Suppose that the true discount rate for next year is r1, for the year after is r2, and so on, the rate for the ith year being ri. No one knows what these will be; the rates will fluctuate from year to year. To calculate the total discount factor over a string of N years, you should multiply the individual factors associated with each year as follows,
where δt = 1 year.
But because the future isn't known, Farmer and Geanakoplos point out, determining the correct discount factor to use over the coming N years means averaging over all possible future paths, i.e. all possible sequences of values r1, r2, ... up through rN. Hence, we need to calculate the value of an "effective" discount factor given by the formula,
[Note: This sum, of course, should be divided by the total number of paths to give the average, effective discount.]
Now, it is tempting to think that when you go through the details of calculating this average, summing up the contributions for all possible paths, and dividing by the number of paths, you will find some kind of simple result in which Deff(T) will be equal to a single exponential factor with an average discount rate for the N years, ravg. In others words, you might think -- and most peoples' intuition would tend this way -- that you would find an equation such as
That is, the effective discount rate over N years takes an exponential form with some constant ravg.
Seems sensible, but turns out to be totally wrong. If you demand the equality reflected in the previous equation, then, to make it work as T gets large, it turns out that in many cases ravg will not be a constant in time, but will take on smaller and smaller values as T gets larger. This is what Farmer and Geanakoplos have shown using computer simulations to do the calculation. They used a so-called geometric random walk for the fluctuating rate r, this being the most common mathematical process used in finance to model interest rate fluctuations (i.e. this isn't a crazy or weird model, but a highly plausible one).
Their simulations show that, as a result, the effective discount factor Deff(T) doesn't have an exponential form at all, but rather a very different "power law" form,
where α and β are constants. This falls off with increasing T much slower than an exponential. In other words, it makes discounting much weaker than the incorrect exponential form would suggest it should be.
In the earlier post I discussed some of the implications of this result and a table indicating just how rapidly, after 200 years or so, the exponential and power law forms give wildly different results, with the exponential discounting the value of the future millions or billions of times too strong.
It's rather frightening that a subtle error could make us mis-value the future so profoundly, but this indeed seems to be what we are currently doing. The incorrect exponential form is in wide and standard use by economists doing cost-benefit analyses of everything.
This comment has been removed by the author.
ReplyDeleteDon't discard exponential discounting just yet. Farmer and Geanakoplos base their 'hyperbolic discount function' on the subjective value test subjects place on events far into the future. Experience and common sense tells me that people are NOT good at making such estimates.
ReplyDeleteBut even if they were, there is still a more fundamental flaw in Farmer and Geanakoplos' argument. Exponential discounting is a way of expressing the value of a future cash flow as the amount of money one would have to deposit in an account earning continuously compounded interest, today. It is not a subjective valuation of the future cash flow! Money in a deposit account, on the other hand, does not accumulate according to a hyperbolic function!
So if Farmer and Geanakoplos care so much about events 500 or 700 years from now, they should simply found an 'environment preservation fund' and deposit one dollar earning the risk-free interest rate. Then they can sit back and let the force of compound interest work its magic.
If you don't believe me, consider what will happen to $1 deposited in an account for 500 years at a (continuously compounded) rate of 4% per annum. Ten compare this with Farmer and Geanakoplos' hyperbolic function using, for instance, alpha = 0.04 and beta = 0.9. Which would you prefer? :)
Jens, you seem to confuse risk-free rate and discount rate. It is common in financial models to discount using the risk-free rate, but the real discount rate includes also premiums for risk, alternativity etc.
ReplyDeleteJens,
ReplyDeleteThis might be true if there was such a thing as a 4% annual return over 500 years - but because of rate fluctuation and risk, there is not. In fact, 30 years is as good as it gets - and that still has default risk.
This can't be right. It seems to completely ignore 30 years of research on consistent yield curve modeling. There are perfectly mathematically well-defined and plausible models of interest rates that will reproduce today's forward rate curve under almost any reasonable dynamics. Geometric brownian motion does give somewhat stupid results in the very long term (because rates get concentrated close to zero), but that model wasn't handed down by god - there are other much more empirically reasonable models widely used in finance. (Almost certainly that includes models that Geanakoplos uses, since he's a partner in a mortgage hedge fund, and those types of investors drove a lot of the model development.)
ReplyDeleteGotta go find the paper...
I am a big fan of New Scientist, but I do not understand why a two year old article is suddenly news-worthy. Did it finally get picked up by a journal?
ReplyDeleteThe main point seems to be that people may be irrational in ignoring the mean-reversion of discounting rates that has been apparent in the pricing of government bonds.
Fine, then we are irrational. But is that a basis for policy?
Fair-enough, the evidence for mean-reversion is only apparent for the next thirty-ish years (and has been for the past hundred) -- so maybe there is a basis for doubt when one speculates about inter-temporal preferences several centuries from now -- but it would seem to be a bit anti-science to ignore the data at hand just because it contradicts an (irrational) belief.
Kursus Teknisi Service HP
DeleteDistributor Kuota
PT Lampung Service
Service Center HP Indonesian
Service Center iPhone Indonesian
PT Lampung ServiceServis HP Metro
The problem with such reasoning is that it doesn't scale properly with dt. Make dt much less than a year (say, 1/1000th of a year), and then run the simulations to price a 30 year bond (30000 time periods). You'll get a miniscule rate, approaching 0 as dt -> 0. Other models don't have this characteristic; they are robust to choice of dt.
ReplyDelete@ D Mac
ReplyDeleteI haven't read the paper. Is the scaling problem you highlight to do with how you calculate the rate of variability of interest rates fluctuations? I.e., the probability an interest rate will move by more than 2 percentage points in a year is greater than the probability is will move by 2 percentage points in a day. That would mean that as you scale down the period and dt, you get increasingly smaller ranges of interest rate realizations that you average over. In the limit, as time goes to zero, there would be no variation in interest rates at all, and you would just have whatever rate you started with.
Isn't the result of the paper driven by assuming non-stationary process for one-period discount rates? It seems to me that if r(t) was stationary, then its average over large number of periods would converge to some mean discount rate E[r], and D(T) would converge to something like exp(-E[r]*T) - a standard exponential discounting formula.
ReplyDelete@ivansml - I agree. I think that's part of it. What's weird is that they picked up from the Ho-Lee paper of the mid-'80s and ignored all of the (much more realistic) models of interest rate dynamics developed since then. The realistic models all have (quasi) stationary processes for the short rate. ("Quasi" because there are always initial transients to match today's forward curve.) A simple example is the Hull-White model, which has gaussian interest rates. That model has certainty-equivalent exponential discount factors.
ReplyDeleteFinance Outlook, the complete financial solution.We provide one of a kind innovative financial remedies to all the financial riddles and puzzles that you tumble upon, every moment.
ReplyDelete"They used a so-called geometric random walk"
ReplyDeleteThat assumption needs to be justified. What is a geometric random walk?
"Geometric Random Walks: A Survey
ReplyDeleteSANTOSH VEMPALA
Abstract. The developing theory of geometric random walks is outlined
here. Three aspects|general methods for estimating convergence (the
\mixing" rate), isoperimetric inequalities in Rn and their intimate connec-
tion to random walks, and algorithms for fundamental problems (volume
computation and convex optimization) that are based on sampling by random walks|are discussed.
1. Introduction
A geometric random walk starts at some point in Rn and at each step, moves to a \neighboring" point chosen according to some distribution that depends only on the current point, e.g., a uniform random point within a xed distance.
The sequence of points visited is a random walk. The distribution of the
current point, in particular, its convergence to a steady state (or stationary) distribution, turns out to be a very interesting phenomenon. By choosing the one-step distribution appropriately, one can ensure that the steady state distribution is, for example, the uniform distribution over a convex body, or indeed any reasonable distribution in Rn."
To make this assumption is scientism, not economics. I'm afraid your committment to physics is ruining your chances of becoming a good economist.
It's called "adjusting the wooden ear phones" (from Feynman's talk on cargo cults)
ReplyDeletehttp://www.lhup.edu/~DSIMANEK/cargocul.htm
I see the superlative contents on your blogs and I perfectly enjoy going through them.
ReplyDeleteunsecured loans online
I Love your articles guys keep it up.
ReplyDeleteonline application
Your computer is rattling instructive and your articles are wonderful.dui attorneys
ReplyDeleteThanks for sharing this information its really nice.
ReplyDeleteadvantages of roll over iras
I would surely give 10 on 10 for such incredible content.
ReplyDeletewhole life insurance quotes
Cool website buddy I am gona suggest this to all my list of contacts.
ReplyDeletewhole life insurance quotes online
I went over this website and I conceive you've got a large number of splendid information,
ReplyDeletepayday loans online direct lenders
I would be appreciating all of your articles and blogs as a result of their suiting up mark.
ReplyDeletepayday loan online
Modern times when internet has so much facility of gossip and stuff, your articles have awfully refreshed me.accident compensation
ReplyDeleteVisit our web and get free Digital Event tickets online. It will be helpful for you.
ReplyDeleteخدمات تسليك المجاري شركة تسليك مجارى بالرياض نقدم اليكم افضل الخدمات علي الاطلاق حيث اننا نقوم بحل مشكلة
ReplyDeleteشركة تسليك مجاري بالرياض انسداد المجاري بافضل الطرق فنحن
ReplyDeleteشركة نقل عفش
اهم شركات مكافحة حشرات بالخبر كذلك معرض اهم شركة مكافحة حشرات بالدمام والخبر والجبيل والخبر والاحساء والقطيف كذلك شركة رش حشرات بالدمام ومكافحة الحشرات بالخبر
شركة مكافحة حشرات بالدمام
شركة تنظيف خزانات بجدة الجوهرة من افضل شركات تنظيف الخزانات بجدة حيث ان تنظيف خزانات بجدة يحتاج الى مهارة فى كيفية غسيل وتنظيف الخزانات الكبيرة والصغيرة بجدة على ايدى متخصصين فى تنظيف الخزانات بجدة
شركة تنظيف خزانات بجدة
شركة كشف تسربات المياه بالدمام
شركة نقل عفش واثاث
ReplyDeleteشركة نقل عفش بالرياض وجدة والدمام والخبر والجبيل اولقطيف والاحساء والرياض وجدة ومكة المدينة المنورة والخرج والطائف وخميس مشيط وبجدة افضل شركة نقل عفش بجدة نعرضها مجموعة الفا لنقل العفش بمكة والخرج والقصيم والطائف وتبوك وخميس مشيط ونجران وجيزان وبريدة والمدينة المنورة وينبع افضل شركات نقل الاثاث بالجبيل والطائف وخميس مشيط وبريدة وعنيزو وابها ونجران المدينة وينبع تبوك والقصيم الخرج حفر الباطن والظهران
شركة نقل عفش بجدة
شركة نقل عفش بالمدينة المنورة
شركة نقل اثاث بالرياض
شركة نقل عفش بالدمام
ReplyDeleteشركة نقل عفش بالطائف
شركة نقل عفش بمكة
شركة نقل عفش بينبع
شركة نقل عفش بالخرج
شركة نقل عفش ببريدة
ReplyDeleteشركة نقل عفش بخميس مشيط
شركة نقل عفش بالقصيم
شركة نقل عفش بتبوك
شركة نقل عفش بابها
شركة نقل عفش بنجران
شركة نقل عفش بحائل
Christmas laser light that u try to find
ReplyDeleteThe distribution of the
ReplyDeletecurrent point, in particular, its convergence to a steady state (or stationary) distribution, turns out to be a very interesting phenomenon.
It is nice blog Thank you porovide important information and i am searching for same information to save my time Data Science online Training
ReplyDelete
ReplyDeleteWe to tell the truth think you would need seriously designed most people within the explanation uncovered plenty of people have the method when it comes to cool off a good storyline using a massive supplemental efficient essential. Page easily are convinced trust working experience good commercialized other individuals about the chance until now have you can expect to have point in time to guarantee everyone on line data backup all suggestion swimming pool. wide-ranging bit more good quality records. Add-ons follow-up Boston red sox tickets Ok bye for good material.
Great artical
ReplyDeleteiphone x powerbank quick charge
iphone powerbank
ReplyDeleteRegards just for excellent knowledge. We will start off an innovative new endeavor around the identical markets as this approach blog. Pretty decent help! Visitor afterward many along with noticed presumes might think these sorts of a tremendous amount understand! Up to now continue on this sort of kitchen remodeling weschester county Very much about section Serious purchased.
Nice blog, keep more updates about this type of information. Visit for the best Website Designing and Development Company in Delhi.
ReplyDeleteTop 5 Website Designing Company in Delhi
Rice Bags Manufacturers
ReplyDeletePouch Manufacturers
wall putty bag manufacturers
we have provide the best ppc service.
ReplyDeleteppc company in gurgaon
website designing company in Gurgaon
PPC company in Noida
we have provide the best fridge repair service.
ReplyDeletefridge repair in faridabad
Videocon Fridge Repair in Faridabad
Whirlpool Fridge Repair in Faridabad
Washing Machine Repair in Noida
godrej washing machine repair in noida
whirlpool Washing Machine Repair in Noida
IFB washing Machine Repair in Noida
LG Washing Machine Repair in Noida
Bali Honeymoon Packages From Delhi
ReplyDeleteBali Honeymoon Packages From Chennai
Hong Kong Packages From Delhi
Europe Packages from Delhi
Bali Honeymoon Packages From Bangalore
Bali Honeymoon Packages From Mumbai
Maldives Honeymoon Packages From Bangalore
travel company in Delhi
Get Mutual Fund Investment Schemes by Mutual Fund Wala and know about the best investment platform for you, to get profit.
ReplyDeleteMutual Fund Agent
Totalsolution is the one of the best home appliances repair canter in all over Delhi we deals in repairing window ac, Split ac , fridge , microwave, washing machine, water cooler, RO and more other home appliances in cheap rates
ReplyDeleteLCD, LED Repair in Janakpuri
LCD, LED Repair in Dwarka
LCD, LED Repair in Vikaspuri
LCD, LED Repair in Uttam Nagar
LCD, LED Repair in Paschim Vihar
LCD, LED Repair in Rohini
LCD, LED Repair in Punjabi Bagh
LCD, LED Repair in Delhi. & Delhi NCR
LCD, LED Repair in Delhi. & Delhi NCR
Washing Machine repair on your doorstep
Microwave repair on your doorstep
Get The Best Repairing Service of office chairs shockers of your sofa or billiards and more other furniturea at best price.For more info visit our site given below :-
ReplyDeleteChair Repairing in nawada
Shocker Repairing in dwarka
Sofa Repairing in delhi ncr
Furniture Repairing in janakpuri
Billiards Repairing in nehru place
chair repair in dwarka
sofa repair in south delhi
shocker repair in nawada
furniture repair in nawada
Are you looking for a maid for your home to care your baby,patient care taker, cook service or a japa maid for your pregnent wife we are allso providing maid to take care of your old parents.we are the best and cheapest service provider in delhi for more info visit our site and get all info.
ReplyDeletemaid service provider in South Delhi
maid service provider in Dwarka
maid service provider in Gurgaon
maid service provider in Paschim Vihar
cook service provider in Paschim Vihar
cook service provider in Dwarka
cook service provider in south Delhi
baby care service provider in Delhi NCR
baby care service provider in Gurgaon
baby care service provider in Dwarka
baby service provider in south Delhi
servant service provider in Delhi NCR
servant service provider in Paschim Vihar
servant Service provider in South Delhi
japa maid service in Paschim Vihar
japa maid service in Delhi NCR
japa maid service in Dwarka
japa maid service in south Delhi
patient care service in Paschim Vihar
patient care service in Delhi NCR
patient care service in Dwarka
Patient care service in south Delhi
We are the best Construction contractors in grater noida if you are planing for any construction in noida contact us and get best offers and all details
ReplyDeletecivil construction contractor in Greater Noida
residential construction contractor in Greater Noida
industrial construction conractors in Greater Noida
building construction contractors in Greater Noida
home renovation works in Greater Noida
painting work service in Greater Noida
wooden work service in Greater Noida
modular kitchen contractor in Greater Noida
almiraha contrcution contractor in Greater Noida
Alleyaaircool is the one of the best home appliances repair canter in all over Delhi we deals in repairing window ac, Split ac , fridge , microwave, washing machine, water cooler, RO and more other home appliances in cheap rates
ReplyDeleteWindow AC Repair in vaishali
Split AC Repair in indirapuram
Fridge Repair in kaushambi
Microwave Repair in patparganj
Washing Machine Repair in vasundhara
Water Cooler Repair in indirapuram
RO Service AMC in vasundhara
Any Cooling System in vaishali
Window AC Repair in indirapuram
In This Summers get the best designer umbrellas for you or for your family members we allso deals in wedding umbrellas and in advertising umbrellas For more info visit links given bellow
ReplyDeleteUMBRELLA WHOLESALERS IN DELHI
FANCY UMBRELLA DEALERS
CORPORATE UMBRELLA MANUFACTURER
BEST CUSTOMIZED UMBRELLA
FOLDING UMBRELLA DISTRIBUTORS
DESIGNER UMBRELLA
GOLF UMBRELLA DEALERS/MANUFACTURERS
TOP MENS UMBRELLA
LADIES UMBRELLA DEALERS
WEDDING UMBRELLA DEALERS
BEST QUALITY UMBRELLA
BIG UMBRELLA
Top Umbrella Manufacturers in India
Umbrella Manufacturers in Mumbai
Umbrella Manufacturers in Delhi
Garden Umbrella Dealers
Garden Umbrella Manufacturers
PROMOTIONAL UMBRELLA DEALERS IN DELHI/MUMBAI
PROMOTIONAL UMBRELLA MANUFACTURERS IN DELHI / MUMBAI
ADVERTISING UMBRELLA MANUFACTURERS
Magnificent data, visit our page way of life magazine to get the best style and way of life magazines.
ReplyDeleteLifestyle Magazine
We are the one of the top blue art pottery manufacturers in jaipur get contact us and get all informations in detail visit our site
ReplyDeleteblue pottery jaipur
blue pottery shop in jaipur
blue pottery manufacturers in jaipur
blue pottery market in jaipur
blue pottery work shop in jaipur
blue pottery
top blue pottery in jaipur
blue pottery wholesale in jaipur
Rihan electronics is one of the best repairing service provider all over india we are giving our service in many different different cities like Noida,Gazibad,Delhi,Delhi NCR
ReplyDeleteAC Repair in NOIDA
Refrigerator Repair Gaziabad
Refrigerator repair in NOIDA
washing machine repair in Delhi
LED Light Repair in Delhi NCR
plasma TV repair in Gaziyabad
LCD TV Repair in Delhi NCR
LED TV Repair in Delhi
We are the one of the top blue art pottery manufacturers in jaipur get contact us and get all informations in detail visit our site
ReplyDeleteblue pottery jaipur
blue pottery shop in jaipur
blue pottery manufacturers in jaipur
blue pottery market in jaipur
blue pottery work shop in jaipur
blue pottery
top blue pottery in jaipur
blue pottery wholesale in jaipur
Rihan electronics is one of the best repairing service provider all over india we are giving our service in many different different cities like Noida,Gazibad,Delhi,Delhi NCR
ReplyDeleteAC Repair in NOIDA
Refrigerator Repair Gaziabad
Refrigerator repair in NOIDA
washing machine repair in Delhi
LED Light Repair in Delhi NCR
plasma TV repair in Gaziyabad
LCD TV Repair in Delhi NCR
LED TV Repair in Delhi
Agar aap chatey hai ki aapka boyfriend ya girlfriend ka rishta tut jae toh aap Rishta todne ki dua Duas in islam is the best dua for love back.
ReplyDeleteARE YOU SEARCHING FOR THE BEST CAR WASHING AND CLEANING SERVICE PROVIDER IN DELHI NCR JUST CLICK ON OUR LINK AND GET ALL DETAILS
ReplyDeleteBest car cleaning service in delhi ncr
Car Coating service in delhi ncr
Best Car Washing center in delhi ncr
car rubbing and polishing service in delhi ncr
best car service provider in delhi ncr
best car seat cover in delhi ncr
car dry cleaning service provider in delhi ncr
best car roof polishing in delhi ncr
This is really nice, thanks again for this wonderful and valuable information sharing with us. Visit Kalakutir for Godown Floor Marking Painting and Base Company Logo Painting.
ReplyDeleteGodown Floor Marking Painting
Good article -Please check this link
ReplyDeleteMynews8
best rechargeable batteries
Thanks for Fantasctic blog and its to much informatic which i never think ..Keep writing and grwoing your self
ReplyDeleteapostille services in ghaziabad
apostille services in faridabad
apostille services in noida
apostille services in gurgaon
apostille services in delhi
uae embassy attestation in gurgaon
uae embassy attestation in ghaziabad
uae embassy attestation in delhi
uae embassy attestation in faridabad
uae embassy attestation in noida
App development company in mumbai
ReplyDeleteUsually I never comment on blogs but your article is so convincing that I never stop myself to say something about it. You’re doing a great job Man,Keep it up.
ReplyDeleteBest study in uk consultants in delhi
ielts coaching in gurgaon