Formula for calculating compound interest:
Where,
P = principal amount (initial investment)
r = annual nominal interest rate (as a decimal)
n = number of times the interest is compounded per year
t = number of years
A = amount after time t
Example: An amount of $1,500.00 is deposited in a bank paying an annual interest rate of 4.3%, compounded quarterly. Find the balance after 6 years.
A. Using the formula above, with P = 1500, r = 4.3/100 = 0.043, n = 4,
And t = 6:
So, the balance after 6 years is approximately $1,938.84.
Translating different compounding periods
Each time unpaid interest is compounded and added to the principal, the resulting principal is grossed up to equal P (1+i %).
A) You are told the interest rate is 8% per year, compounded quarterly. What is the equivalent effective annual rate?
The 8% is a nominal rate. It implies an effective quarterly interest rate of 8%/4 = 2%. Start with $100. At the end of one year it will have accumulated to:
$100 (1+ .02) (1+ .02) (1+ .02) (1+ .02) = $108.24
We know that $100 invested at 8.24% will give you $108.24 at year end. So the equivalent rate is 8.24%. Using a financial calculator or a table is simpler still. Using the Future Value of a currency function, input
PV = 100
n = 4
i = .02
Solve for FV = 108.24
B) You know the equivalent annual interest rate is 4%, but it will be compounded quarterly. You need to find the interest rate that will be applied each quarter.
$100 (1+ .009853) (1+ .009853) (1+ .009853) (1+ .009853) = $104
the mathematics to find the 0.9853% is discussed at time value of money, but using a financial calculator or table is easier. Input
PV = 100
n = 4
FV = 104
Solve for interest = 0.9853%
C) You sold your house for a 60% profit. What was the annual return? You owned the house for 4 years, paid $100,000 originally, and sold it for $160,000.
$100,000 (1+ .1247) (1+ .1247) (1+ .1247) (1+ .1247) = $160,000
Find the 12.47% annual rate the same way as B.) Above, using a financial calculator or table. Input
PV = 100,000
n = 4
FV = 160,000
Solve for interest = 12.47%
Simple interest via compound interest:-
Is = IC*n
(100 +1)ⁿ
R
Compound interest via simple interest:-
IC = Is * 100 + 1
N r
Main measures of financial risk:-
Trading in assets whole future outcomes are unnecessarily involves risk for the investor. The management of such risk is of fundamental concern for the operation of financial market. For example
- Financial regulator seek minimise the occurrence and impact of the collapse of financial institutions by placing restrictions on the types and sizes of permitted markets. Such as limits on short sales,
Value at risk:-
Value at Risk (VaR) is defined with respect to a specific portfolio of financial assets, at a specified probability and a specified time horizon. The probability that the mark to market loss on the portfolio over the time horizon is greater than VaR, assuming normal markets and no trading, is the specified probability level.
For example, if a portfolio of stocks has a one-day 5% VaR of $1 million, there is a 5%probability that the portfolio will decline in value by more than $1 million over the next day, assuming markets are normal and there is no trading. Such an event is termed a “VaR break.” The 10% Value at Risk of a normally distributed portfolio
VaR has five main uses in finance, risk management, risk measurement, financial control, financial reporting and computing regulatory capital. VaR is sometimes used in non-financial applications as well.
The confidence interval is the probability (or level of confidence) that the actual maximum loss experienced will not exceed the maximum expected loss (value-at-risk) generated by the model. For example, using a value-at-risk method, a bank estimates that the maximum potential loss on its financial instrument portfolio will be $10,000 for a 99% confidence interval. The 99% confidence interval indicates that the bank is 99% sure that the maximum expected loss on its financial instrument portfolio will not exceed $10,000.
The last factor used in determining the value-at-risk assessment is the holding period of the instrument or portfolio. This is the period over which the value-at-risk is to be measured. For illustration purposes, this paper will focus on a holding period of one day. However, value-at-risk calculations can also focus on longer holding periods. As the holding period increases, it becomes more difficult to estimate the value-at-risk because of the additional complexities that longer time periods have on the model.
Value-at-Risk Illustration -- single financial instrument
To illustrate the basic value at risk model, consider an entity that holds only one financial instrument, a Treasury bond position. The following information is known about the Treasury bond position.
The following model assesses the value at risk for a single financial instrument
VAR = Current Market Value * Confidence * sigma * sq. root of Holding-Period
Using the factors identified above, the calculation of value-at-risk for a 99% confidence interval is:
VAR = $15 Million * 2.33 * .0004 * sq. root of 1 = $13,980
This result indicates that the entity is 99% confident that they will not lose more than $13,980 from this individual Treasury bond position. Managers can use this information when assessing their firm's overall risk exposure and making resource allocation decisions
Present and Future Value of Annuities:-
Annuities differ from ordinary simple and compound interest problems in that payments are made on a regular basis. For example, monthly, quarterly, semi-annual or yearly payments. This topic illustrates future and present value of annuities using several examples. Also discussed are sinking funds: if a borrower makes periodic deposits that will produce a specified amount on a later specified date, then this borrower has established a sinking fund.
Definition (Future Value of an Ordinary Annuity) If R dollars is invested at the end of each period for n periods in an annuity that earns interest at a rate of per period, the future value of the ordinary annuity will be﴾
.
Example (Future Value of an Ordinary Annuity) someone qualifies to invest $5000 in an IRA each June 30 for the next 20 years. If they make these investments, and if the certificates pay 12%, compounded semi-annually, how much will they have at the end of 20 years?
We use the formula
With r = 5000, I =.12/2 = .06 and N = 2 (20) = 40 and so we have
If they did not invest their money they would only have $5000 * 40 =$200.000 instead of the $773,810.
Example (Payment for an Ordinary Annuity) what size payments must be put into an account at the end of each month to establish an ordinary annuity that has future value of $20,000 in 7 years, if the investment pays 7.3%, compounded monthly?
We use the formula
With s = 20,000, I = 0.o73/12 and N = 7(12) = 84 and so we have
If the payments are not invested then $183.13 * 84 = $15,382.9 is obtained which is not as good as the investment which obtains $20,000.
Definition (Sinking Fund) if a borrower makes periodic deposits that will produce a specified amount on a later specified date, and then this borrower has established a sinking fund.
Example (Sinking Fund) a small company establishes a sinking fund to discharge a debt of $30,000 due in 10 years by making semi-annual payments, the first due in 6 months. If the deposits are placed into an account that pays 6%, compounded semi-annual, what is the size of the deposits?
We use the formula
with s = 30,000 I = 0.06/2 =.03 and n = 2 (10) =20and so we have
Therefore, payments of $1,116.47 will discharge a debt of $30,000 even though $1116.47 * 20 = $22,329.4 So the point is to invest the money rather than paying the debt at once.
Definition (Present Value of an Ordinary Annuity) If a payment of dollars is to be made at the end of each period for periods from an account that earns interest at a rate of per period, then the account is an ordinary annuity, and the present value is
Example (Present Value of an Ordinary Annuity) Find the present value of an annuity that pays $500 at the end of each month for 3 years, if the interest rate is 6%, compounded monthly.
We use the formula
With r =500 I = 0.06/12 =0.005 and n = 3 (12) = 36 and so we have
If the 36 payments of $500 were not invested it would take
$500*36 = $ 18,000.
Example (Payments from an Ordinary Annuity) (a) If $1,000,000 is invested in an annuity that earns 5.8% compounded monthly, what size of payments will it provide at the end of each month for the next 30 years?
We use the formula
with A = 1000000, I = 0.058/12 and n = 30 (12) = 360 and so we have
Now the payments of $ 107,032.00 for 360 payments leads to
* 107, 032 * 360 = $38, 531, 520.
Example (Using Present and Future Values) Is it more economical to buy an automobile for $29,000 cash or to pay $8000 down and $3000 at the end of each quarter for 2 years, if money if worth 8% compounded quarterly?
The automobile can be bought now for $29,000 or can be bought for $8000 plus the present value of the investment. The present value is given by
The formula
Where r = 3000, I = 0.08 / 4 =0.02 and n=4(2) and so we have
= 21976.4 Thus the automobile can be bought for $29,000 or for $ 21,976.4 + $ 8000 = 29,976.4. Thus, it is cheaper to pay cash.
Exercises (Present and Future Values)
(1) $10000 is deposited for 10 years in an account paying 8% compounded quarterly. At the end of the 10 year period, I want to make 20 quarterly withdrawals. What is the size of each withdrawal?
We can find the future value of the first investment, using the formula
Where p = 1000, n = 10(4), and I = 0.08/4 = 0.02, so we have
to find the quarterly withdrawals, we use the formula
With
A= 22080.4, I = 0.08/4 = 0.02, and n = 20 and so we have
R = $1,350.36
Thus, R = $1,350.36 is the size of each withdrawal.
(2) $500 is deposited each six months for 5 years into an account paying 6% compounded semi annually. No more deposits are made but the account still earns the interest. How much is in the account 10 years after the last deposit?
To find the future value after 5 years, we use the formula with R = 500, I = 0.06 / 2 = 0.03, and n = 2(5) = 10 and so we have
We can find the future value of the second investment, using the formula
Where p = 5731.94, n = 10(2) = 20, and I = 0.06/2 = 0.03,
So we have
Thus, $ 10,352.5 is in the account 10 years after the last deposit?
(3) $2000 is deposited each year for 20 years into an IRA account paying 6% compounded annually. Then 20 annual withdrawals are made from the account. (a) How much is in the account just after the 20th deposit? (b) How much was deposited? (c) What is the size of each withdrawal? (d) How much is withdrawn?
For part (a) we find the future value of the annuity, we use the formula
with r = 2000, I 0.06, and n = 20, and so we have
Thus, $73,571.2 is in the account 20 years after the last deposit?
For part (b), we want to know how much was deposited. Since we made 20 deposits of 2000, we have 20 (2000) = $ 40,000
For part (c), to find the annual withdrawals, we use the formula
with A = 73571.2, I = 0.06 and n = 20 and so we have
R = $ 6, 414.27
Thus, R= $6,414.27 is the size of each withdrawal.
For part (d) the amount that is withdrawn is $ 6, 414.27 for a total of 20 withdrawals and so the total amount that is withdrawn is
6,414.27 (20) = 128,285.
Bonds: how to calculate current yield, yield to maturity (YTM), price of the bond for various types of bonds: -
Bonds, also known as fixed-income securities, are debt instruments created for the purpose of raising capital. Essentially loan agreements between an issuer and an investor, the terms of a bond obligate the issuer to repay the amount of principal at maturity. Most bonds also require that the issuer pay the investor a specific amount of interest on a semi-annual basis.
The return of a bond is largely determined by its interest rate. The interest that a bond pays depends on a number of factors, including the prevailing interest rate and the creditworthiness of the issuer, which, of course, is what is assessed by the credit rating companies, such as Standard & Poor’s and Moody’s. The higher the credit rating of the issuer, the less interest the issuer has to offer to sell its bonds. The prevailing interest rate—the cost of money—is determined by the supply and demand of money. Like virtually anything else, the greater the supply and the lower the demand, the lesser the interest rate, and vice versa. An often used measure of the prevailing interest rate is the prime rate charged by banks to their best customers.
Most bonds pay interest semi-annually until maturity, when the bondholder receives the par value of the bond back. Zero coupon bonds pay no interest, but are sold at a discount to par value, which is paid when the bond matures.
Current Yield
Because bonds trade in the secondary market, they may sell for less or more than par value, which will yield an interest rate that is different from the nominal yield, called the current yield, or current return. The price of bonds moves in the opposite direction of interest rates. If rates go up, the price of bonds decrease; if the rates go down, then the bonds increase in value. To see why, consider this simple example. You buy a bond when it is issued for $1,000 that pays 8% interest. Suppose you want to sell the bond, but since you bought it, the interest rate has risen to 10%. You will have to sell your bond for less than what you paid, because why is somebody going to pay you $1,000 for a bond that pays 8% when they can buy a similar bond of equal credit rating and get 10%. So to sell your bond, you would have to sell it so that the $80 that is received per year in interest will be 10% of the selling price—in this case, $800, $200 less than what you paid for it. (Actually, the price probably wouldn’t go this low, because the yield-to-maturity is greater in
Such a case, since if the Current yield example:-
$60 Annual interest payment = 8% current yield
$800 for bond
Bondholder keeps the bond until maturity; he will receive a price appreciation which is the difference between $1,000, the bond’s par value and what he paid for it.) Bonds selling for less than par value are said to be selling at a discount. If the market interest rate of a new bond issue is lower than what you are getting, then you will be able to sell your bond for more than par value—you will be selling your bond at a premium.
Current yield formula for bonds:-
Annual interest payment = current yield
Price of bond
Note that if the market price for the bond is equal to its par value, then:
Current Yield = Nominal Yield
Example: -
Assume you bought a one-year maturity bond with a coupon rate of 8.1% and a face value of $10,000 for a price of $9,000. Calculate the current yield.
$ 810 = 9% current yield.
$ 10000
Yield to maturity: -The most accurate measure of the long-term yield on a bond investment is its yield to maturity (YTM). That figure is calculated using the bond's interest rate, the current price, the par value and the years to maturity.
The yield to maturity on a bond is the rate of return that an investor would earn if he bought the bond at its current market price and held it until maturity. It represents the discount rate which equates the discounted value of a bond's future cash flows to its current market price. This is illustrated by the following equation:
In an equation,
1. C (1 + r)-1 + c (1 + r)-2 + . . . + c (1 + r)-Y + B (1 + r)-Y = P
where
c = annual coupon payment (in dollars, not a percent)
Y = number of years to maturity
B = par value
P = purchase price
You should try to form a mental picture of what this equation is saying. The left side represents Y+1 different compound interest curves, all starting out now, and each one ending at the moment that the payout it corresponds to takes place. Most of these curves will lie pretty low to the axis, because they only grow to a value of c, the coupon payment. The very last curve will be a lot taller, and end up at the par value B. And if you add up the present values of all these curves (that's the left side of the equation), the sum will exactly equal the purchase price of the bond (that's the right side).
As with most composite payout problems, equation 1 can't be solved exactly, in general. The nice part is that all yield-to-maturity problems have basically the same form, so people have been able to create programmable calculators and computer programs (and even tables back in the old days) to help you find r.
Example: Suppose your bond is selling for $950, and has a coupon rate of 7%; it matures in 4 years, and the par value is $1000. What is the YTM?
The coupon payment is $70 (that's 7% of $1000), so the equation to satisfy is
70(1 + r)-1 + 70(1 + r)-2 + 70(1 + r)-3 + 70(1 + r)-4 + 1000(1 + r)-4
= 950 $
Price of the bond for various types of bonds: -
Bonds are long-term debt securities that are issued by corporations and government entities. Purchasers of bonds receive periodic interest payments, called coupon payments, until maturity at which time they receive the face value of the bond and the last coupon payment. Most bonds pay interest semi annually. The Bond Indenture or Loan Contract specifies the features of the bond issue. The following terms are used to describe bonds.
The price or value of a bond is determined by discounting the bond's expected cash flows to the present using the appropriate discount rate. This relationship is expressed for a semi-annual coupon bond by the following formula:
Where
B0 = the bond value,
C = the annual coupon payment,
F = the face value of the bond,
r = the required return on the bond, and
t = the number of years remaining until maturity.
Bond valuation example: - Find the price of a semi annual coupon bond with a face value of $1000, a 10% coupon rate, and 15 years remaining until maturity given that the required return is 12%.
Yield to call
Many bonds, especially those issued by corporations, are callable. This means that the issuer of the bond can redeem the bond prior to maturity by paying the call price, which is greater than the face value of the bond, to the bondholder. Often, callable bonds cannot be called until 5 or 10 years after they were issued. When this is the case, the bonds are said to be call protected. The date when the bonds can be called is referred to as the call date.
The yield to call is the rate of return that an investor would earn if he bought a callable bond at its current market price and held it until the call date given that the bond was called on the call date. It represents the discount rate which equates the discounted value of a bond's future cash flows to its current market price given that the bond is called on the call date. This is illustrated by the following equation:
Where
B0 = the bond price,
C = the annual coupon payment,
CP = the call price,
YTC = the yield to call on the bond, and
CD = the number of years remaining until the call date.
Like the yield to maturity, the yield to call usually cannot be solved for directly. It generally must be determined using trial and error or an iterative technique. Fortunately, financial calculators make the task of solving for the yield to
Maturity quite simple.
Yield to Call Example:- Find the yield to call on a semi annual coupon bond with a face value of $1000, a 10% coupon rate, 15 years remaining until maturity given that the bond price is $1175 and it can be called 5 years from now at a call price of $1100.
Main models to estimate the value of a stock:-
Valuing common stock:-
As with bonds and preferred stock, the intrinsic value of a share of common stock is the present value of its future cash flows. Its future cash flows are (1)
Expected cash dividends and (2) an expected sale price, or disposal value.
Unlike most bond and preferred stock, common stock offers investor cash flows that may change, increasing and decreasing over time. For this reason estimating the intrinsic value of common stock is more difficult then doing so for a bond or preferred stock. The equation for intrinsic value of a common stock is:
For Example: - Assume that you are considering the purchase of a stock which will pay dividends of $3 next year, and $3.16 the following year. After receiving the second dividend, you plan on selling the stock for $33.33. What is the intrinsic value of this stock if your required return is 15%?
Vcs = Div1 + Div2 + S
(1 = R) ¹ (1+R) ²
- Vcs = valuation of common stock.
-
Div1 = dividend for next year Div2 = div for following.
- R = rate of return
- S = Stock price.
Vcs = 3 + 3.16+33.33
(1+ .15) (1.15)²
= 0.93 + 36, 49
1.32
= 0.93 + 27.64
= 28.57
Note that the interest factors in appendix b can be used as an alternative to the equation above. The intrinsic value of a stock calculated with the equation or with the tables will the same.
Valuing preferred stock:-
Calculating the intrinsic value of a share of preferred stock required the equation for perpetuity. Perpetuity is a perpetual annuity, an annual dollar amount an investor receives through infinity. A preferred stock is a perpetuity because it pays a stated, level dividend indefinitely.
Where Ct = cash flow at time t
T = time of cash flow
Kd = interest rate
N = years to maturity
VB = market value of the bond
As the duration of a bond increase, its interest rate risk increases as well. If a bond returns only one cash flow at maturity, then its duration is the same as its years to maturity. For a coupon bond with more then one cash flow, duration is less then year to maturity.
The expected annual dividend divided by the required rate of return yields the intrinsic value. 4 this method assumes that the investor acquired the stock at the beginning of the first period and receives dividends at the end of each year including the first one. The equation is.
Vp = Dp
Kp
Where: vp = intrinsic value of a share of preferred stock on the day following
The most recent dividend
dp = preferred share’s annual end of year dividend
kp = investor’s annual required rate of return
A share of preferred stock offers a $5 annual dividend. If you require a return of 8 percent year and buy the stock at the beginning of the year, then the stock’s intrinsic value is:
VP = $5
0.08
= $62.50
Constant Growth Stock Valuation:-
Stock Valuation is more difficult than Bond Valuation because stocks do not have a finite maturity and the future cash flows, i.e., dividends, are not specified. Therefore, the techniques used for stock valuation must make some assumptions regarding the structure of the dividends. A constant growth stock is a stock whose dividends are expected to grow at a constant rate in the foreseeable future. This condition fits many established firms, which tend to grow over the long run at the same rate as the economy, fairly well. The value of a constant growth stock can be determined using the following equation:
Where
P0 = the stock price at time 0,
D0 = the current dividend,
D1 = the next dividend (i.e., at time 1),
G = the growth rate in dividends, and
r = the required return on the stock, and
g < r.
Constant growth stock valuation example:-
Find the stock price given that current dividend is $2 per share dividends are expected to grow at a rate of 6% in the foreseeable future, and required retur is 12%.
Po = 2(1 + .06)
.12- .06
=$35.33
Nonconstant Growth Stock Valuation:-
Many firms enjoy periods of rapid growth. These periods may result from the introduction of a new product, a new technology, or an innovative marketing strategy. However, the period of rapid growth cannot continue indefinitely. Eventually, competitors will enter the market and catch up with the firm.
These firms cannot be valued properly using the Constant Growth Stock Valuation approach. This section presents a more general approach which allows for the dividends/growth rates during the period of rapid growth to be forecast. Then, it assumes that dividends will grow from that point on at a constant rate which reflects the long-term growth rate in the economy.
Stocks which are experiencing the above pattern of growth are called nonconstant, supernormal, or erratic growth stocks.
The value of a nonconstant growth stock can be determined using the following equation:
Where
P0 = the stock price at time 0,
Dt = the expected dividend at time t,
T = the number of years of nonconstant growth,
gc = the long-term constant growth rate in dividends, and
r = the required return on the stock, and
gc < r.
Nonconstant Growth Stock Valuation Example:-
The current dividend on a stock is $2 per share and investors require a rate of return of 12%. Dividends are expected to grow at a rate of 20% per year over the next three years and then at a rate of 5% per year from that point on. Find the price of the stock.
Solution:
There are 3 years of nonconstant growth, thus, T = 3. Before substituting into the formula given above it is necessary to calculate the expected dividends for years 1 through 4 using the provided growth rates.
D1 = 2(1 + .20) = $2.40
D2 = 2.40(1 + .20)2.88
D3 = 2.88 (1 .20) = $3.456
D4 = 3.456(1+.05) = $3.6288
Po = 2.40 + 2.88 + 3.456 + 3 .6288 (1+.12) ³
(1+.12)¹ (1+.12)² (1+.12)³ . 12-.05
= $43.80
Reference:-
.com
Business financial management (Philip l Cooley)
