Comparing the present value of total payment today and partial payments over 3 months
Its kind of a dumb question because no one believes that you can earn 8% in the short term in the market, but for arguments sake the math is painfully easy.  Keep in mind I am an engineer not a finance guy. So the first payment will earn you one month at 8%, the second, two.  In effect three months at 8% on 997.  You can do it that way because the payments are equal: 997 * (.08 /12) *3 = earnings ~= 20 So with the second method you pay: 997 * 3 - 20 =  2971 "I got $3394.83 The first problem with this is that it is backwards.  The NPV (Net Present Value) of three future payments of $997 has to be less than the nominal value.  The nominal value is simple:  $2991.   First step, convert the 8% annual return from the stock market to a monthly return.  Everyone else assumed that the 8% is a monthly return, but that is clearly absurd.  The correct way to do this would be to solve for m in  But we often approximate this by dividing 8% by 12, which would be .67%.  Either way, you divide each payment by the number of months of compounding.   Sum those up using m equal to about .64% (I left the calculated value in memory and used that rather than the rounded value) and you get about $2952.92 which is smaller than $2991.   Obviously $2952.92 is much larger than $2495 and you should not do this.  If the three payments were $842.39 instead, then it would about break even.   Note that this neglects risk.  In a three month period, the stock market is as likely to fall short of an annualized 8% return as to beat it.  This would make more sense if your alternative was to pay off some of your mortgage immediately and take the payments or yp pay a lump sum now and increase future mortgage payments.  Then your return would be safer.   Someone noted in a comment that we would normally base the NPV on the interest rate of the payments.  That's for calculating the NPV to the one making the loan.  Here, we want to calculate the NPV for the borrower.  So the question is what the borrower would do with the money if making payments and not the lump sum.   The question assumes that the borrower would invest in the stock market, which is a risky option and not normally advisable.  I suggest a mortgage based alternative.  If the borrower is going to stuff the money under the mattress until needed, then the answer is simple.  The nominal value of $2991 is also the NPV, as mattresses don't pay interest.  Similarly, many banks don't pay interest on checking these days.  So for someone facing a real decision like this, I'd almost always recommend paying the lump sum and getting it over with.  Even if the payments are ""same as cash"" with no premium charged." What's the present value of using the payment plan? In all common sense the present value of a loan is the value that you can pay in the present to avoid taking a loan, which in this case is the lump sum payment of $2495.  That rather supposes the question is a trick, providing irrelevant information about the stock market. However, if some strange interpretation is required which ignores the lump sum and wants to know how much you need in the present to pay the loan while being able to make 8% on the stock market that can be done. I will initially assume that since the lender's APR works out about 9.6% per month that the 8% from the stock market is also per month, but will also calculate for 8% annual effective and an 8% annual nominal rate. The calculation If you have $x in hand (present value) and it is exactly enough to take the loan while investing in the stock market, the value in successive months is $x plus the market return less the loan payment.  In the third month the loan is paid down so the balance is zero. I.e. So the present value of using the payment plan while investing is $2569.37. You would need $2569.37 to cover the loan while investing, which is more than the $2495 lump sum payment requires.  Therefore, it would be advisable to make the lump sum payment because it is less expensive: If you have $2569.37 in hand it would be best to pay the lump sum and invest the remaining $74.37 in the stock market.  Otherwise you invest $2569.37 (initially), pay the loan and end up with $0 in three months. One might ask, what rate of return would the stock market need to yield to make it worth taking the loan? The APR proposed by the loan can be calculated.   The present value of a loan is equal to the sum of the payments discounted to present value.  I.e. with  ∴ by induction  So by comparing the $2495 lump sum payment with $997 over 3 x monthly instalments the interest rate implied by the loan can be found. Solving for r If you could obtain 9.64431% per month on the stock market the $x cash in hand required would be calculated by This is equal to the lump sum payment, so the calculated interest is comparable to the stock market rate of return.  If you could gain more than 9.64431% per month on the stock market it would be better to invest and take the loan. Recurrence Form Solving the recurrence form shows the calculation is equivalent to the loan formula, e.g. becomes v[m + 1] = (1 + y) v[m] - p where v[0] = pv where In the final month v[final] = 0, i.e. when m = 3 Compare with the earlier loan formula: s = (d - d (1 + r)^-n) / r They are exactly equivalent, which is quite interesting, (because it wasn't immediately obvious to me that what the lender charges is the mirror opposite of what you gain by investing). The present value can be now be calculated using the formula. Still assuming the 8% stock market return is per month. If the stock market yield is 8% per annum effective rate and if it is given as a nominal annual yield, 8% compounded monthly
"Concise answers to your questions: Depends on the loan and the bank; when you ""accelerate"" repayment of a loan by applying a pre-payment balance to the principal, your monthly payment may be reduced. However, standard practice for most loan types is that the repayment schedule will be accelerated; you'll pay no less each month, but you'll pay it off sooner. I can neither confirm nor deny that an internship counts as job experience in the field for the purpose of mortgage lending. It sounds logical, especially if it were a paid internship (in which case you'd just call it a ""job""), but I can't be sure as I don't know of anyone who got a mortgage without accruing the necessary job experience post-graduation. A loan officer will be happy to talk to you and answer specific questions, but if you go in today, with no credit history (the student loan probably hasn't even entered repayment) and a lot of unknowns (an offer can be rescinded, for instance), you are virtually certain to be denied a mortgage. The bank is going to want evidence that you will make good on the debt you have over time. One $10,000 payment on the loan, though significant, is just one payment as far as your credit history (and credit score) is concerned. Now, a few more reality checks: $70k/yr is not what you'll be bringing home. As a single person without dependents, you'll be taxed at the highest possible withholdings rate. Your effective tax rate on $70k, depending on the state in which you live, can be as high as 30% (including all payroll/SS taxes, for a 1099 earner and/or an employee in a state with an income tax), so you're actually only bringing home 42k/yr, or about $1,600/paycheck if you're paid biweekly. To that, add a decent chunk for your group healthcare plan (which, as of 2014, you will be required to buy, or else pay another $2500 - effectively another 3% of gross earnings - in taxes). And even now with your first job, you should be at least trying to save up a decent chunk o' change in a 401k or IRA as a retirement nest egg. That student loan, beginning about 6 months after you leave school, will cost you about $555/mo in monthly payments for the next 10 years (if it's all Stafford loans with a 50/50 split between sub/unsub; that could be as much as $600/mo for all-unsub Stafford, or $700 or more for private loans). If you were going to pay all that back in two years, you're looking at paying a ballpark of $2500/mo leaving just $700 to pay all your bills and expenses each month. With a 3-year payoff plan, you're turning around one of your two paychecks every month to the student loan servicer, which for a bachelor is doable but still rather tight. Your mortgage payment isn't the only payment you will make on your house. If you get an FHA loan with 3.5% down, the lender will demand PMI. The city/county will likely levy a property tax on the assessed value of land and building. The lender may require that you purchase home insurance with minimum acceptable coverage limits and deductibles. All of these will be paid into escrow accounts, managed by your lending bank, from a single check you send them monthly. I pay all of these, in a state (Texas) that gets its primary income from sales and property tax instead of income, and my monthly payment isn't quite double the simple P&I. Once you have the house, you'll want to fill the house. Nice bed: probably $1500 between mattress and frame for a nice big queen you can stretch out on (and have lady friends over). Nice couch: $1000. TV: call it $500. That's probably the bare minimum you'll want to buy to replace what you lived through college with (you'll have somewhere to eat and sleep other than the floor of your new home), and we're already talking almost a month's salary, or payments of up to 10% of your monthly take-home pay over a year on a couple of store credit cards. Plates, cookware, etc just keeps bumping this up. Yes, they're (theoretically) all one-time costs, but they're things you need, and things you may not have if you've been living in dorms and eating in dining halls all through college. The house you buy now is likely to be a ""starter"", maybe 3bed/2bath and 1600 sqft at the upper end (they sell em as small as 2bd/1bt 1100sqft). It will support a spouse and 2 kids, but by that point you'll be bursting at the seams. What happens if your future spouse had the same idea of buying a house early while rates were low? The cost of buying a house may be as little as 3.5% down and a few hundred more in advance escrow and a couple other fees the seller can't pay for you. The cost of selling the same house is likely to include all the costs you made the seller pay when you bought it, because you'll be selling to someone in the same position you're in now. I didn't know it at the time I bought my house, but I paid about $5,000 to get into it (3.5% down and 6 months' escrow up front), while the sellers paid over $10,000 to get out (the owner got married to another homeowner, and they ended up selling both houses to move out of town; I don't even know what kind of bath they took on the house we weren't involved with). I graduated in 2005. I didn't buy my first house until I was married and pretty much well-settled, in 2011 (and yes, we were looking because mortgage rates were at rock bottom). We really lucked out in terms of a home that, if we want to or have to, we can live in for the rest of our lives (only 1700sqft, but it's officially a 4/2 with a spare room, and a downstairs master suite and nursery/office, so when we're old and decrepit we can pretty much live downstairs). I would seriously recommend that you do the same, even if by doing so you miss out on the absolute best interest rates. Last example: let's say, hypothetically, that you bite at current interest rates, and lock in a rate just above prime at 4%, 3.5% down, seller pays closing, but then in two years you get married, change jobs and have to move. Let's further suppose an alternate reality in which, after two years of living in an apartment, all the same life changes happen and you are now shopping for your first house having been pre-approved at 5%. That one percentage point savings by buying now, on a house in the $200k range, is worth about $120/mo or about $1440/yr off of your P&I payment ($921.42 on a $200,000 home with a 30-year term). Not chump change (over 30 years if you had been that lucky, it's $43000), but it's less than 5% of your take-home pay (month-to-month or annually). However, when you move in two years, the buyer's probably going to want the same deal you got - seller pays closing - because that's the market level you bought in to (low-priced starters for first-time homebuyers). That's a 3% commission for both agents, 1% origination, 0.5%-1% guarantor, and various fixed fees (title etc). Assuming the value of the house hasn't changed, let's call total selling costs 8% of the house value of $200k (which is probably low); that's $16,000 in seller's costs. Again, assuming home value didn't change and that you got an FHA loan requiring only 3.5% down, your down payment ($7k) plus principal paid (about another $7k; 6936.27 to be exact) only covers $14k of those costs. You're now in the hole $2,000, and you still have to come up with your next home's down payment.  With all other things being equal, in order to get back to where you were in net worth terms before you bought the house (meaning $7,000 cash in the bank after selling it), you would need to stay in the house for 4 and a half years to accumulate the $16,000 in equity through principal payments. That leaves you with your original $7,000 down payment returned to you in cash, and you're even in accounting terms (which means in finance terms you're behind; that $7,000 invested at 3% historical average rate of inflation would have earned you about $800 in those four years, meaning you need to stick around about 5.5 years before you ""break even"" in TVM terms). For this reason, I would say that you should be very cautious when buying your first home; it may very well be the last one you'll ever buy. Whether that's because you made good choices or bad is up to you." The monthly repayments of the initial $ 300,000 loan can be calculated using this formula:  source: Finance Formulas The monthly payment is It is not readily apparent how the formula works, but it is derived by induction from this summation, in which the sum of the discounted future payments are set equal to the present value of the loan:  For the second part of the question, reinvestments are stopped after 9 months, after four investments of $ 26,374.77 * 3 = $ 79,124.31.  And presumably each loan is repaid in 3 years, since 45 - 9 = 36 months. Calculating the repayments for these loans: The total returned for all four loans is: The main point to consider is that your payments toward your own home replace your rent. Any house or apartment you buy will have changes in value; the value is generally going slowly up, but there is a lot of noise, and you may be in a low phase at any time, and for a long time. So seeing it as an investment is not any better than buying share or funds, and it has a much worse liquidity (= you cannot as easily make it to cash when you want to), and not in parts either. However, if you buy for example a one-room apartment for 80000 with a 2% mortgage, and pay 2% interest = 1600 plus 1% principal = 800, for a total of 2400 per year = 200 per month, you are paying less than your current rent, plus you own it after 30 years. Even if it would be worth nothing after 30 years, you made a lot of money by paying half only every month, and it probably is not worthless. You need to be careful not to compare apples with oranges - if you buy a house for 200000 instead, your payments would be higher than your rent was, but you would be living in your house, not in a room. For most people, that is worth a lot. You need to put your own value to that; if you don't care to have a lot more space and freedom, the extra value is zero; if you like it, put a price to it. With current interest rates, it is probably a good idea for most people to buy a house that they can easily afford instead of paying rent. The usual rules should be considered - don't overstretch yourself, leave some security, etc. Generally, it is rather difficult to buy an affordable house instead of renting today and not saving a lot of money in the process, so I would say go for it. "Basically, the easiest way to do this is to chart out the ""what-ifs"". Applying the amortization formula (see here) using the numbers you supplied and a little guesswork, I calculated an interest rate of 3.75% (which is good) and that you've already made 17 semi-monthly payments (8 and a half months' worth) of $680.04, out of a 30-year, 720-payment loan term. These are the numbers I will use. Let's now suppose that tomorrow, you found $100 extra every two weeks in your budget, and decided to put it toward your mortgage starting with the next payment. That makes the semi-monthly payments $780 each. You would pay off the mortgage in 23 years (making 557 more payments instead of 703 more). Your total payments will be $434,460, down from $478.040, so your interest costs on the loan were reduced by $43,580 (but, my mistake, we can't count this amount as money in the bank; it's included in the next amount of money to come in). Now, after the mortgage is paid off, you have $780 semi-monthly for the remaining 73 months of your original 30-year loan (a total of $113,880) which you can now do something else with. If you stuffed it in your mattress, you'd earn 0% and so that's the worst-case scenario. For anything else to be worth it, you must be getting a rate of return such that $100 payments, 24 times a year for a total of 703 payments must equal $113,880. We use the future value annuity formula (here): v = p*((i+1)n-1)/i, plugging in v ($113880, our FV goal), $100 for P (the monthly payment) and 703 for n (total number of payments. We're looking for i, the interest rate. We're making 24 payments per year, so the value of i we find will be 1/24 of the stated annual interest rate of any account you put it into. We find that in order to make the same amount of money on an annuity that you save by paying off the loan, the interest rate on the account must average 3.07%. However, you're probably not going to stuff the savings from the mortgage in your mattress and sleep on it for 6 years. What if you invest it, in the same security you're considering now? That would be 146 payments of $780 into an interest-bearing account, plus the interest savings. Now, the interest rate on the security must be greater, because you're not only saving money on the mortgage, you're making money on the savings. Assuming the annuity APR stays the same now vs later, we find that the APR on the annuity must equal, surprise, 3.75% in order to end up with the same amount of money. Why is that? Well, the interest growing on your $100 semi-monthly exactly offsets the interest you would save on the mortgage by reducing the principal by $100. Both the loan balance you would remove and the annuity balance you increase would accrue the same interest over the same time if they had the same rate. The main difference, to you, is that by paying into the annuity now, you have cash now; by paying into the mortgage now, you don't have money now, but you have WAY more money later. The actual real time-values of the money, however, are the same; the future value of $200/mo for 30 years is equal to $0/mo for 24 years and then $1560/mo for 6 years, but the real money paid in over 30 years is $72,000 vs $112,320. That kind of math is why analysts encourage people to start retirement saving early. One more thing. If you live in the United States, the interest charges on your mortgage are tax-deductible. So, that $43,580 you saved by paying down the mortgage? Take 25% of it and throw it away as taxes (assuming you're in the most common wage-earner tax bracket). That's $10895 in potential tax savings that you don't get over the life of the loan. If you penalize the ""pay-off-early"" track by subtracting those extra taxes, you find that the break-even APR on the annuity account is about 3.095%." In the US, our standard fixed rate mortgages would show no difference. My payment is calculated to be due on the 1st of each month.  When I first got a mortgage, I was intrigued by this question, and experimented. I paid early, on the 15th, 2 weeks early, and looked at my next statement. It matched the amortization, exactly. Mortgages at the time were over 12%, so I'd imagine having seen the benefit of that 1/2% for the early payment.  Next I paid on the last day before penalty, in effect, 2 weeks late. I expected to see extra interest accrue, again, just a bit, but enough to see when compared to the amortization table. Again, no difference, the next statement showed the same value to the penny. "Of course it can. This is a time value of money calculation. If I knew the maturity date, or current yield to maturity I'd be able to calculate the other number and advise how much rates need to rise to cause the value to drop from 18 to 17. For a 10 year bond, a rise today of .1% will cause the bond to drop about 1% in value. This is a back of napkin calculation, finance calculators offer precision. edit - when I calculate present value with 34 years to go, and 5.832% yield to maturity, I get $14.55. At 5.932, the value drops to $14.09, a drop of 3.1%.  Edit - Geo asked me to show calculations. Here it goes - A) The simplest way to calculate present value for a zero coupon bond is to take the rate 5.832%, convert it to 1.05832 and divide into the face value, $100. I offer this as the ""four function calculator"" approach, so one enters $100 divided by 1.05832 and repeat for the number of years left. A bit of precision is lost if there's a fractional year involved, but it's close. The bid/ask will be wider than this error introduced.  B) Next - If you've never read my open declaration of love for my Texas Instruments BA-35 calculator, here it is, again. One enters N=34 (for the years) FV = 100, Rate = 5.832, and then CPT PV. It will give the result, $14.56.  C) Here is how to do it in Excel -  The numbers in lines 1-3 are self evident, the equation in cell B4 is =-PV(B3/100,B1,0,B2) - please note there are tiny differences in the way to calculate in excel vs a calculator. Excel wants the rate to be .05832, so I divided by 100 in the equation cell.  That's the best 3 ways I know to calculate present value. Geo, if you've not noticed, the time value of money is near and dear to me. It comes into play for bonds, mortgages, and many aspect of investing. The equations get more complex if there are payments each year, but both the BA-35 and excel are up to it." "The first thing that it is important to note here is that the examples you have given are not individual bond prices.  This is what is called the ""generic"" bond price data, in effect a idealised bond with the indicated maturity period.  You can see individual bond prices on the UK Debt Management Office website. The meaning of the various attributes (price, yield, coupon) remains the same, but there may be no such bond to trade in the market.  So let's take the example of an actual UK Gilt, say the ""4.25% Treasury Gilt 2019"".  The UK Debt Management Office currently lists this bond as having a maturity date of 07-Mar-2019 and a price of GBP 116.27.  This means that you will pay 116.27 to purchase a bond with a nominal value of GBP 100.00.  Here, the ""nominal price"" is the price that HM Treasury will buy the bond back on the maturity date.   Note that the title of the bond indicates a ""nominal"" yield of 4.25%.  This is called the coupon, so here the coupon is 4.25%.  In other words, the treasury will pay GBP 4.25 annually for each bond with a nominal value of GBP 100.00.  Since you will now be paying a price of GBP 116.27 to purchase this bond in the market today, this means that you will be paying 116.27 to earn the nominal annual interest of 4.25.  This equates to a 3.656% yield, where 3.656% = 4.25/116.27.   It is very important to understand that the yield is not the whole story.  In particular, since the bond has a nominal value of GBP100, this means that as the maturity date approaches the market price of the bond will approach the nominal price of 100. In this case, this means that you will witness a loss of capital over the period that you hold the bond.  If you hold the bond until maturity, then you will lose GBP 16.27 for each nominal GBP100 bond you hold.  When this capital loss is netted off the interest recieved, you get what is called the gross redemption yield.  In this case, the gross redemption yield is given as approximately 0.75% per annum. NB.  The data table you have included clearly has errors in the pricing of the 3 month, 6 month, and 12 month generics." From the description, you have a post-1998 income contingent loan. The interest rate on those is currently 1.5% but it has varied quite a bit in the last few years due to the formula used to calculate it, which is either the inflation rate (RPI), or 1% + the highest base rate across a group of banks - whichever is smaller. This is indeed really cheap credit compared to any commercial loan you could get, though whether you should indeed just repay the minimum depends on making a proper comparison with the return on any spare money you could get after tax elsewhere. There is a table of previous interest rates. From your description I think you've had the loan for about 4 years - your final year of uni, one year of working without repayments and then two years of repayments. A very rough estimate is that you would have been charged about £300 of interest over that period. So there's still an apparent mismatch, though since both you and I made rough calculations it may be that a more precise check resolves it. But the other thing is that you should check what the date on the statement is. Once you start repaying, statements are sent for a period ending 5th April of each year. So you may well not be seeing the effect of several months of repayments since April on the statement. Finally, there's apparently an online facility you can use to get an up to date balance, though the administration of the loans repaid via PAYE is notoriously inefficient so there may well be a significant lag between a payment being made and it being reflected in your balance, though the effect should still be backdated to when you actually made it.