How to calculate the price of a bond based with a yield to Maturity, term and annual interest?
"Like all financial investments, the value of a bond is the present value of expected future cash flows.  The Yield to Maturity is the annualized return you get on your initial investment, which is equivalent to the discount rate you'd use to discount future cash flows. So if you discount all future cashflows at 6% annually*, you can calculate the price of the bond: So the price of a $1,000 bond (which is how bond prices are typically quoted) would be $1,097.12. The current yield is just the current coupon payment divided by the current price, which is 70/1,097.12 or 6.38% Question 3 makes no sense, since the yield to maturity would be the same if you bought the bond at market price Question 4 talks about a ""sale"" date which makes me think that it assumes you sold the bond on the coupon date, but you'd have to know the sale price to calculate the rate of return." "The answer to almost all questions of this type is to draw a diagram.  This will show you in graphical fashion the timing of all payments out and payments received.  Then, if all these payments are brought to the same date and set equal to each other (using the desired rate of return), the equation to be solved is generated. In this case, taking the start of the bond's life as the point of reference, the various amounts are: Pay out = X Received = a series of 15 annual payments of $70, the first coming in 1 year.  This can be brought to the reference date using the formula for the present value of an ordinary annuity. PLUS Received = A single payment of $1000, made 15 years in the future.  This can be brought to the reference date using the simple interest formula. Set the pay-out equal to the present value of the payments received and solve for X I am unaware of the difference, if any, between ""current rate"" and ""rate to maturity"" Finding the rate for such a series of payments would start out the same as above, but solving the resulting equation for the interest rate would be a daunting task..."
The reason the market value is low is because the market does not believe that the company or country will pay. Another reason for it to go down is lack of liquidity in the market. However if you believe that the conditions would improve by the time bond matures, and you don't need money right now, then you can wait for maturity and get the maturity value. "Compounding is just the notion that the current period's growth (or loss) becomes the next period's principal. So, applied to stocks, your beginning value, plus growth (or loss) in value, plus any dividends, becomes the beginning value for the next period. Your value is compounded as you measure the performance of the investment over time. Dividends do not participate in the compounding unless you reinvest them.  Compound interest is just the principle of compounding applied to an amount owed, either by you, or to you. You have a balance with which a certain percentage is calculated each period and is added to the balance. The new balance is used to calculate the next period's interest, which again adds to the balance, etc. Obviously, it's better to be on the receiving end of a compound interest calculation than on the paying end. Interest bearing investments, like bonds, pay simple interest. Like stock dividends, you would have to invest the interest in something else in order to get a compounding effect. When using a basic calculator tool for stocks, you would include the expected average annual growth rate plus the expected annual dividend rate as your ""interest"" rate. For bonds you would use the coupon rate plus the expected rate of return on whatever you put the interest into as the ""interest"" rate. Factoring in risk, you would just have to pick a different rate for a simple calculator, or use a more complex tool that allows for more variables over time. Believe it or not, this is where you would start seeing all that calculus homework pay off!" Usually the market.  I'm a company issuing a 5-year bond with 5% coupon payments. It goes on the market to whoever is willing to pay the most for it. The prices that those investors pay implies what the required yield is. For instance, if they're willing to pay exactly face value for the bond, then that shows they have a required return of (in this case) 5%. Paying more or less for the bond implies a require rate less than or greater than 5%, with the exact amounts derivable with basic algebra.  The same principle can be applied to any other asset. "Below I will try to explain two most common Binomial Option Pricing Models (BOPM) used. First of all, BOPM splits time to expiry into N equal sub-periods and assumes that in each period the underlying security price may rise or fall by a known proportion, so the value of an option in any sub-period is a function of its possible values in the following sub period. Therefore the current value of an option is found by working backwards from expiry date through sub-periods to current time. There is not enough information in the question from your textbook so we may assume that what you are asked to do is to find a value of a call option using just a Single Period BOPM. Here are two ways of doing this: First of all let's summarize your information: Current Share Price (Vs) = $70 Strike or exercise price (X) = $60 Risk-free rate (r) = 5.5% or 0.055 Time to maturity (t) = 12 months Downward movement in share price for the period (d) = $65 / $70 = 0.928571429 Upward movement in share price for the period (u) = 1/d = 1/0.928571429 = 1.076923077 ""u"" can be translated to $ multiplying by Vs => 1.076923077 * $70 = $75.38 which is the maximum probable share price in 12 months time. If you need more clarification here - the minimum and maximum future share prices are calculated from stocks past volatility which is a measure of risk. But because your textbook question does not seem to be asking this - you probably don't have to bother too much about it yet. Intrinsic Value: Just in case someone reading this is unclear - the Value of an option on maturity is the difference between the exercise (strike) price and the value of a share at the time of the option maturity. This is also called an intrinsic value.  Note that American Option can be exercised prior to it's maturity in this case the intrinsic value it simply the diference between strike price and the underlying share price at the time of an exercise. But the Value of an option at period 0 (also called option price) is a price you would normally pay in order to buy it. So, say, with a strike of $60 and Share Price of $70 the intrinsic value is $10, whereas if Share Price was $50 the intrinsic value would be $0. The option price or the value of a call option in both cases would be fixed.  So we also need to find intrinsic option values when price falls to the lowest probable and rises to the maximum probable (Vcd and Vcu respectively) (Vcd) = $65-$60 = $5 (remember if Strike was $70 then Vcd would be $0 because nobody would exercise an option that is out of the money) (Vcu) = $75.38-$60 = $15.38 1. Setting up a hedge ratio:  h = Vs*(u-d)/(Vcu-Vcd) h = 70*(1.076923077-0.928571429)/(15.38-5) = 1 That means we have to write (sell) 1 option for each share purchased in order to hedge the risks. You can make a simple calculation to check this, but I'm not going to go into too much detail here as the equestion is not about hedging. Because this position is risk-free in equilibrium it should pay a risk-free rate (5.5%). Then, the formula to price an option (Vc) using the hedging approach is: (Vs-hVc)(e^(rt))=(Vsu-hVcu) Where (Vc) is the value of the call option, (h) is the hedge ratio, (Vs) - Current Share Price, (Vsu) - highest probable share price, (r) - risk-free rate, (t) - time in years, (Vcu) - value of a call option on maturity at the highest probable share price. Therefore solving for (Vc): (70-1*Vc)(e^(0.055*(12/12))) = (75.38-1*15.38) => (70-Vc)*1.056540615 = 60 => 70-Vc = 60/1.056540615 => Vc = 70 - (60/1.056540615)  Which is similar to the formula given in your textbook, so I must assume that using 1+r would be simply a very close approximation of the formula above. Then it is easy to find that Vc = 13.2108911402 ~ $13.21 2. Risk-neutral valuation: Another way to calculate (Vc) is using a risk-neutral approach. We first introduce a variable (p) which is a risk-neutral probability of an increase in share price. p = (e^(r*t)-d)/(u-d) so in your case: p = (1.056540615-0.928571429)/(1.076923077-0.928571429) = 0.862607107 Therefore using (p) the (Vc) would be equal: Vc = [pVcu+(1-p)Vcd]/(e^(rt)) => Vc = [(0.862607107*15.38)+(0.137392893*5)]/1.056540615 => Vc = 13.2071229185 ~ $13.21 As you can see it is very close to the hedging approach. I hope this answers your questions. Also bear in mind that there is much more to the option pricing than this. The most important topics to cover are: Multi-period BOPM Accounting for Dividends Black-Scholes-Merton Option Pricing Model" what do you mean exactly?  Do you have a future target price and projected future dividend payments and you want the present value (time discounted price) of those? Edit: The DCF formula is difficult to use for stocks because the future price is unknown.  It is more applicable to fixed-income instruments like coupon bonds.  You could use it but you need to predict / speculate a future price for the stock. You are better off using the standard stock analysis stuff: Learn Stock Basics - How To Read A Stock Table/Quote The P/E ratio and the Dividend yield are the two most important.  The good P/E ratio for a mature company would be around 20.  For smaller and growing companies, a higher P/E ratio is acceptable.  The dividend yield is important because it tells you how much your shares grow even if the stock price stays unchanged for the year. HTH A bond has a duration that can be easily calculated. It's the time weighted average of all the payments you'll receive and helpful to understand the effect a change in rates will have on that instrument.  The duration of a stock, on the other hand, is a forced construct to then use in other equations to help calculate, say, the summation of a dividend stream. I can calculate the duration of a bond and come up with an answer that's not up for discussion or dispute. The duration of a stock, on the other hand, isn't such a number. Will J&J last 50 more years? Will Apple? Who knows? "Thanks!  Do you know how to calculate the coefficients from this part?:  ""The difference between the one-year rate and the spread coefficients represents the response to a change in the one-year rate. As a result, the coefficient on the one-year rate and the difference in the coefficients on the one-year rate and spread should be positive if community banks, on average, are asset sensitive and negative if they are liability sensitive. The coefficient on the spread should be positive because an increase in long-term rates should increase net interest income for both asset-sensitive and liability-sensitive banks.""  The one-year treasury yield is 1.38% and the ten-year rate is 2.30%.  I would greatly appreciate it if you have the time!" But it also can't be 1.46%, because that would imply that a 30Y US Treasury bond only yields 2.78%, which is nonsensically low. The rates are displayed as of Today. As the footnote suggests these are to be read with Maturities. A Treasury with 1 year Maturity is at 1.162% and a Treasury with 30Y Maturity is at 2.78%. Generally Bonds with longer maturity terms give better yields than bonds of shorter duration. This indicates the belief that in long term the outlook is positive. Hi guys, I have a difficult university finance question that’s really been stressing me out....  “The amount borrowed is $300 million and the term of the debt credit facility is six years from today  The facility requires minimum loan repayments of $9 million in each financial year except for the first year.  The nominal rate for this form of debt is 5%. This intestest rate is compounded monthly and is fixed from the date the facility was initiated.  Assume that a debt repayment of $10 million is payed on 31 August 2018 and $9million on April 30 2019. Following on monthly repayments of $9 million at the end of each month from May 31 2019 to June 30 2021.  Given this information determine the outstanding value of the debt credit facility on the maturity date.”  Can anyone help me out with the answer? I’ve been wracking my brain trying to decide if I treat it as a bond or a bill.  Thanks in advance, "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." The periodic rate (here, the interest charged per month), as you would enter into a finance calculator is 9.05%. Multiply by 12 to get 108.6% or calculate APR at 182.8%. Either way it's far more than 68%.  If the $1680 were paid after 365 days, it would be simple interest of 68%. For the fact that payment are made along the way, the numbers change.  Edit - A finance calculator has 5 buttons to cover the calculations:  N = number of periods or payments %i = the interest per period PV = present value PMT = Payment per period FV= Future value In your example, you've given us the number of periods, 12, present value, $1000, future value, 0, and payment, $140. The calculator tells me this is a monthly rate of 9%. As Dilip noted, you can compound as you wish, depending on what you are looking for, but the 9% isn't an opinion, it's the math. TI BA-35 Solar. Discontinued, but available on eBay. Worth every cent.  Per mhoran's comment, I'll add the spreadsheet version.  I literally copied and pasted his text into a open cell, and after entering the cell shows,  which I rounded to 9.05%. Note, the $1000 is negative, it starts as an amount owed.  And for Dilip - 1.0905^12 = 2.8281 or 182.8% effective rate. If I am the loanshark lending this money, charging 9% per month, my $1000 investment returns $2828 by the end of the year, assuming, of course, that the payment is reinvested immediately. The 108 >> 182 seems disturbing, but for lower numbers, even 12% per year, the monthly compounding only results in 12.68% The calculation can be made on the basis that the loan is equal to the sum of the repayments discounted to present value.  (For more information see Calculating the Present Value of an Ordinary Annuity.) With Deriving the loan formula from the simple discount summation.  As you can see, this is the same as the loan formula given here. In the UK and Europe APR is usually quoted as the effective interest rate while in the US it is quoted as a nominal rate.  (Also, in the US the effective APR is usually called the annual percentage yield, APY, not APR.) Using the effective interest rate finds the expected answer. The total repayment is £30.78 * n = £1108.08 Using a nominal interest rate does not give the expected answer. "When we calculate the realized yield of a bond, we assume the coupons are invested at an interest rate. I assume it is some kind of vehicle that guarantees a return, thinking it is government bond, savings account or something. Investing in a benchmark bond index might be risky though for this ""interest rate""." "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." Adding a couple more assumptions, I'd compute about $18.23 would be that pay out in 2018.  This is computed by taking the Current Portfolio's Holdings par values and dividing by the outstanding shares(92987/5100 for those wanting specific figures used). Now, for those assumptions: Something to keep in mind is that bonds can valued higher than their face value if the coupon is higher than other issues given the same risk.  If you have 2 bonds maturing in 3 years of the same face value and same risk categories though one is paying 5% and the other is paying 10% then it may be that the 5% sells at a discount to bring the yield up some while the other sells at a premium to bring the yield down.  Thus, you could have bonds worth more before they mature that will eventually lose this capital appreciation.