By repeating the same consistent design method, the quality and reliability of models improves and this reduces development and audit time. Companies and individuals face the problem of investing today in the hope of some financial gain or deferring expenditure. There is a risk in investing so companies expect a return for these risks. Similarly, investors require some extra return for investing in risky assets. Money today is more certain than the promise of funds in the future.
They can be said to be risk averse and require some sort of gain to give up the certainty of money today. This may not be significant in Europe at present, however, there have been recent periods where infla- tion has been in excess of 10 per cent. In theory, the risk premium should reflect the timescale and relative risk reflected such that the decision to delay pro- vides a greater gain.
Ratings agencies grade institutions and countries according to the perceived risk they represent. These three factors underpin the concepts of time value of money and dis- counted cash flows, which drive the financial concepts in this book. This method is calculated only on the principal and does not take into account the fact that interest can be calculated on interest. Nevertheless, simple interest is usually used for periods of less than a year and there are an array of conventions for bills, deposits, bonds, etc.
Figure 2. Three conventions show the number of days and the standardized number of days in the year. The inter- est, future value and present value are calculated using the formulas above. Example 2 Figure 2. The addition of the standard number of days in the year results in three different levels of interest based on the simple rate of 10 per cent.
Simple interest could be used for calculating loan payments and is fre- quently used for hire purchase US - dollar out or lease purchase contracts. These are often structured as transactions with deposits and the balance of payments over a period.
The annual inter- est is 90, at 10 per cent so the total interest payable is , The total payable is 1,, after the deposit, which can be divided by 60 monthly payments. The monthly rental is 22, and the total payable with the deposit 1,, Notice that the total payable does not vary as you select monthly, quarterly, semi-annual or annual payments.
This is because the method does not recognize the effect of time value of money. Cell C10 is validated using a data list in the range BB32 see Figure 2. Compounding means that interest is calculated not only on the initial investment but also on the interest of previous periods.
Time value of money calculations underpins many applications in finance such as investment analysis, bonds, options, etc. Typically this is a ten-year government bond as a proxy to a risk-free rate. In the previous section, the amount payable was the same irrespective of the payment period, however compounding would lead to different results due to the intervals between payments.
The terminology often used in these calculations is: N — number of periodic payments I — periodic interest PV — present value or capital value PMT — periodic payment FV — future value It is often useful to draw time lines of problems to understand the timing of cash flow or alternatively to draw a grid of the known and unknown parameters.
As with all financial modelling, the cash flow nota- tion is used: cash out is negative and cash in is positive. The answer is, of course, 1, The example spread- sheet uses both the formula and the PV function to derive the answer see Figure 2. It makes a difference whether you receive the payments at the beginning of the end of a period.
The model works out the periodic interest and the answer section uses condi- tional IF statements to show an answer solely for the zero variable in the inputs section. The model is set up to calculate any vari- able when four are input. For example, the payment is only 90 so what is the future value left unamortized over the period?
Twelve payments of The future value is calculated as As a further example, you could calculate the number of periods needed to write off 1, at This is not an integer number of payments with the answer as Therefore the calculator is flexible since the initial formula must hold.
If you change the payment and all other variables remain the same, solving for the number of payments will produce the number of payments required to write off the loan. This means that a rate of 3. The rate of return that is actually paid or charged depends on the number of compounding periods in the year and this is the effective or effective annual rate EAR. Adjustments have to be made for the number of com- pounding periods so the rate will vary between monthly, quarterly, semi-annual and annual payments.
The example below uses 14 per cent as a nominal rate and calculates the effective rates and back again. The example in Figure 2. The inputs are the rate and the number of periods and the function calculates the result without the need for a formula.
You can test the model by work- ing backwards and forwards. Annual effective and nominal rates will always be the same since there is only one period in the year. This con- stant is the inverse of LN, the natural logarithm of a number. Continuous discounting Figure 2. This method of compounding arises in option pricing and is an option on most financial calculators. The simple interest rate is calculated based on the charges and the true period.
This is 11 quarters in the first case and 36 months in the second case rentals are in arrears. The simple interest rate is higher for the quarterly trans- action since the periodicity is not taken into account. The effective rate is higher for the monthly option since there are 12 compounding periods against four. The loan repayment is 2, per month and the nominal rate 10 per cent. How much capital remains at the end of five years? Build a spreadsheet using the functions demonstrated in this chapter.
Where you have a forecast of future cash flows, you need to know the value today to compare the possible gain against the per- ceived risk. The models in this chapter use a simple grid for calculating either the value or the internal rate. The number of periods is five and on expiry the salvage is The cash flow today is not discounted. Net present value Figure 3. You select the interest rate and the outstanding cash flows. The answer here is 1, and then you add the cash flow today.
If you include all the cash flows, Excel assumes that the first cash flow is also to be discounted and will assume that the subsequent cash flows occur one period later. Figure 3. The net present value will vary depending on the discount rate: as the discount rate increases, the net present value falls. The schedule contains a simple data table to show the effect of interest rate changes see Figure 3. The data table is a built in sensitivity function accessed through Data Table on the menus see Figure 3.
The single variable grid has to be set out as above with variables across and the answer in the next row down on the left. This is the maximum percentage that could be afforded before the net present value becomes negative. Alternatively this is the rate at which the net present value is zero. The adjustment using the formula is 7. In this case, the interest rate guess is left blank since the default is 10 per cent.
This is simple cash flow with only one crossing from positive to negative cash flows and therefore there can be only one solution. Where there are multi- ple sets of positive and negative cash flows, there can be more than one potential result. An alternative method uses a chart to plot the interest rates and the net present values see Figure 3.
The net present value is zero at a rate of The data table shows the results at rates above and below the correct answer. As input with the chart series reversed, the known Y series is the interest rates and the known X series the net present values. This means that there are multiple solutions as shown by the sensitivity table.
The trend line crosses twice, meaning that IRR results from these figures are unreliable with two possible answers. It would be better to use a known discount rate and discount the cash flows to a net present value. This would be advantageous if you wanted to rank or compare results from two sets of cash flows.
The internal rate also assumes that any cash receipts are invested at the IRR rate which may not be possible in times of falling interest rates and a sinking fund method might be needed to account for the funds received. The standard functions assume that each period is the same length, which is rarely the case.
Annual payments have leap years whereas monthly periods are punctuated with months of 28, 29, 30 and 31 days. The example in Figure 3. Note that these functions are in the Analysis ToolPak and you have to install this add-in as in the installation instructions. If you do not, then you will see errors on the schedule. The answers on the standard functions are Answers using the day-to-day functions are normally lower than the stan- dard functions. This is a loan or rental of The XIRR function yields Similarly, the XNPV function using the effective rate produces a net present value close to the present value of 1, Note that you include all the cash flows including the opening cash flow on the XNPV function in contrast to the NPV function where you only include the outstanding cash flows.
This uses a separate finance and reinvestment rate in order to deal with multiple positive and negative cash flows. Using the same data as the Multiple IRR sheet with a finance rate of 10 per cent and a reinvestment rate of 5 per cent the answer is 4. As with net present values projects or loans can be ranked based on this measure.
For example, leveraged leases need to be measured using this more advanced internal rate function, since you are not sure that rentals received can be reinvested at the same rate as the initial lease. MIRR function Figure 3. The highest rate is at the bottom right of the table whereas the lowest MIRR is at the top left-hand corner. Using these functions you can assess the value or the return on series of cash flows. The period or maturity is typically five to fifteen years and perhaps longer for governmen- tal bodies.
The upper limit is set by what investors will accept rather than any rules. The mathematics for bonds concerns pricing, the yield and vari- ous measures of risk. The majority of bonds are issued on a fixed-rate basis but a floating rate is also possible. This is borrowing from the capital markets and dealing directly without the intervention of a bank. A bond is a form of security which means that it is transferable and the issuer needs to keep a register of the owners. The borrower pays interest in the form of coupons and the final principal to the registered holder.
Therefore the bond in its simplest form consists of a series of cash flows, which can be valued using the building blocks from the last two chapters. The price of the bond is the present value of all the cash flows coupons and principal cal- culated using normal discounted cash flow techniques. There is an inverse relationship between the price and discount rate since the present value goes down as the discount rises.
It therefore follows that the price of the bond falls as interest rates rise. Money market instruments use the exact number of days for simple interest rate calculations. If the pricing is required on the date a coupon is due, then there are no problems. The price is simply the present value of the coupons and princi- pal.
Between periods, a seller expects to receive the accrued coupon within the period, while the buyer will only pay the present value of the future payments. Interest on the coupon is payable using simple interest calculations. The first period could be less than the coupon periods depending on the purchase date, but thereafter coupons are payable annu- ally, semi-annually or sometimes quarterly.
The dates are the same, for example 17 January and 17 July for a semi-annual bond, and are not based on the exact number of days. Day and year conventions vary and they are used in the various Excel functions. The methods are the number of days in the month and days in the year. The sheet called Price sets out the flows for an example bond.
The price is calculated using a yield of 10 per cent and this is a simple net present value function. The interest rate is divided by the number of coupons per annum since the function requires a periodic interest rate. The cash flows are nine periods of 50 Figure 4. The periodic nominal rate is ten per cent divided by 2 or 5 per cent. Note that the principal of is repayable with the last coupon and the interest payments occur at the end of each period. This can be proved with the TVM calculator from Chapter 2 which confirms the value of the cash flows Figure 4.
TVM calculator Figure 4. Such options protect the issuer from paying too much if the option is exercised. The discount rate in Figure 4. In reality the rate should be the combination of a risk-free rate such as a ten- year government bond plus a premium for perceived risk borne by the investor. It follows therefore that the bond must be priced at a deep discount to compensate for the lack of coupons during the lending period.
The zero coupon element reduces some uncertainty since there are no coupons to reinvest and the final prin- cipal is known. The example in Figure 4. Figure 4. It fol- lows that you can perform the same calculations the other way around and you can derive the yield from the price see Figure 4. These measures are similar to the net present value and the internal rate of return in discount- ing. Since the cash flows for an option-free bond are fixed it follows that any changes to yield will be reflected in the price of the bond.
The model finds the internal rate of return in the cash flows as 10 per cent. Notice that the effective rate is Yield Figure 4. This confirms a price of 1, and a yield of 10 per cent. The bond is callable after two and a half years or five coupons. The price offered is 1, which equates to an internal rate of This is confirmed by the IRR in cell H Rather it is slightly curved or convex.
Higher prices are linked to lower yields. As the bond yield increases, prices fall at a decreasing rate. Alternatively, as bond yields fall prices rise at a decreasing rate. This is positive convexity which means that bond prices go up faster than they decrease. The table in lines 22 to 25 demon- strates the yield price matrix. Line 24 shows the difference to a yield of 10 per cent while line 25 provides the difference from one data point to the next on the right. The chart at the bottom plots line 23 against line 24 as a curve.
Price yield relationship Figure 4. In prac- tice, the yield curve represents the rates on a zero coupon bond plus a premium for the perceived credit risk on reinvestment. The first part of the formula checks that the period number is less than the total number of periods. The answer here is 9. Note that the RATE function produces the periodic interest rate and this has to be multiplied by two as a semi-annual bond. The bond functions sheet demonstrates the func- tions. The example is brought forward from previous sections but now the settlement and maturity dates do not fall into exact periods.
The settlement date is 1st April The model calculates the clean price as There have been 91 coupon days based on a day year since the last coupon date. Therefore half of one coupon is accrued and added to form the dirty price of Yield measures therefore vary given the uncertainty in the reinvestment income.
The method ignores the time value of money and therefore it cannot be used for comparing bonds with differing maturity dates and coupon periods. Also it ignores any capital gain or loss arising from the difference between the amount paid and the principal received on expiry. There are further inputs for the days convention and the number of coupons per annum below the visible entries.
With prices below par, the order of values is current yield, yield to maturity and adjusted coupon yield. Above par, this order is reversed. Calculate the prices and using a data table in Excel check the relevant prices if the yield falls or rises by 1 per cent. This should confirm the bond which is more responsive to changes in yield. Period 6. Using the basic building blocks, you can find the price or the yield by setting out and analysing the relevant cash flows.
Excel also has a number of built-in functions in the Analysis ToolPak for simplifying the calculations. The next chapter reviews bond risks and sets out the main calculations for understanding the responsiveness of bonds to changes in interest rates. Since the value of investments can vary due to a number of factors and the value of the coupon payments are normally fixed, investors need to understand the potential risks before investment.
Risk factors can also change during the investment period so investors need to make decisions on whether to hold, reduce or increase the investment. Interest rates Since bonds consist of a series of cash flows, their value reduces as interest rates rise. If market interest rates increase coupons on existing bonds will be worth less. Since lower coupons will be less attractive to investors then their value has to drop to a market value in a freely traded market. The opposite is true as interest rates fall as the relationship is inverse.
When market yields fall below the coupon rate, the bond price will increase above par value and therefore it will trade at premium prices. When the yield and coupon rate are the same the bond will trade at par value. It follows therefore that the bond will trade at a discount when market yields rise above coupon rates. The direction of the change in yields and the size of the change has an effect on bond prices.
In reality, reinvestment rates reduce when yields decline. This means that bonds with high coupon rates are risky since the investor cannot be sure of the reinvestment rate. The value of the coupons can be eroded by inflation and uncertainty.
Investors may therefore wish to balance reinvestment against price risk. Yield curve risk The relationship between the term to maturity and yield to maturity is known as the yield curve. The term nature of interest rates mean that the rate could change at each coupon date.
Yield curves could be flat, rising or falling and this affects the value of the fixed bond cash flows. Prepayment and call risk Embedded options affect pricing as the uncertainty makes the future cash flows harder to predict. Since callable bonds can be redeemed before maturity, the bond holder can lose potential gains with early redemption.
There are effectively two elements: the value of the bond without any termination rights less the value of the embedded call option. Where yields fall the call option becomes more valuable to the issuer and as the yield rises towards the coupon rate, callable bonds approach the call price and do not rise any further.
Credit risk defined as default and downgrading This can be defined as default risk, credit spread or downgrade risk. Default risk means that the issuer becomes unable to meet its obligations such as a failure to pay regular interest rates or a breach of other terms. Since investors need to be rewarded for accepting increased risks, rational investors should receive an increased return. This increases the required yield and therefore decreases the value of the bond.
The ratings agency may of course assign an increased rating and therefore the opposite would be true. At the lower end of the scale, there is a higher probability of default within a defined time period. Grading explanation Figure 5. Dealers in bonds post a bidding and asking price and as liquidity falls the spread between the two prices increases. Liquidity can of course vary over the life of a bond.
This is in effect a transaction cost and must be considered part of the cost of holding the bond. Exchange rates If the coupon and principal payments are denominated in a foreign cur- rency then the cash flows may be worth more or less when translated into the home currency.
If the home currency appreciates against the foreign currency then the payments will be worth less and vice versa. If the coupon is 10 per cent and inflation is 4 per cent, then the real return is only 6 per cent. This loss of value has to be fac- tored into the evaluation of returns.
Macro and exterior risk There are a number of exterior events which cannot be controlled or per- haps foreseen at the time of investment. For corporate bonds non-control- lable risks can include corporate restructuring such as buyouts, mergers and disposals. The simple maturity of a bond is not a suitable indicator for a bond since the cash flows occur during the period to and at maturity.
A bond with a longer maturity may be more risky since an investor is exposed to changes in yield rates for a longer period. Therefore this measure provides a measure of risk in the future cash flows which can be used to compare different bonds. Alternative definitions are the weighted average maturity or the present value weighted number of years to maturity. Yield, maturity date and the coupon rate affect duration.
Duration decreases with an increasing yield as distant cash flows become worth less. Normally duration increases with a longer maturity date whereas a low coupon bond will have a higher duration than a high coupon bond. Bond example Figure 5. The present value of the cash flow is multiplied by its period number and added.
The sum of these cash flows is then divided by the price. The results in cells G5 and G6 are the same. Duration can be applied to any groups of cash flows to find the average maturity. If you have cash flows received and paid out, you can immunize the cash flows by obtaining cash flows with equal duration.
At some point between the date and maturity, the loss of interest returns and the capital gain from a higher bond price balance or cancel each other out. There is a simple formula on the sheet, which calculates the price movement based on a 1 per cent yield change see Figure 5. The effect of convexity is discussed in the next section. The formula equates approximately to the slope to the change in yield against the price.
CI47 There is a further variant of duration called the modified duration also called volatility on the Model sheet. The calcu- lated change varies from the actual change due to the curvature or convexity of the relationship.
The actual change depends on the amount of curvature and this is known as convexity. There are a number of different formulas for computing the convexity and these are set out on the Model sheet. The formula calculates the price move- ment based on a 1 per cent yield change. The coupon rate is the periodic rate rather than the annual rate. C44 This is derived as —3. The answer is revised price 1 , Again this assumes a linear relationship and will become progressively inaccurate.
The bond therefore is not expected to reduce as much as the simple linear formula. The convexity means nothing on its own but can be useful for comparison purposes since a higher figure means more price volatility than a lower figure. The workings are on a data table at the bottom of the Model sheet see Figure 5.
Formula five The model see Figure 5. The duration is the period number multiplied by the weighting. The convexity is Formula six The schedule also includes data table workings for comparison with the calculated values and this is Figure 5.
This can then be multi- plied out against the existing price. Column I uses formula five to obtain the percentages and these are added in column J. You can see that the dif- ferences are small close to the current yield to maturity, but, due to convexity, become more pronounced as you move further and further from the existing price see Figure 5.
Summary table Figure 5. The first plots the actual and pre- dicted changes in amounts from formulas two and five see Figure 5. The convexity formula five rather than the duration-based formula is more accurate in tracking the actual price changes. The periodic convexity fol- lows the actual change closely. The duration-based formula becomes progressively less accurate away from the current yield of 10 per cent.
By contrast, formula four based on the convexity cash flows remains more accurate further from the current yield. Interest rates are cur- rently 8 per cent and a company wishes to invest an amount in bonds, which will grow to , at the time of maturity.
Two bonds are avail- able see Figure 5. These include risks from reinvestment risk, changes in the yield curve, prepayment and call risk, credit, liquidity, exchange rates, inflation and exterior factors. Since the calculation needs to include the convexity of the price yield relationship, further measures of convexity are needed to predict the changed price more accurately.
Several measures of convexity are pre- sented in this chapter along with duration. Other terms are floating rate notes, floating rate certificates of deposit or variable rate notes. Whilst this is more complex than fixed coupons, floating rates are generally advantageous for lenders when interest rates are rising.
The British Bankers Association is the most widely used benchmark or reference rate for short-term interest rates and is the rate of interest at which banks borrow funds from other banks, in marketable size, in the London interbank market. A floater is a fixed income instrument whose coupon fluctuates with some designated reference rate.
A floating rate note FRN is a floater issued by a corporate or agency borrower. Typically, FRNs have maturities of about five years. The three-month or six-month LIBOR are two com- monly-used reference rates, as are Treasury bill yields, the prime rate or the Federal funds rate. Collateralized mortgage obligations CMOs are also sometimes structured to have floating rate coupons. For FRNs, the coupon rate is usually reset each time interest is paid.
A typical arrangement might be to pay interest at the end of each quarter based on the value of three-month LIBOR at the start of the quarter. One feature that can affect the spread is a provision that places a cap or floor on the floating coupon rate.
For example, an FRN might be issued with a cap of 7. Calculated for a one-year horizon, this may be termed the expected default frequency. Note that every risk comprises two elements: exposure and uncertainty. For credit risk, credit exposure represents the former, and credit quality represents the latter. To investors, holding an FRN is similar to investing in money market instruments and continuously reinvesting as those instruments mature.
FRNs tend to have stable market values. If the floating rate is reset with each coupon payment as is typically done the duration of an FRN is simply the time until the next interest payment. The most active issuers of bonds today are governments and government agencies government bonds , banks and corporations corpo- rate bonds. Face value Face value is the amount that is to be paid to an investor at the maturity date of a bond. Bonds can be issued at different face values, however float- ing rate securities typically have a unit face value of Interest coupon The coupon represents an interest payment paid at regular intervals by the issuer to owners of interest rate securities.
The coupon rate is the interest rate paid to investors during the life of the bond and is set when the issuer first sells the securities into the market. A floating rate note has a coupon that varies in line with a benchmark rate, usually at a margin above the bank-bill rate, and is different at each payment date. Since the amount of coupon interest is known in advance, its accumulation is spread over the relevant period. This is referred to as the daily accrued interest.
This con- trasts with a share dividend which is only known shortly before it is paid. Floating rate notes nor- mally pay interest quarterly. Yield The yield is the return an investor receives on a bond. The yield is based on the price paid by an investor for a bond and the payments coupons received if the bond is held to maturity. The most important types of yield are the nominal yield and the yield to maturity. Maturity date The final coupon and the face value of a bond is repaid to the investor on its maturity date.
The time to maturity can vary greatly, although it is typi- cally between two and twenty years. Purchase price The price of a bond is stated as a percentage of its face value. For example, a price of means per cent of face value; a price of The purchase price also known as the gross price is the total amount that an investor pays for a bond. The total purchase price comprises the number of bonds that an investor buys times the price paid for a bond.
Because interest is paid at regular intervals the bond price increases daily by the amount of inter- est accruing. Immediately following the coupon payment the price should fall by the amount of that coupon payment. Since floating rate securities are linked to an index, an investor is con- cerned with the margin above or below the index.
A further position is the implied coupon date price, which is a price-based evaluation against the price in the market together with the cost of carrying the position. This is a way of marking the floating investment to market. Effective margin This is the total marginal return over the index to maturity and comprises a combination of margin negative or positive over the index plus capital growth or depreciation see Figure 6.
This result can be compared with similar bonds as a basis for evaluation since the variables for the calcu- lation are the life, margin over the index, price and par value. The effective margin is increased by a larger margin over LIBOR and conversely decreased by a smaller margin.
The main factor is the price since the margin increases as the price falls Figure 6. Effect of a fall in price Figure 6. This simple method does not take into account the present value of future cash flows in the same way as the simple yield on a fixed bond. Using standard bond calculations you can calculate the bond as a fixed bond at par and then compare the yield on the variable bond. Column C in Figure 6. Column D evaluates an 8.
The coupon is therefore 4. The effective margin is therefore 8. This figure can be used for evaluating a potential investment where the key factors are price, margin, period and index value. In other words this is the difference between the cost of the investment and the benefit derived.
Figure 6. Current marginal income Figure 6. It is one method of marking the instrument to market the current holding in a specific security. Implied coupon date price Figure 6. The Excel sheet breaks the calculation down into three stages: A. LIBOR could be different from the current rate depending when the rate was set. This could arise if the floating rate security was purchased between coupons and LIBOR has moved since the last coupon was reset.
The bond without the coupons is a zero coupon bond. The arbitrage possibility arises from the difference from reselling each of the cash flows separately and the bond itself. The individual profits will depend on the yield curve from selling a one, two, three, etc. The first part of the spreadsheet confirms the present value of the cash flows at The second part of the schedule shows the input yields and in line 25 the calculations for the individual discount factors.
The stripped cash flows add up to The difficulty of realizing a profit depends primarily on the size of the coupon values to make it worthwhile in terms of time and effort. Nevertheless the difference in yields could give rise to a potential profit. The model see Figure 6. Exercise answer detail Figure 6. Relevant yield measures can be calculated such as the effective margin, current marginal income and the implied coupon date price.
As a further example of discounting mathematics, coupon stripping shows a potential profit based on the dif- ference in interest rates and the yield curve. Amortization is the reduction of the value of an asset by prorating its initial cost over a number of periods.
You can calculate the payment required under an annuity and then you need to split each payment into capital and interest. Due to the compounding nature of the time value of money, more capital is outstanding in the earlier periods and therefore more interest is payable on the outstanding balance. As the loan progresses less interest is payable on the notional balance outstanding and more capi- tal can be repaid in each period. Typical applications include house mortgages, bank loans or leases.
To solve these problems, you need to con- struct a cash flow grid with the period number, rental payment, interest paid, capital repayment and balance outstanding. The example below uses a capital value of , lent at a nominal rate of 8 per cent over three years with quarterly rentals.
One rental is due on signing. Since there is no final rental, this results in a rental of 9, The amount lent is , less the first rental which is 90, To calculate the interest for the first period, you multiply the capital outstand- ing by the periodic interest rate. The first calculation is: 90, The capital of 7, The process is then repeated with the period two interest calculated from the capital outstanding at the end of period one and so on.
Progressing through the 12 periods means that the interest per period declines while the capital repayable grows. In the final period, the capital outstanding reduces to zero. The checks are that the total interest equals the total charges and the capital repayments add up to the original capital. As seen in Figure 7. There are also functions in Excel for calculating directly the interest or the capital for any period. The functions take the same arguments as the PMT and PV procedures with the addition of the current period number.
Figure 7. As an example, Figure 7. There is a final payment or residual value of 20, The same calculations are used to multiply out the outstanding capital by the periodic interest rate. Since there is a rental due on signing, the first capital value is the net advance of 98, Ten-year amortization Figure 7.
The cash flow is payments and grouped in years in order to show the summary. Instead of the capital reducing it rises due to the lack of repayments, since the outstanding balance has to be notionally financed. Whilst you could use Solver or Goal Seek, this procedure will compute a rental or payment directly.
This is B. The next stage is to present value the final payment of 10, that is due in months see Figure 7. The TVM calculator from Chapter 2 can be used to find the answer of 7, PV of future value Figure 7. This then needs to be reduced by a further 12 months. The basis is the , advance followed by the eleven payments of zero, followed by payments of 1, Since the IRR is an iterative formula, a guess of the input yield is used in the formula.
With a grid of the rentals and cash flow in place the amortization is multiplied out. Whilst the capital outstanding increases during the initial 12 months it is paid off during the rental period such that the capital is completely recovered by the expiry rental. The method allocates the interest and for loans or leases with regular structures will pro- vide a charges figure in line with the capital outstanding.
The example below shows a quarter transaction with 11 quarters outstanding on commencement. The charges are the sum of the factors which is 66 and the charges are 11, As in the amortization table, the capi- tal drops out as the payment minus the interest. Again the check is the totals at the bottom which must come back to the capital value, charges and total payable. Excel has a function called SYD which saves having to calculate the factor. The inputs are cost, salvage value, total periods and current period and you will obtain the same answers as the manual calculations.
Cost is the amount you want to separate into periods such as the total interest payable to be spread over the outstanding periods. The simplest is straight line depreciation which divides the equipment less any salvage value by the number of periods and writes it off at an equal amount per period.
Whilst you can do this manually, Excel has a function called SL to perform this automatically. The fixed-declining balance method computes depreciation at a fixed rate. DB uses the following formulas to calculate depreciation for a period. This method results in a declining amount of depreciation per period that can be factored for the number of periods in the first tax year. The example in Figure 7. Both result in a final salvage value of 2, For example, plant machinery gener- ally attracts an allowance of 25 per cent per annum while land and buildings attract 4 per cent per annum.
Accounting depreciation is replaced with tax depreciation or capital allowances for the purposes of computing the tax payable. The curve of the depreciation is asymptotic in that it will mathematically never reach exactly zero. The method gives rise to a UK tax depreciation Figure 7. The short life column includes an IF statement to take in the previous balance if the period number is equal to the total number of periods.
The relevant function is DDB. Again the method needs a factor which is the rate at which the balance declines. If the factor is omitted, it is assumed to be two the double declining balance method. The double declining balance method derives depreciation at an acceler- ated rate. Depreciation is highest in the first period and decreases in successive periods.
You can change the factor if you do not want to use exactly the double declining balance method. The US tax system uses a method of double declining balance while allowing a switch to straight line when the straight line depreciation is higher than the double declining balance.
The VDB function can also be used for the double declining balance and there is a switch true or false to allow you to choose where to change or not. Column F calculates the straight line depreci- ation on the remaining balance to show why the function continues to choose the double declining balance. In the next year the balance is brought for- ward and multiplied by 25 per cent. In the last periods, the balance is written off using the straight line method see Figure 7.
The correct method is to fix the start dates and multiply out the number of periods from the start date. The last period contains the balance Figure 7. Again the function has a switch for selecting the formalized days in the month and year.
As you can see the choice of method exerts a significant effect on the speed of write-off of the underlying asset. Comparison Figure 7. The cash flow is: Six rentals of 1, on signing Six further monthly rentals of 1, Sixty further rentals of X starting in the next month after the initial rentals A final rental is payable on expiry of 20, Amortization and sum of digits can be used to split loan payments for accounting purposes.
There are a number of depreciation methods for accounting or tax purposes which use a variety of declining balance methods to deal effectively with remaining balances. Somebody could borrow fixed but may prefer floating and another party may borrow floating but prefer fixed. This may seem an odd thing for each party to undertake, but this chap- ter will show that there can be advantages to both parties in entering into swap arrangements. To show the size of the market since their introduction in the early s, swaps have expanded rapidly into a multi-trillion dollar market.
Total outstanding interest rate swaps currently exceed five trillion dollars in notional principal amount. US dollar denominated transactions account for roughly 50 per cent of all interest rate swaps outstanding. Interest rate swaps are agreements between two parties to exchange a vari- able interest rate payment for a fixed interest rate payment for a specific maturity on a notional amount of principal see Figure 8. The underly- ing transaction remains in place and the cash flows are swapped for a specific period.
No principal usually changes hand and the cash flows for interest are overlaid and netted. Interest rate swaps are generally priced at a specific fixed rate swap rate which counterbalances the implied forward interest of some floating rate index.
Like any commodity, a swap has a bid side and an offer side. Obviously, the bid side swap rate is lower than the offer side swap rate. The difference between the two is called the bid-offer spread. A swap spread is the margin above the underlying treasury where the swap rate is set.
A ten-year swap, bid at 7. Nevertheless, the capital markets do possess a considerable body of information about the relationship between interest rates and future periods of time and can price accordingly. In many countries, there is a large and liquid market in government interest bearing securities issued. These securities pay coupon interest on a periodic basis and are issued with a wide range of maturities. Principal is repaid only at maturity and at any given point in time the market values these securities based on current interest rates.
It is possible to plot a graph of the yields of such securities having regard to their varying maturities. This graph, discussed in the next chapter is known generally as a yield curve as the relationship between future interest rates and time. A chart showing the yield of securities displaying the same characteristics as government securities is known as the par coupon yield curve.
The classic example of a par coupon yield curve is the US Treasury yield curve. A different kind of security to a government security or similar interest bearing note is the zero coupon bond. The zero coupon bond does not pay interest at periodic intervals and is issued at a discount from its par or face value.
A graph of the internal rate of return IRR of zero coupon bonds over a range of maturities is known as the zero coupon yield curve and will be used in val- uation later in the chapter. Finally, at any time the market is prepared to quote an investor forward interest rates. The month forward deposit rate is a mathematically derived rate which reflects an arbitrage relationship between current or spot interest rates and forward interest rates. Therefore, the month for- ward interest rate will always be the precise rate of interest which eliminates any arbitrage profit.
The pricing picture is now complete. Since the floating rate payments due under the swap can be calculated as explained above, the fixed-rate payments will be of such an amount that when they are deducted from the floating rate payments and the net cash flow for each period is discounted at the appro- priate rate given by the zero coupon yield curve, the net present value of the swap will be zero.
It might also be noted that the actual fixed rate produced by the above calculation represents the par coupon rate payable for that maturity if the stream of fixed-rate payments due under the swap are viewed as being a hypothetical fixed-rate security. This could be proved by using the fixed-rate bond valuation techniques from previous chapters. Any up-front payments are generally made two days after the trade date.
Interest payments are made on the period end-date. Floating rates are usually set two days in advance of the period begin-date. These statements are fully consistent and describe the mechanics of swap transactions based on comparative advantage and information. The essen- tial reason is lowering the cost of borrowings as will be shown in the worked examples later in the chapter. Other theories provided from economics include the theory of comparative advantage and asymmetric information. The AA company therefore raises funds in the floating rate market where it has an advantage, an advantage which is also possessed by company BB in the alternative fixed-rate market.
The international capital markets are, however, fully mobile and companies can raise funds in different mar- kets. In the absence of barriers to capital flows, theory suggests that arbitrage would eliminate any comparative advantage that exists within such markets and therefore this theory cannot explain the continued exis- tence of the markets. Asymmetric information Whilst capital markets are thought to be efficient there may also exist cer- tain information asymmetries.
In this situation a company is able to exploit its information asymmetry by issuing short-term floating rate debt and to protect itself against future interest rate risk by swapping such floating rate debt into fixed-rate debt. As time passes, however, this will cease to be the case, since the shape of the yield curves used to price the swap initially will change over time. Assume, for example, that shortly after an interest rate swap has been completed there is an increase in forward interest rates: the forward yield curve rises.
Since the fixed-rate payments due under the swap are, by definition, fixed, this change in the prevailing interest rate environ- ment will affect future floating rate payments only since the market expects that the future floating rate payments due under the swap will be higher than those originally expected on inception.
This benefit will accrue to the fixed- rate payer under the swap and will represent a cost to the floating rate payer. If the new net cash flows due under the swap are computed and if these are discounted at the appropriate new zero coupon rate for each future period i. This demonstrates how the value of the swap to the floating rate payer has declined from zero to a negative amount.
The above example marks the interest rate swap to market. Alternatively, if the floating rate payer wishes to cancel the swap by entering into a reverse swap with a new counterparty for the remaining term of the original swap, the net present value figure represents the payment that the floating rate payer will have to make to the new counterparty in order for him to enter into a swap which precisely mirrors the terms and conditions of the original swap.
A swap is a notional principal contract and therefore no credit risk arises in respect of the principal unlike a loan. The periodic cash flows under a swap will, by definition, be smaller than the periodic cash flows due under a comparable loan. A would prefer a floating rates and B would prefer fixed.
The relative advantage is 8. The model builds up the cash flows to show the position of the parties at each stage see Figure 8. Initial inputs Figure 8. A pays fixed 7 per cent and receives 7. The full set of cash flows in Figure 8. The benefit is split 0. As the differential in the rates change the net benefit reduces or increases. As the company A fixed rate and company B floating increase the potential savings decline based on the bank offered rate see Figure 8.
The correct rates to use for each of the cash flows are the zero coupon rates. The problem is that the future floating rate cash flows are not yet known. The solution is to calculate the forward-forward interest rates for each cash flow interval and then to discount each cash flow at this rate.
The objective is to value the swap during period two. The model in Figure 8. The net position is the sum of the interest out and in for each period. The overall position is therefore Figure 8. The customer is able to lock in a specific exchange rate for the life of an asset or liability. The issuer is able to obtain lower rate funding in curren- cies where it perhaps has weaker market recognition. The swap provides access to funds in currencies where an end-user has already saturated the primary debt markets.
The currency swap market has evolved as an extension of forward cur- rency exchange contracts. US dollars account for more than 30 per cent of all currency swap notional amounts up from one-third in Japanese yen notional amounts to one-fifth of all transactions. No other currency constitutes more than 10 per cent of the market.
Cross currency swaps are generally priced at a specific rate or at some spread to a major index in both currencies. The exchange rate is set as part of the trade. Interest payments are made on the period end-date and floating rates are usually set two days in advance of the period date.
The main difference between a single currency and multi-currency swap is that the principal amounts may be exchanged on inception and must then be re-exchanged on termination. Show more World link World. Show more US link US. Show more Companies link Companies. Show more Markets link Markets. Show more Opinion link Opinion. Show more Personal Finance link Personal Finance.
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