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Time Value:Compounding and Discounting Dr. Julio Huato SFC - jhuato.sfc@gmail.com Fall 2012
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Questions• Why is time value important in macro and ﬁnance?• What are the present value (PV) and future value (FV) of an asset?• What are the PV and FV of a simple asset (one-shot cash ﬂow in the future)?• What is the PV of a perpetuity?• What are the PV and FV of an ordinary annuity and of a mixed-stream asset? 1
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• How much do we need to deposit periodically in order to accumulate a ﬁxed sum of money in the future?• What are the diﬀerent types of bonds or loans and how do we determine their value?
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Time value of moneyWealth generates beneﬁts and entail costs over their lifetime. Netbeneﬁts = total beneﬁts - total costs. In a market economy,beneﬁts and costs can be measured as cash ﬂows.USD 1 on 10-5-2012 = USD 1 on 10-5-2022Time line diagrams.Financial tables/calculators/spreadsheets. 2
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One-shot cash ﬂowsLet V0 = $100 today (i.e. at t = 0).Principal: Amount on which interest is paid. Compound interest:Interest earned on previous interest that has increased the previousprincipal.Say the banks pay an annual interest rate i = 5%, which is thenadded to the principal. 3
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FV and compounding Vt = V0 (1 + i)tIf V0 = $100, t = 10, and i = 5% = 0.05, then V1 0 = $100 × (1 + 0.05)10 = $162.89(1 + i) is called the gross interest rate. (1 + i)t is called the futurevalue interest factor. What’s the meaning of the FVIF in plainterms?In general, compound growth of any variable means that the valueof the variable increases each period by the factor (1 + g). Whenmoney is invested at the compound interest rate i, the growth rateis i. 4
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FV and compounding Vt = V0 (1 + i)tNote that the greater V0 , the greater i, and the greater t, then Vtwill also be greater. And vice versa. 5
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PV and discounting V0 = Vt (1 + i)−tIf V10 = $162.89 and i = 5% = 0.05, then V0 = $162.89 (1 + 0.05)−10 = $100.Here, the interest rate i is called the discount rate and (1 + i)−t iscalled the discount factor. What’s the meaning of the DF in plainterms?Note that the greater Vt , the smaller i, and the smaller t, thegreater will V0 be. 6
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Finding iSuppose a company needs to borrow. It issues bonds for $129each promising to pay its holder $1,000 at the end of 25 years.No coupons. One single payment at the end of 25 years. What(ﬁxed) annual interest rate is the bond paying?We know the PV ($129), the FV ($1,000). We don’t know theinterest rate i. Let’s ﬁgure it out: V0 = Vt (1 + i)−t Vt (1 + i)t = V0 Vt 1/t 1+i= V0 Vt 1/t i= −1 V0 $1, 000 1/25 i= −1 $129 7
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Frequent compoundingSo far, we have implicitly assumed that the interest rate com-pounds annually (once a year). What if it compounds more often?Suppose you have $100 and earn an annual interest rate of 6%to be compounded monthly. Each month the bank pays you 1/12of the annual rate, 0.06/12 = 0.005 or a half percent (50bp).∗Since the interest is compounded monthly, your $100 earn [1 +(i/12)]12 − 1 = (1.005)12 − 1 = 1.0617 − 1 = 0.617. Your trueannual interest rate is not 6% but 6.17%!We need to modify the FV formula when dealing with more fre-quent compounding. Let m be the number of periodic paymentsper year (e.g. 2 semi-annually, 4 quarterly, 12 monthly,etc.).Then: i mt Vt = V0 1 + m∗ 1%=0.1=100bp. 8
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WARNING!Do NOT ever compare, add up, or subtract cash ﬂowsthat occur at diﬀerent times without previously dis-counting them (or compounding them) to a commondate. They are apples and oranges! 9
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FV and PV of multiple cash ﬂowsIf an asset generates multiple cash ﬂows, how do we ﬁnd its FV(or PV)? One way is to use computational brute force, i.e. wecalculate the FV (or PV) of each cash ﬂow with the formulas weknow and then add up all those FVs (or PVs).Example: You save $300 each year for 3 years to buy a computerstarting today (3 deposits of $300 each). You earn 2% annuallyon your savings balance. How much money will you have in 3years? (Answer: You will have $993.04. Show how that resultwas obtained by applying the formulas learned so far.)We better ﬁnd a way to simplify the computation of asset thatgenerate multiple cash ﬂows. 10
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PV of a consol or perpetuityAn asset that promises to pay a ﬁxed annual payment foreverstarting at the end of the present year (or beginning of the second).The principal is not repaid, you just receive these annual paymentsforever. Say the annual payment is C. What is the annual interestrate on this perpetuity? Assume the market decides that the right Cvalue of this security is P V P0 .∗ Then obviously: i = P V P0 .Therefore, P V P0 = C/P V P0 . Usually, we know i of assets ofsimilar risk and C. We can then ﬁnd the PV of the perpetuity as:P V P0 = C/i. This is the PV of the perpetuity at zero or, to bemore precise, P V P0 .∗ How does the market do it? 11
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PV of a perpetuityWhat is its PV at t some year in the future (P V Pt )? This perpe-tuity would begin generating a cash ﬂow C starting in year t + 1.The value of the perpetuity at n will be the same as today’s!P V Pt = P M T . iNote that P V Pt is a FV (PV at t), not really a PV! An actualpresent value is P V P0 . So, to ﬁnd the PV of that perpetuitythat generates an eternal cash ﬂow beginning in year t + 1, weneed to discount P V Pt to the present to ﬁnd its P V P0 . That is, P V PtP V P0 = (1+i)t = C (1 + i)−t . iSee how this lego toy works? 12
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PV of a consol or perpetuityExample: Find the P V P at the beginning of 2012 of a consolthat yields $100 annually starting at the end of 2012. The mar-ket interest rate of equally safe investments is 5%. (Show thatP V P0 = $2, 000.Example: Find the P V P at the beginning of 2012 of a perpetuitythat yields $100 annually starting at the end of 2016. The mar-ket interest rate of equally safe investments is 5%. (Show thatP V P4 = $1, 645.40.) 13
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Annuities (FV)What will be the balance of a savings account in 10 years if onedeposits $1,000 at the end of each year and i is 5%? What willthe balance be if the deposits are made at the beginning of eachyear?These are annuities. The payments are the same each year. Theformer (payments at the end of each year) is known as an ordinaryannuity. The latter (payments at the beginning of each year) isknown as an annuity due.An annuity is just a ﬁnite number of periodic constant ﬂows. Ap-plying the FV formula we know, we construct a formula for theFV of annuities. 14
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FV of an ordinary annuityLet C be the annual deposit (or payment), i the interest rate, andt the term of the annuity. The FVA will be the sum of the FV’sof each annual deposit (or payment). F V At = C(1 + i)t−1 + C(1 + i)t−2 + . . . + C(1 + i)2 + C(1 + i) + C (1) t−1 F V At = C (1 + i)n (2) n=0 t−1The interest factor is F V IF Ai,t = t=0 (1 + i)n . 15
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PV of an ordinary annuityBut there’s an easier way: The PV of an ordinary annuity can beviewed as the PV of a perpetuity that begins to produce cash ﬂowsat the end of year 1 minus the PV of a perpetuity that producesits ﬁrst cash ﬂow at the end of year (t + 1). Hence: C P V At = 1 − (1 + i)−t i 16
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MiscellaneousHow do we ﬁgure out the FV and PV of mixed-stream assets(diﬀerent cash ﬂows each period)? Mix and match the previousformulas as may be required. If no fancy formula can be applied,use the basic formulas for FV and PV of single cash ﬂows. Thatalways works!Compounding and discounting in continuous time: F V = P V einand P V = F V e−in 17
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And?The principles learned about FV and PV, compounding and dis-counting, applies to all types of bonds. Not only to regular loansor bonds with a ﬁxed interest rate (ﬁxed return), but to variable-return securities as well (e.g. stocks, variable interest-rate creditinstruments, etc.)With variable-return securities, the return is uncertain. But eventhe most certain (‘ﬁxed’) return is in fact uncertain. The futureis essentially unknown and no human institution is eternal.When we compare FV’s and PV’s of diﬀerent investments, we canonly make a meaningful comparison when the degree of risk in-volved in the investments we compare are is similar. If the degreesof risk are diﬀerent, then we are comparing apples and oranges.We need to ﬁnd an analytical way to translate risk into return.Then we will be able to convert uncertain cash ﬂows into ‘certain’cash ﬂows (i.e. adjusted for risk) and thus use what we’ve learnedto compare these diﬀerent instruments on an apples-to-apples ba-sis. 18
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What did we learn?• Why is time value a big deal in ﬁnance?• What’s the PV and FV of a security?• What’s the PV and FV of a simple bond (one-shot cash ﬂow in the future)?• What’s the PV of a perpetuity?• What’s the PV and FV of an annuity (ordinary- and -due) and those of a mixed-stream asset? 19
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• What’s frequent compounding and how does that aﬀect asset valuation?• How much do we need to deposit periodically in order to accumulate a ﬁxed sum of money in the future?• How are loans amortized – i.e., how do we determine equal periodic payments to repay a loan principal plus a stipulated interest?• How are interest or growth rates found? How do we ﬁnd the number of periods it takes for an initial deposit to grow to a certain future amount, given the interest rate?
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