Issuing bonds at discount - computing effective interest rate
In this case the market interest rate is the discount rate that sets equal the market price (current value) of the bond to its present value. To find the market interest rate which is also referred to as promised yield YTM you would have solve for the interest rate in the bond price formula A market price of bond is the sum of discounted coupons and the terminal value of the bond. Most spreadsheet programs and calculators have a RATE function that makes possible finding this market interest rate. First see this for finding a coupon paying bond price The coupon payments are discounted so is the par value of the bond and sum of such discounts is the market price of the bond. The TVM functions in Excel and calculators make this possible using the following equation Let us take your data, 9% $100,000 coupon with 5 years remaining to maturity with market interest rate of 10%. Bonds issued in the US mostly pay two coupons per year. Thus we are finding the present value of 10 coupons each worth $4500 and par value of $100,000. The semi-annual market interest rate is 10%/2 or 5% The negative sign indicate money going out of hand Now solving for RATE is only possible using numerical methods and the RATE function is programmed using Newton-Raphson method to find one of the roots of the bond price equation. This rate will be the periodic rate in this case semi-annual rate which you have to multiply by 2 to get the annual rate. Do remember there is a difference between annual nominal rate and an annualized effective rate. To find the market interest rate  If you don't have Excel or a financial calculator then you may opt to use my version of these financial functions in this JavaScript library tadJS "Yes, the ""effective"" and ""market"" rates are interchangeable. The present value formula will help make it possible to determine the effective interest rate. Since the bond's par value, duration, and par interest rate is known, the coupon payment can be extracted. Now, knowing the price the bond sold in the market, the duration, and the coupon payment, the effective market interest rate can be extracted.  This involves solving large polynomials. A less accurate way of determining the interest rate is using a yield shorthand. To extract the market interest rate with good precision and acceptable accuracy, the annual coupon derived can be divided by the market price of the bond." If the market rate and coupon were equal, the bond would be valued at face value, by definition. (Not 100% true, but this is an exercise, and that would be tangent to this discussion).  Since the market rate is higher than the coupon rate, the value I am willing to pay drops a bit, so my return is the same as the market rate.  This can be done by hand, a time value of money calculation for each payment. Discount by the years till received at the market rate to get the present value for each payment, and sum up the numbers.  The other way is to use a finance calculator and solve for rate.  The final payment of $10,000 (ignore final coupon just now) is $10,000/(1.1^5). In other words, that single chunk of cash is worth 10% less if it's one year away, (1.1)^2 if 2 years away, etc.  Draw a timetable with each payment and divide by 1.1 for each year it's away from present. If the 9% coupon is really 4.5% twice a year, it's $450 in 6 month intervals, and each 6 mo interval is really 5% you discount.  Short durations like this can be done by hand, a 30 year bond with twice a year payments is a pain.  Welcome to Money.SE.
"Taking the last case first, this works out exactly. (Note the Bank of England interest rate has nothing to do with the calculation.) The standard loan formula for an ordinary annuity can be used (as described by BobbyScon), but the periodic interest rate has to be calculated from an effective APR, not a nominal rate.  For details, see APR in the EU and UK, where the definition is only valid for effective APR, as shown below. 2003 BMW 325i £7477 TYPICAL APR 12.9% 60 monthly payments £167.05 How does this work? See the section Calculating the Present Value of an Ordinary Annuity.  The payment formula is derived from the sum of the payments, each discounted to present value. I.e. The example relates to the EU APR definition like so.  Next, the second case doesn't make much sense (unless there is a downpayment). 2004 HONDA CIVIC 1.6 i-VTEC SE 5 door Hatchback £6,999 £113.15 per month ""At APR 9.9% [as quoted in advert], 58 monthly payments"" 58 monthly payments at 9.9% only amount to £5248.75 which is £1750.25 less than the price of the car. Finally, the first case is approximate. 2005 TOYOTA COROLLA 1.4 VVTi 5 door hatchback £7195 From £38 per week ""16.1% APR typical, a 60 month payment, 260 weekly payments"" A weekly payment of £38 would imply an APR of 14.3%." Basically you need to use a time-value-of-money equation to discount the cashflows back to today. The Wikipedia formula will likely work fine for you, then you just need to pick an effective interest rate to use in the calculation. Run each of your amounts and dates though the formula (there are various on-line calculators to pick from, and sum up the values. You did not mention your location or jurisdiction, but a useful proxy for the interest rate would be the average between the same duration mortgage rate and fixed-deposit rate at your bank; it should be close enough for your purposes - although if an actual lawsuit is involved and the sums high enough to have lawyers, it might be worth engaging an accountant as well to defend the veracity of both the calculation and the interest rates chosen. "These rates are so low because the cost of money is so low. Specifically, two rates are near zero.  The Federal Reserve discount rate, which is ""the interest rate charged to commercial banks and other depository institutions on loans they receive from their regional Federal Reserve Bank's lending facility--the discount window."" The effective federal funds rate, which is the rate banks pay when they trade balances with each other through the Federal Reserve. Banks want to profit on the loans they make, like mortgage loans. To do so, they try to maximize the difference between the rates they charge on mortgages and other loans (revenue), and the rates they pay savings account holders, the Federal Reserve or other banks to obtain funds (expenses). This means that the rates they offer to pay are as close to these rates as possible. As the charts shows, both rates have been cut significantly since the start of the recession, either through open market operations (the federal funds rate) or directly (the discount rate). The discount rate is set directly by the regional Federal Reserve banks every 14 days.   In most cases, the federal funds rate is lower than the discount rate, in order to encourage banks to lend money to each other instead of borrowing it from the Fed. In the past, however, there have been rare instances where the federal funds rate has exceeded the discount rate, and it's been cheaper for banks to borrow money directly from the Fed than from each other." "A (very) simplified bond-pricing equation goes thus: Fair_Price: {Face_Value * (1 + Interest - Expected_Market_Return) ^ (Years_To_Maturity)} * P(Company_Will_Default_Before_Maturity) To reiterate, that is a very simplified model. But it allows us to demonstrate the 3 key factors that drive ""Fair"" Value: The interest relative to the current market rate. If your AAA bond yields 1%, but an equally-good AAA bond currently sells at 3% in the market, then the ""Equivalent"" value is the face value minus 2% (1% - 3%) for every year to maturity.  Years to maturity. Because 1) is multiplied for every year to maturity, longer-dated bonds are more sensitive to changes in market rates. If your bond yields 2% less than market but matures in a year, then it's worth $98, but if it matures in 56 years, then it's only worth 0.98^56 = $32. Conversely, if your bond yields more than the market rate, then its' price will be greater than face value. The company might default on the debt. If a Bond has a ""Fair"" Value of $100, but you think there's a 50% chance that the company will default, then it's only worth $50. In fact, it can be worth even less because getting paid on a defaulted bond can often take time and/or money and/or lawyers. In your case, because your bond matures in 56 years but yields ~5% (well above the current market rate), for it to be below Face value implies a strong probability of default, or a strong belief that market returns will be above 5% over the next 56 years." You've got the right idea, except that the stated interest rate is normalized for a 1-year investment.  Hence if you buy a 4-week bill, you're getting something closer to 4/52 of what you've computed in your question.  More precisely, the Treasury uses a 360 day year for these calculations, so you multiply the stated rate by (number of days until maturity)/360 to get the actual rate of return. "The rate of the bond is fixed. But there is a risk known as ""interest rate risk"". Basically, if you have a 2 percent bond and market rates are 4 percent, you'll have to offer your bond at a discount or nobody would buy it. So if you ever needed to sell it, you'd lose a bit of money." When you buy a bond - you're giving a loan to the issuer. The interest rate on the bond is the interest rate on the loan. Usually (and this is also the case with the treasury bonds), the rate is fixed for the term of the loan. Thus, if the market rate for similar loans a year later is higher, the rate for the loan you gave - remains the same. "The name of the Gilt states the redemption date, but not the original issue date. A gilt with 8.75% yield and close to its redemption date may have been issued at a time when interest rates were indeed close to 8.75%. For example in the early 1990s, the UK inflation rate was about 8%. One reason for preferring high or low coupon gilts is the trade off between capital gains and income, and the different taxation rules for each. If you buy a gilt and hold it to its maturity date, you know in advance the exact price that it will be redeemed for (i.e. £100). You may prefer to take a high level of income now, knowing you will make a capital loss in future (which might offset some other predictable capital gain for tax purposes) or you may prefer not to take income that you don't need right now, and instead get a guaranteed capital gain in future (for example, when you plan to retire from work). Also, you can use the change in the market value of gilts as a gamble or a hedge against your expectation of interest rate changes in future, with the ""government guaranteed"" fallback position that if your predictions are wrong, you know exactly what return you will get if you hold the gilts to maturity. The same idea applies to other bond investments - but without the government guarantee, of course." Your calc is spot on, the output is small because it's just 5 days worth of interest, and at today's low rates that's practically 0. Also the rate you would want to use is money market rates as that's typically where companies will park cash to earn interest since its a highly liquid market. "There is no formula that can be applied to most variations of the problem you pose.  The reason is that there is no simple, fixed relationship between the two time periods involved: the time interval for successive payments, and the time period for successive interest compounding. Suppose you have daily compounding and you want to make weekly payments (A case that can be handled).  Say the quoted rate is  4.2% per year, compounded daily Then the rate per day is 4.2/365, or 0.0115068 % So, in one week, a debt would grow through seven compoundings.  A debt of $1 would grow to 1 * (1+.000225068)^7, or 1.000805754 So, the equivalent interest rate for weekly compounding is 0.0805754% Now you have weekly compounding, and weekly payments, so the standard annuity formulas apply. The problem lies in that number ""7"", the number of days in a week.  But if you were trying to handle daily / monthly, or weekly / quarterly, what value would you use? In such cases, the most practical method is to convert any compounding rate to a daily compounding rate, and use a spreadsheet to handle the irregularly spaced payments." I wrote one to check against the N3 to N6 bonds: http://capitalmind.in/2011/03/sbi-bond-yield-calculator/ Things to note: I'm not too familiar with the Bank of England's objectives, but it seems similar to the FED's QE program. The interest rate the BOE sets, similar to the FED rate, affects mainly the short term (the left side) of the [interest rate curve](http://www.bankofengland.co.uk/statistics/pages/yieldcurve/default.aspx). However, in order to bring down intermediate and long term rates, central banks will buy intermediate and long dated government and corporate bonds. The government's added demand will drive those bond prices up, which will drive yields down.  But like I said, I'm not too familiar with the BOE's bond purchasing program, so I could be way off base here. EDIT: After reading one of the comments on the original question, I realized that there is a much more intuitive way to think about this. If you look at it as a standard PV calculation and hold each of the cashflows constant. Really what's happening is that because of inflation the discount rate isn't the full value of the interest rate. Really the discount rate is only the portion of the interest rate above the inflation rate. Hence in the standard perpetuity PV equation PV = A / r r becomes the interest rate less the inflation rate which gives you PV = A / (i - g). That seems like a much better way to get to the answer than all the machinations I was originally trying. Original Answer: I think I finally figured this out. The general term for this type of system in which the payments increase over time is a gradient series annuity. In this specific example since the payment is increasing by a percentage each period (not a constant rate) this would be considered a geometric gradient series. According to this link the formula for the present value of a geometric gradient series of payments is: Where P is the present value of this series of cashflows.   A_1 is the initial payment for period 1 (i.e. the amount you want to withdraw adjusted for inflation).   g is the gradient or growth rate of the periodic payment (in this case this is the inflation rate)   i is the interest rate   n is the number of payments This is almost exactly what I was looking for in my original question. The only problem is this is for a fixed amount of time (i.e. n periods). In order to figure out the formula for a perpetuity we need to find the limit of the right side of this equation as the number of periods (n) approaches infinity. Luckily in this equation n is already well isolated to a single term: (1 + g)^n/(1 + i)^-n}. And since we know that the interest rate, i, has to be greater than the inflation rate, g, the limit of that factor is 0. So after replacing that term with 0 our equation simplifies to the following: Note: I don't do this stuff for a living and honestly don't have a fantastic finance IQ. It's been a while since I've done any calculus or even this much algebra so I may have made an error in the math. How would you compute the earnings for governments that are some of the main issuers of bonds and debt? When governments run deficits they would have a negative earnings ratio that makes the calculation quite hard to evaluate. "Usually... I think that's overstating the case.  You CAN get a bargain (especially if the place is in not-so-great condition), but not every foreclosure will be a good deal even if it is priced well below its most recently appraised value. As the buyer it's your responsibility to determine whether it's priced well or not, and to decide whether you're willing and able to repair its deficiencies after you buy it. The same's true when purchasing any house; foreclosures just make it more likely that there are problems and (hopefully) wind up being priced to allow for them. I don't know of a single website which lists all foreclosures. Some of the home listing websites do have a ""show me foreclosure listings"" filter, and I'm sure that the better tools available to real estate agents can select these. But if that's the direction you're interested in going, you should be looking at distressed properties generally, NOT just foreclosures; you may get a better deal, in the long run, by going for the one that has been mechanically maintained but is just plain ugly rather than the one with a pretty skin whose heating system hasn't been serviced for the last decade. Do your homework, shop around, don't fall in love with any one house... all the same rules apply at this end of the spectrum just as strongly as they do in the mid or upper ranges. Perhaps more so. Happy hunting!"