Bond Valuation Tutorial
1a. Show the cash flows for the following four bonds, each of which has a par value of $1,000 and
pays interest semi-annually.
Bond
Coupon Rate(%) Number of Years to Maturity
Price
W
7
5
$884.20
X
8
7
$948.90
Y
9
4
$967.70
Z
0
10
$456.39
Bond W has cash flows of 0.07($1,000) / 2 = $35 for semiannual periods from periods 1 to 10. At the
end of period 10, Bond W pays back the par of $1,000 and its semiannual interest for
a total payment of $1,000 + $35 = $1,035.
Bond X has cash flows of 0.08($1,000) / 2 = $40 for semiannual periods from periods 1 to 14. At the
end of period 14, Bond X pays back the par of $1,000 and its semiannual interest for a total payment
of $1,000 + $40 = $1,040.
Bond Y has cash flows of 0.09($1,000) / 2 = $45 for semiannual periods from periods 1 to 8. At the
end of period 8, Bond Y pays back the par of $1,000 and its semiannual interest for a total payment
of $1,000 + $45 = $1,045.
Bond Z has cash flows of 0($1,000) / 2 = $0 for semiannual periods from periods 1 to 20. At the end
of period 20, Bond Z pays back the $1,000 and its semiannual interest for a total payment of $1,000
+ $0 = $1,000.
Below we show these cash flows in table format.
Period
Cash Flow
Cash Flow
Cash Flow
Cash Flow
for Bond W
for Bond X
for Bond Y
for Bond Z
1
$35
$40
$45
$0
2
$35
$40
$45
$0
3
$35
$40
$45
$0
4
$35
$40
$45
$0
5
$35
$40
$45
$0
6
$35
$40
$45
$0
7
$35
$40
$45
$0
8
$35
$40
$1,045
$0
9
$35
$40
$0
10
$1,035
$40
$0
11
$40
$0
12
$40
$0
13
$40
$0
14
$1,040
$0
15
$0
16
$0
17
$0
18
$0
19
$0
20
$1,000
1.b Calculate the Rate of return on these bonds for the present year
𝑹=
W
X
Y
Z
1000
1000
1000
1000
884.2
948.9
967.7
456.39
70
80
90
0
𝑪𝒐𝒖𝒑𝒐𝒏 + 𝑪𝒂𝒑𝒊𝒕𝒂𝒍 𝑮𝒂𝒊𝒏
𝑷𝒖𝒓𝒄𝒉𝒂𝒔𝒆 𝑷𝒓𝒊𝒄𝒆
-0.0458
0.0289
0.0577
0.54361
-4.58
2.89
5.77
-54.361
2. What is the cash flow of a 8-year bond that pays coupon interest semiannually, has a coupon
rate of 6%, and has a par value of $100,000?
The principal or par value of a bond is the amount that the issuer agrees to repay the bondholder at
the maturity date. The coupon rate multiplied by the principal of the bond provides the dollar
amount of the coupon (or annual amount of the interest payment). An 8-year bond with a 6% annual
coupon rate and a principal of $100,000 will pay semi-annual interest of (0.06/2)($100,000) = $3,000
for 8(2) = 16 periods. Thus, the cash flow is $3,000. In addition to this periodic cash, the issuer of the
bond is obligated to pay back the principal of $100,000 at the time the last $3,000 is paid.
3. What is the cash flow of a 4-year bond that pays no coupon interest and has a par value of
$1,000?
There is no periodic cash flow as found in the previous problem. Thus, the only cash flow will be the
principal payment of $1,000 received at the end of six years. This type of cash flow resembles a zerocoupon bond. The holder of such a bond realizes interest by buying the bond substantially below its
principal value. Interest is then paid at the maturity date, with the exact amount being the
difference between the principal value and the price paid for the bond.
4. Give three reasons why the maturity of a bond is important.
There are three reasons why the maturity of a bond is important. First, the maturity gives the time
period over which the holder of the bond can expect to receive the coupon payments and the
number of years before the principal will be paid in full. Second, the maturity is important because
the yield on a bond depends on it. The shape of the yield curve determines how the maturity affects
the yield. Third, the price of a bond will fluctuate over its life as yields in the market change. The
volatility of a bond’s price is dependent on its maturity. More specifically, with all other factors
constant, the longer the maturity of a bond, the greater the price volatility resulting from a change in
market yields.
5. Government Gilts are not selling well in Europe, countries like Spain, Italy and Greece are
finding it very difficult to sell treasury bonds and are relying on money from the Eurozone. Using
the supply and demand theory of Bonds, explain this situation using Graphs.
Due to the Financial Crisis and the Monetary and Fiscal stimulus carried out by Governments, the
interest rate has been held very low by many countries. Households expect this to change rapidly in
the future and so therefore they expect the interest rate and inflation to rise sharply. Thus the
Fisher effect is seen in bond markets.
Students should outline the Fisher Effect of expected Inflation as follows
6. You are a financial consultant. At various times you have heard comments on interest rates
from one of your clients. How would you respond to each comment?
(a) Respond to: “The yield curve is upward-sloping today. This suggests that the market consensus
is that interest rates are expected to increase in the future.”
This is not necessarily true because investors demand a greater return as the maturity increases. The
maturity premium results from the fact that more uncertainty exists for longer term maturity. Other
factors causing the yield curve to be upward-sloping include liquidity considerations and supply and
demand concerns. For example, if investors wanted fewer longer term bonds than were currently
being supplied, then this would drive up the yield on longer term bonds.
(b) Respond to: “I can’t make any sense out of today’s term structure. For short-term yields (up to
three years) the spot rates increase with maturity; for maturities greater than three years but less
than eight years, the spot rates decline with maturity; and for maturities greater than eight years
the spot rates are virtually the same for each maturity. There is simply no theory that explains a
term structure with this shape.”
There are various theories that can account for any slope that the yield curve might take. First, there
is the pure expectations theory where the forward rates exclusively represent the expected future
rates. Since these rates can either increase or decrease for any time period, the yield curve can be
sloped upward or downward for that time period.
Second, there is the liquidity preference theory which asserts that investors do not like uncertainty
and so much be offered a higher rate of return for longer term maturities. Thus, the forward rate will
not only reflect expectations about future interest rates but also a “liquidity” premium that will be
higher for longer term securities. Ceteris paribus, an increasing liquidity premium implies that the
yield curve will be upward sloping.
The preferred habitat theory also adopts the view that the term structure reflects the expectation of
the future path of interest rates as well as a risk premium. However, the preferred habitat theory
rejects the assertion that the risk premium must rise uniformly with maturity. The preferred habitat
theory asserts that to the extent that the demand and supply of funds in a given maturity range do
not match, some lenders and borrowers will be induced to shift to maturities showing the opposite
imbalances. However, they will need to be compensated by an appropriate risk premium whose
magnitude will reflect the extent of aversion to either price or reinvestment risk. Thus this theory
proposes that the shape of the yield curve is determined by both expectations of future interest
rates and a risk premium, positive or negative, to induce market participants to shift out of their
preferred habitat. Thus, according to this theory, yield curves sloping up, down, flat, or humped are
all possible.
The market segmentation theory also recognizes that investors have preferred habitats dictated by
the nature of their liabilities. This theory also proposes that the major reason for the shape of the
yield curve lies in asset-liability management constraints (either regulatory or self-imposed) and/or
creditors (borrowers) restricting their lending (financing) to specific maturity sectors. However, the
market segmentation theory differs from the preferred habitat theory in that it assumes that neither
investors nor borrowers are willing to shift from one maturity sector to another to take advantage of
opportunities arising from differences between expectations and forward rates. Thus for the
segmentation theory, the shape of the yield curve is determined by supply of and demand for
securities within each maturity sector.