/r)*(1 DF T ) In our example: Annuity factor = (1 / 0.10) * ( ) =

27 Annuity Factors Interest rate per year # years 1% 2% 3% 4% 5% 6% 7% 8% 9% 10%

28 Growing annuity C t = C 1 (1+g) t-1 for t = 1, 2,, T r? g PV = C r g 1+ g r T This is again the difference between two growing annuities: Starting at t = 1, first cash flow = C 1 Starting at t = T+1 with first cash flow = C 1 (1+g) T Example: What is the NPV of the following project if r = 10%? Initial investment = 100, C 1 = 20, g = 8%, T = 10 NPV= [20/(10% - 8%)]*[1 (1.08/1.10) 10 ] = = This derivation of the formula for a growing annuity is similar to the derivation of the formula for the constant annuity. Start from a growing perpetuity the present value of which is C 1 /(r g) Subtract from this growing perpetuity the present value of a second growing perpetuity starting in T+1. The annual cash flow at that time is C 1 (1+g) T. The present value at time 0 of this second growing perpetuity is [C 1 (1+g) T /(r g)] / (1+r) T. Note that this formula works for any g? r. In particular, it will work for g>r. In this case, r-g is negative but simultaneously the ratio (1+g)/(1+r) is greater than 1 so that the difference between brackets is also negative. 28

29 Compounding interval Up to now, interest paid annually If n payments per year, compounded value after 1 year : r 1+ n Example: Monthly payment : r = 12%, n = 12 Compounded value after 1 year : ( /12)12= Effective Annual Interest Rate: 12.68% Continuous compounding: [1+(r/n)] n? e r (e= ) Example : r = 12% e 12 = Effective Annual Interest Rate : 12.75% n 29 Here we introduce one additional complication: the compounding interval. Up to now, we have assumed that interest are paid and reinvested once a year. But this not always the case. Suppose that you borrow for 3 months. The stated annual interest rate (or the annual percentage rate) is the interest quoted by the bank. Suppose it is 6%. This means that the interest you will pay after 3 months is 6% / 4 = 1.5% Now imagine that you want to borrow for 1 year. The bank tells you that it will charge an annual percentage rate of 6% but that interest are payable quarterly. This means that you will have to borrow at the end of each quarter the principal and the interest. The future value after 1-year will be: FV 1 = (1+6%/4) 4 = The 1-year borrowing rate is effectively 6.136%. This is the effective annual interest rate. 29

30 Juggling with compounding intervals The effective annual interest rate is 10% Consider a perpetuity with annual cash flow C = 12 If this cash flow is paid once a year: PV = 12 / 0.10 = 120 Suppose know that the cash flow is paid once a month (the monthly cash flow is 12/12 = 1 each month). What is the present value? Solution 1: 1. Calculate the monthly interest rate (keeping EAR constant) (1+r monthly ) 12 = 1.10? r monthly = % 2. Use perpetuity formula: PV = 1 / = Solution 2: 1. Calculate stated annual interest rate = % * 12 = 9.568% 2. Use perpetuity formula: PV = 12 / = Here I want to show how to take into account the timing of cash flows when the time interval between cash flows is less than a year. The most obvious solution consists in changing the discount factor to take into account the spacing of cash flows. But this should be done carefully while keeping the values of annual cash flows unchanged. In our example, we want to value a perpetuity of 1 per month. We calculate an interest per month while keeping the effective annual interest rate unchanged. We then use the formula for valuing a perpetuity. We get: PV = (C / 12) /(r monthly ) But this is the same as: PV = C / (12 * r monthly ) The product 12 * r montly is the stated annual interest rate with monthly compounding with the same effective annual interest rate. Note that the stated annual rate with monthly compounding is lower than the effective annual rate. 30

31 Interest rates and inflation: real interest rate Nominal interest rate = 10% Date 0 Date 1 Individual invests \$ 1,000 Individual receives \$ 1,100 Hamburger sells for \$1 \$1.06 Inflation rate = 6% Purchasing power (# hamburgers) H1,000 H1,038 Real interest rate = 3.8% (1+Nominal interest rate)=(1+real interest rate) (1+Inflation rate) Approximation: Real interest rate Nominal interest rate - Inflation rate 31 31

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