Tie Value of Money Suary Notation and Forulae Liuren Wu May 6, 2014 1 Coonly used notations Present value, PV Future value, FV n, where the subscript n is used as an indicator for the tie of the future, for exaple, n periods later. soe ties, we use n to denote nuber of years and to denote the nuber of copounding per year so that n is the nuber of copounding periods. For exaple, for onthly copounding for ten years, =,n = 10 and the nuber of copounding periods is n = 0. is 1 for annual copounding, 2 for sei-annual copounding, 4 for quarterly copounding, for onthly copounding, 52 for weekly copounding, and 365 for daily copounding. rate: it links PV and FV, and is often denoted as either r or i, or R. Note that all rates are quoted in annualized per year) ters. Their difference is copounding frequency. In this write-up, I will use N = n to denote nuber of periods, r as the annualized rate. APR: Annual percentage rate. It is an annualized rate quoted by the arket. You not only need to know this rate, but also its copounding frequency. EAR: Effective annual copounding rate. For any quoted rate at any copounding frequency, you can convert it into effective an annual copounding rate to facilitate coparison. 1
2 Forulae The ost basic tie value of oney forula that links PV with FV is FV N = PV 1 + r ) n One can solve for PV fro FV : FV N PV = 1 + r ) n Solving for copounding rate r, r = ) 1 FVN n 1 PV Solving for nuber of years n, Converting APR into EAR n = 1 ln FV N ) PV ln 1 + r ) EAR = 1 + APR ) 1 Converting fro one quote to another. Let R 1 ) and R 2 ) denote the rates copounded at two different frequencies, 1 and 2 ties per year. For the to be equivalent, they ust generate the sae future value if the starting present value is the sae. Use PV = 1 as a noralization, we ask the future value for one year to be equal 1 + R ) 1 1) = 1 + R ) 2 2) 1 2 Hence, [ R 1 ) = 1 1 + R ) 2 ] 1 ) 2) 1 1. 2 A logically siple way to think about the proble is: If you are given one rate, say, R 2 ), copute ) the one-year future value of one dollar first, FV 1 = 1 + R 2) 2. 2 Then, you can copute the rate 2
fro the future value based on whatever copounding, R 1 ) = 1 FV1 PV ) 1 ) 1 ) 1 = 1 FV 1 ) 1 1 1. Continuous copounding: When, you copound infinite nuber of ties per year, it is called continuous copounding. In this case, the valuation forula becoes FV = PVe rn PV = FVe rn r = 1 lnfv /PV ) n n = 1 lnfv /PV ), r EAR = e rn 1, where n is nuber of years, r is the continuous copounding rate, and e is the natural nuber approxiately equal to 2.718282). In your calculator, it ay be written as EXP. Reeber that copouding frequency is just a quotation ethod. The rate together with the copounding frequency deterines how uch oney you ake, you can always convert the quote into an annual copounding rate EAR). Present value of an ordinary annuity with N payents C, payents per year for n years) PV = C 1 1 + r ) ) n r/) Future value of the ordinary annuity, FV N = PV 1 + r ) n C = 1 + r ) n ) 1 r/) 3
One can solve for the payent C given the present value or future value) and the rate/copounding/years, C = C = PV ) r 1 1 + r ) ) n FV ) r ) 1 + r n ) 1 One can also solve for the nuber of periods, given the payent C, the interest rate per period r), and present value, ) PV r C n = ln 1 ln1 + r) The present value of a perpetuity annuity with infinite nuber of payents N ) PV = C r/). 3 Exaples 1. When you graduate three years fro now, you expect to find a job that pays $80,000 per year and you expect your pay to go up 10% per year. According to your expectation, how uch oney you will be aking at your ten-year working anniversary? Answer: You are essentially asked to copound $80,000 for 10 years at 10% annual copounding rate. It does not atter when you graduate because the question asks about 10 years after your graduation. The answer is 80 1 + 0.1) 10 = 207.50 thousands In applying the TVM forula, you can regard 80000 as PV present value, starting value) and regarding the question as asking for FV 10 = PV 1 + r) 10. Of course, you can also regard 80,000 as FV 3 future value 3 years later) and you are solving for FV 13 = FV 3 1 + r) 13 3. What atters is the length of tie nuber of years in this case) in between starting and ending. 2. Reverse the question: If you expect to ake $100,000 at your 10-year working anniversary and you expect your pay to grow at 10% per year, what should be your initial salary? 4
Answer: PV = FV /1 + 10%) 10 = 100000/1.1) 10 = 38,554. 3. You want to ake a saving today and let it to grow to 100,000 in ten years. How uch do you need to save if a) you can ake 10% per year annual copounding? Answer: You are coputing the present value of a future value: PV = FV = 100000 = 38554. 1+r) 10 1+.1) 10 b) a bank is willing to pay you 9.8% onthly copounding? Answer: PV = 100000 1+ 0.098 ) 10 = 37681. c) Assuing the two investent opportunities have the sae risk or no risk at all), which one should you take? 10% annual copounding or 9.8% onthly copounding? Answer: There are several ways to look at this proble: 1) Since you want to reach the sae target, whichever asks yuo to pay less now is the better choice. Hence, it is the bank s 9.8% onthly copounding offer that is better because you just need to put in 37681 today instead of 38554 to get 100,000 ten years later. This eans that although 9.8% is a lower return nuber, but the faster copounding frequency ore than akes up for it. 2) Hence, in coparing returns, you ust convert the into the sae copounding frequency before you can copare the. The tradition/convention is to convert everything into annual copounding rates: 10% annual copounding is still 10% annual copounding. To convert the 9.8% onthly copounding rate into an annual copounding rate, we have 1 + 0.098 ) 1 = 10.252%, which is higher and hence better) than 10%. The textbook call this annual copounding rate as Effective Annual Rate EAR) and the quoted rate as Quoted Annual Rate APR). In this exaple, the quoted rate APR) is 9.8% onthly copounding, and we can convert it to an EAR as 10.252%. The conversion is EAR = 1 + APR ) 1, 1) 5
where is nuber of copounding per year for the APR. d) It is iportant to understand the origin of this conversion: If you want to convert everything to annual copounding return EAR), all you need to ake sure is that they ake the sae aount of oney over the sae tie interval, hence the equality: PV 1 + APR ) n = PV 1 + EAR) n, where n denotes n years into the future. The equality says that starting fro the sae PV, whether you use APR rate at your -ties per year copounding or use EAR at annual copounding, you should be aking the sae aount of future value for any n years down the road. Then, you can see PV and n cancels out, and you can get to the forula:ear = 1 + APR ). e) Note that the conversion works even if the quoted rate copounds slower than once per year. For exaple, if the quoted rate copounds once every two years, = 1/2 once every two years). 4. A newly elected President clais that he will strive to double the GDP gross doestic output) during his 8-year ters yes, he wants to be re-elected). What annual copounding GDP growth rate he needs in order to reach his target? Answer: We don t know exactly what the current or future GDP is, but we know FV 8 /PV = 2 FV is twice as uch as PV). Use the TVM forula, FV 8 = PV 1 + r) 8, we have r = ) 1/8 FV8 1 = 2 1/8 1 = 9.05%. PV Siilar questions: A fund/bank/soebody offers to pay you back twice as uch in 10 years, what kind of return you are getting if the return is copounded annually, quarterly, or weekly? Answer: The general equation is r = [ FV PV ) 1 n 1 ], with n nuber of years and nuber of copounding per year. For the rate per period, all 6
you need is nuber of periods n, but then you need to convert it into an annual rate. The solutions are Annual: 2 1/10 1 = 7.18% ) Quarterly: 2 1/40 1 4 = 6.99% ) Weekly : 2 1/520 1 52 = 6.94%. You need a lower return if the copounding frequency is higher. 5. If your investent akes 10% per year annual copounding, how any years does it take to double? Answer: PV=1,FV=2,r = 10, = 1, n = lnfv /PV ) ln1 + r) = ln2 = 7.2725 years. ln1 + 0.1) How any years does it take if the 10% rate is copounded weekly? Answer: = 52 and n = 1 lnfv /PV ) ln1 + r/) = 1 52 ln2 = 6.938 years. ln1 + 0.1/52) At the sae quoted rate, copounding faster akes ore oney and reaches your goal earlier. 6. If you ake a onthly saving of $1000 and earn a 10% onthly copounding rate of return, how uch oney will you have in your account after 10 years of consecutive savings? Answer: 10% is per year annualized rate) even though it is onthly copounding. The question asks for the FV of an annuity, FV = 1000 0.1/) 1 + 0.1 ) 0 1) = 2.0484 10 5. If you start savings 10 years fro now, this is how uch you get 20 years fro now that is, after you save for 10 years or 0 onthly savings). As long as the question is asking for the value 7
at your last saving, it does not atter when it starts, it just atters on how any consecutive savings you have ade by then. 7. You need to take a $500,000 ortgage to buy a house. The bank charges you a 6% onthly copounding rate for a 30-year loan of $500,000 that pays every onth for 360 onths. What will be your onthly payent? Answer: You know PV of the annuity and are asked to solve for the payent, PV ) r C = 1 500000 0.06 1 + r ) ) n = 1 1 + 0.06 ) 360 ) = 2997.8 The onthly payent is about $3000. The $3000 onthly payent covers both the aount you borrow $500,000) and the 6% interest the bank charges you. In the U.S., interest payent is tax deductable but the principal payent payent for the $500,000) is not. So you want to figure out: Out of the $3000 onthly payent, which part is considered as interest payent, and which part covers your loan. This separation is also iportant in case you want o payback everything in the iddle Then, you want to figure out how uch you still owe. The calculation starts as follows: a) Figure out how uch you still owe the bank your reaining debt). Initially, you owe the $500,0000, but as you pay the back, your debt will reduce. b) Your reaining debt ties 6%/ = 0.005) onthly rate) is how uch you owe the in interest This is your interest payent. c) Whatever left in your $3000 payent reduces your debt. Let e use a table to show the separation: 8
Month Debt Before Payent Interest Expense Debt Reduction Debt After Payent 1 500000 500000 0.005 = 2500.0 3000-2500=500 500000 500 = 499500 2 499500 499500 0.005 = 2497.5 3000 2497.5 = 502.5 499500-502.5= 498, 997.50 3 498,997.50 2492.99 505.01 498,492.49 4 498,492.49 2492.46 507.54 497,984.87 5... Suppose the bank gives you the loan today, but kindly allows you start your first payent five years later instead of right now, how will your onthly payent change? Answer: The bank is still charging you 6% rate, so there is nothing kind about it. If you start late, you need to pay extra interest for the five years you delay back. Hence, a $500,000 loan today becoes a 500000 1 + 0.06 ) 5 = 6.7443 10 5 debt 5 years down the road. Then, you are aking onthly payent on an annuity that has a present value at the start of the annuity) of $674,430. The onthly payent increases accordingly to, C = 500000 1 + 0.06 ) 5 0.06 1 1 + 0.06 ) 360 ) = 4043.5, which can also be coputed directly fro the original onthly payent: 2997.8 1 + 0.06 ) 5 = 4043.5. The idea is that either you can start to ake the onthl payent now or you can delay the payent for five years, but you need to pay interest for your delay on each payent, which is LaterPayent=CurrentPayent 1 + 0.06 ) 5. 8. An apartent s rental is $3000 per onth. Suppose the rent is fixed forever. At a 6% onthly copounding rate, what s the fair value for this apartent? Answer: People disagree on what constitutes fair. It could be based on cost or deand. In our case, let s define fair relative to the rent. That is, we want to know how uch all these rent payents are worth in today s dollar ters. We are then essentially calculating the present value of a perpetuity, PV = 3000 0.06/) = 600,000. 9
This is a quick way to get a rough idea on how uch an apartent is worth: 200 ties onthly rent. 9. You are evaluating an investent project that is projected to generate $100 illion in one year, $300 in year 2, $700 in year 3, and $200 in year 4 The negative cash outflow is because you need to pay to clean up the site, for exaple). At a 10% annual copounding return, please copute how uch this project is worth today. The question asks you to copute the PV of ultiple, irregular cashflows. You just need to copute the PV of each cash flow and su the together: PV = 100 1 + 0.1) 1 + 300 1 + 0.1) 2 + 700 1 + 0.1) 3 + 200 1 + 0.1) 4 = 728.16 The project is worth $728.16 illion in today s dollar ter. How uch is this project worth in two years later s dollar? Answer: Just copound the PV for two years, FV 2 = PV 1 + 0.1) 2 = 728.16 1 +.1) 2 = 881.07. You can also directly convert each cash flow into year-2 dollar: FV 2 = 1001 + 0.1) 1 + 300 + 700 1 + 0.1) + 200 1 + 0.1) 2 = 881.07. To convert year-1 dollar 100) into year-2 dollar, you need to copound one year. To convert year-3 dollar 700) into year-2 dollar, you need discount one year. To convert year-4 dollar -200), you need to discount two years. 10. Let e use a series of questions to highlight the idea that with a fixed rate, tie value coes only in relative ters the nuber of years between the conversion), not in absolute ters e.g., 2014 or 2054). Suppose you plan to invest $100 in a fund 10 years later and expect to ake 10% a year annual 10
copounding, how uch will your investent grow into one year after your investent? How will your answer change if you decide to ake the investent 5 years later, or 50 years later? Answer: You invest $100 for one year at 10% rate and you will end up with $1001+0.1)=$110. Since the question is asking you the value one year after your investent, it does not atter when which year) you ake the investent. What atters is that your investent will ake one year of return. Suppose you plan to ake an annual saving of $500 for 10 consecutive years for an annual copounding return of 6%. You want to know how uch oney you will have at the tie of the last saving. Say you plan to ake your first saving 15 years fro now. How uch will you have at the end of the last saving? Answer: Again, the answer does not depend on whether you start your saving today or 5, 10, or 15 years later because the question asks you about the value of your savings at the end of your saving. It is the sae as the previous question except here we have ultiple cash flows/savings. This is a 10-year annuity and you want to know its future value: FV = C r ) 1 + r) 10 1 = 500 1.06 10 1 ) = 6590.4 0.06 Your friend wants to borrow $2000 fro you now and proises to pay you back in 10 consecutive annual payents of $500 each, with the first payent ade 15 years fro today. If you want a 6% annual copounding return, are you willing to lend hi $2000 today? Answer: At 6% discount rate, you want to fugure out whether the present value of the 10 payents of $500 each is worth $2000. So you are coputing the present value of an annuity. The annuity is the sae as in the previosu question. This tie, when the first payent is ade atters because you are coputing its today s value and you have to easure the tie distance between today and the payent tie. There are a few ways of doing this: a) Fro the answer to the previous question, you know its future value FV = 6590.4. This is the value at the end of the last payent, which is 15+9=24 years fro now Count carefully! I always ake istake here. The first payent is in year 15. The 10th payent 11
is in year 24. So you need to discount it back for 24 years PV = 6590.4 = 1627.7 24 1 + 0.06) The present value is only 1627.7, not worth $2000. So you should not lend hi $2000 if you want 6% return. b) Use the present value of annutiy forula. Since the first payent is in year 15. If you look at the proble in year 14, it is an ordinary annuity. You can use the present value of annuity forula to solve the value in year 14 and then discount it back to today. Value in year 14 = PV of annuity = C 1 1 + r) n ) = 500 1 1.06 10 ) = 3680.0 r 0.06 Value in year 14 PV = 1 +.06) 14 = 3680 = 1627.7. 1.0614