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book chapter 3 financial calculations (calculate mortgage payments…
book chapter 3 financial calculations
calculate bill prices and
bond prices & calculate holding period yields on bills and bonds sold prior to maturity
Discount Securities
introduction
bond
a security with an initial term (i.e. its term when it is issued) that is
greater than
one year is called a bond
bills or notes
a security with an initial term (i.e. its term when it is issued) that is
less than
one year is called a bond
discount securities
Securities with a term of less than one year are usually discount securities.
A discount security is sold at a price
below
its face value and the return to the investor is the difference between the face value and the price.
yield实际收益率
e.g assume that an investor buys a 90-day bill with a face value of $100 000 for $98 000. The return to the investor is $2000, but this return has been earned from a 90-day investment. The
yield
can be calculated (in nominal per annum terms) as:
(2000/98000)/(90/365)x100%=8.28%
yield=(reture/price)/(days/365)x100%
Price
A bill with a given face value can be priced by calculating the price that must be paid to give an investor the market yield. We do this by finding the present value of the face value at the market yield.
For example
assume that the market yield is 5.50% p.a. Then the price of a 90-day security with a face value of $500 000 is:
price=500,000/(1+90/365x0.055)=$493 309.91
1 more item...
price=face value/[1+(days/365) x yield]
discount rate贴现率
the price of a bill can also be described by stating the percentage discount off the face value that the price represents.This consideration leads us to calculate
the discount rate
For example, if a 90-day bill with a face value of $100 000 sells for $98 000, the discount rate is given by
discount rate=(2000/100000)/(90/365)x100%=8.11%
discount rate=(reture/future value)/(90/365)x100%
the formula that defines the relationship between the discount rate and the yield, are given below
price value of a basis point (PVBP)
The price value of a basis point (PVBP) of a security is the change in its price when the yield on it changes by one basis point, which is a change of one unit in its second decimal place. As yields are normally stated to two decimal places, this is the smallest change that can occur in a market yield.
PVBP = future value - price
the sensitivity of the price of a security to changes in the yield increases as the term of a security increases
holding period yield 持有期收益率
definition
If an investor buys the 90-day bill discussed (in Section 3.2 above) at 5.50% and holds it until maturity, the investor will earn 5.50% p.a. over the 90 days of the investment. However, buyers of bills often sell them before maturity and it is then necessary to work out the return that has been earned over the life of the investment (the holding period yield).
HPY=[P(sell them before maturity) - P(holds it until maturity)]/P(holds it until maturity)
If the bill had been sold at a yield above the purchase yield, there would have been a capital loss—this would have reduced the holding period return below the purchase yield.
Pricing Bonds
zero-coupon bonds
definition
Zero-coupon bonds give the owner title to a single payment at the end of their life, and they, therefore, have the same cash flow pattern as discount securities, which were discussed in Section 3.3 above.
Their price is obtained in the same way—it is the present value of the final payment at the current market yield.
An important difference is that zero-coupon bonds have an initial term
greater than one year
. This means that although the holder of the bond receives no cash payments, the bond accrues compound interest—no cash flows occur and all returns are, in effect, reinvested at the same yield.
price
Consider a five-year zero coupon bond which pays $500 000 at maturity. The five-year market yield is 5.00% p.a., which, as per the convention for bond returns, is compounded biannually. The price of the security is
price=500,000/[(1+5%/2)^(5x2)]=$390 599.20
price=face value/[(1+yield)^(years x compounding times)]
Price = F × V^M
M=years x compounding times
V=1/(1+yield)
annuity 年金
definition
An annuity provides the same payment every period, but no face value payment at the end of its life.
price
Consider an annuity which pays $2000 every half-year for two years. The market yield is 5.00% p.a. As usual, the price of the annuity is calculated as the present value of the cash flows.
price=2000/(1+5%/2)+2000/[(1+5%/2)^2]+2000/[(1+5%/2)^3]+2000/[(1+5%/2)^4]=$7523.95
P=C(1-V^M)/yield
V=1/(1+yield)
M=years x compounding times
a^M
is the price of an annuity of $1 paid for M periods when the per
period interest rate is r(yield).
a^M=(1-V^M)/r
coupon bond
A coupon bond can be regarded as the combination of an annuity and a zero-coupon bond.
To illustrate this point, consider a 10-year bond with a face value of $1 million and a semi-annual coupon payment of 4.00% p.a. (i.e. $20 000 per half year). The market yield is 5.00% p.a. The price of the bond is
V=1/(1+5%/2), M=2 X10=20,r(yeild)=5%/2
P=20,000(1-V^20)/(5%/2)+1,000,000xV^20
Assume that we buy this bond, hold it for a year and then
sell it at a yield of
4.50% p.a.
In this case:
V=1/(1+4.5%/2), M=2x10-2x1=18,r=4.5%/2
P=20,000(1-V^18)/(4.5%/2)+1,000,000xV^18=$963,330.85
In addition, we received two coupon payments of $20 000. The
holding period yield
on this investment is 4.359619% per half-year or 8.72% p.a. (calculated on a financial calculator).
Price=20,000/(1+4.359619%)+20,000/[(1+4.359619%)^2]+963,330.85/[(1+4.359619%)^2]
The high return was obtained because the bond was sold at a lower yield than its purchase yield, which generated a capital gain
yield to maturity 到期收益率
the yield on a bond (sometimes called the yield to maturity) is the discount rate that makes the present value of all the cash flows equal to the price of the bond.
It is the yield that an investor will earn if:
• the bond is held to maturity
• all the coupon payments are invested at the same yield for the remainder of the term of the bond.
price value of a basis point (PVBP)
PVBP = future value(with yield to maturity) - price(market yield)
calculate an appropriate prepayment penalty on fixed-rate mortgages
prepayment
Many lenders providing fixed-rate mortgages impose a penalty for the prepayment of the mortgage.
The reason for this penalty is that mortgage holders have an incentive to prepay their loan (and refinance it)
when interest rates fall
. Lenders will have offset 抵消 the fixed rate in some way (e.g. by funding it from deposits which have a fixed rate for the same period) and, with an interest rate fall, they will have suffered a loss on that hedge. If the mortgage is prepaid, they lose the offsetting gain for that loss; that is, they have
a net loss
.
Prepayment penalties have two basic forms
a fixed dollar amount
The problem with this approach is that there will be some fall in interest rates that makes it profitable for the borrower to prepay the loan.
prepayment can be allowed only at the market value of the loan.
the lender cannot lose from the prepayment.
The problem with this approach is the difficulty in explaining it to customers, particularly when the new market value exceeds the amount originally borrowed.
calculate mortgage payments
definition
A mortgage is a loan to fund the purchase of a house, in which the lender takes security over the house in the form of a legal document known as a mortgage.
The cash flow pattern of a mortgage is similar to that of
an annuity
and we can use the formula developed earlier to calculate mortgage payments
In Australia, most housing mortgages are principal-and-interest loans; that is, they are repaid in a large number of
equal payments
that include both repayment of principal and the interest charged on the outstanding balance. There is no large final repayment.
P=C(1-V^M)/yield
V=1/(1+yield)
M=years x compounding times
you will reduce the interest you pay by making earlier principal repayments.
do basic interest rate
calculations
3.2.1 NOMINAL AND EFFECTIVE INTEREST RATES
The first is that interest rates are usually stated in terms of the nominal per annum interest rate. However, interest payments may be compounded more than once a year.
assume that the nominal per annum interest rate is 5%. Then the perperiod interest rates for different periods are:
half-yearly：5/2=2.5
daily：5/365=0.0137
monthly：5/12=0.4167
quarterly：5/4=1.25
Effective Interest Rates for Different Compounding Periods
the effective interest rate is always above the nominal per annum interest rate, and that the same nominal per annum interest rate produces very different effective interest rates, which increase with the frequency of compounding
instantaneous compounding
one dollar amounts to e^0.08 after one year (‘e’ is the exponential constant 2.718282), which is equal to 1.0513—that is, the return is equivalent to that received from daily compounding and not that much higher than the return produced by monthly compounding
discount securities
discount securities which have an initial term of one year or less. In spite of this, their yield is also stated on a per annum basis and this value has to be converted to one that is appropriate to the term of the security. This is done as follows
Per annum interest rate ×（Term of security in days/365）
e.g if we have a 90-day security and the nominal per annum interest rate is 5% p.a., the rate for 90 days is
0.05x(90/365)=1.23%
3.2.2 THE TIME VALUE OF MONEY
definition
A sum of money to be received today is more valuable than the same sum of money to be received at a future date. The reason for the lower value of future cash flows is that they can be generated by investing a smaller amount of money now.
time value of money
The amount of money to be invested now depends on the interest rate. Therefore, we can regard the interest rate as the time value of money
present value (PV)
The present value (PV) of a future cash fl ow is the amount that needs to be invested now to produce that cash fl ow at the time that it occurs
PV=Future cash flow/(1 + r)^n
For example, if we are to receive $10 000 in 10 years and the 10-year interest rate is 5.50% p.a., the present value is
PV =$10,000/(1.055)^10= $5854.31
