Tutorial 3: fixed income securities
Bonds:
Bonds typically pay coupons on a semi-annual basis and yields are usually reported as a
stated rate with semi-annual compounding. You should assume that these hold unless
told otherwise.
Bonds can be viewed as the sum of two sources of money, an annuity (the coupon
payments) and lump sum (the face value). The price is therefore PVIFA x the coupon
payment plus the face value x PVIF.
Problem 1:
A bond with a face value of $100, 8 years to maturity and an 11% coupon rate is
currently trading at a yield to maturity of 8%.
a) What is the current price of that bond?
b) What would the price be if the yield were 7%?
F = 100
C = 11/2 = 5.5
k = 8/2 = 4%
Price = CxPVIFA(4%, 16) + FxPVIF(4%, 16)
= 64.088 + 53.391
= 117.478
B)
Change k to 7/2 or 3.5%
Price = 5.5xPVIFA(3.5%, 16) + 100xPVIF(3.5%, 16)
= 66.518 + 57.671
= 124.188
n = 8x2 = 16 periods
Tutorial 3: fixed income securities
Problem 2:
You bought a $1,000 face value bond for $1,200 six months ago. The bond has just paid
a coupon of $60 and has 12 years left until it matures.
a) What is your return, stated on an annual effective basis, if the bond is currently priced
at a yield to maturity of 9%?
b) What was the YTM when you bought the bond?
F = 1000
C = 60
k = 9/2 = 4.5%
n = 12x2 = 24
Price = 60xPVIFA(4.5%, 24) + 1000xPVIF(4.5%, 24)
= 869.73 + 347.70
= 1217.43
Return = ((1217.43 – 1200) + 60) / 1200 = 6.45% per 6 months
EAR = (1 + 0.0645)2 –1 = 13.32%
B) Price = 1200
F = 1000
C = 60
n = 25
k=?
There is no simple formula to find k, so you use trial and error. First try the
9% that gives a current price of $1217.
@YTM of 9%
Price = $1222.42
Too high, so raise discount rate to lower the price.
@YTM of 9.5%
Price = $1180.67
Too low, so lower the discount rate.
@YTM of 9.25%
Price = $1201.29
Close enough for most purposes.
@YTM of 9.26546%
Price = $1200.00
Tutorial 3: fixed income securities
Term Structure of interest rates:
If you examine the quoted yields on bonds or the interest rates advertised for various
loans and deposits you will see that the length of term has an effect on the interest rate.
This is the starting point for looking at the term structure of interest rates. A simple
graph of comparable securities yields vs. their maturity will generate a yield curve. To be
able to hold all other aspects constant, we should use pure discount instruments if this is
possible.
There are 2 main theories about the term structure, the Unbiased Expectations Theory
(UET) and the Liquidity Preference Theory (LPT). The UET proposes that the interest
rates that are implied by the yield curve are the market consensus prediction of what
interest rates are going to be in that period. LPT assumes that these rates are higher than
the predictions and that they embody a reward for bearing risk.
If we assume that UET and pure discount bonds, we can restate an n period rate as:
(1+r)n = (1+r1)(1+r2)(1+r3)…(1+rn)
Where r is the rate for the n period bond and ri is the expected rate in period i.
Problem 3:
If the current one year interest rate is 7% and the market expects interest rates to rise to
8% for the next two years before falling back to 7% in year 4, what rates should you see
on 2, 3 an 4 year pure discount bonds if UET holds?
2 year FVIF = (1 + r)2 = (1.07)(1.08)
r = .0750
3 year FVIF = (1 + r)3 = (1.07)(1.08)(1.08)
r = .0767
4 year FVIF = (1 + r)4 = (1.07)(1.08)(1.08)(1.07)
r = .0750
Tutorial 3: fixed income securities
Forward Rates:
By using the above formulas you can find the rates that are effectively being offered for
holding a bond in a particular year. You can also find the rates that should be expected
on multi-period bonds at a certain point in the future.
Problem 4:
You have observed the following yields on stripped government bonds.
(1 + r)n - 1
1 year forward
2 year forward
6%
6.00%
n. a.
n. a.
2 Year
7%
14.49%
f1, 1 = 8.01%
f1, 2 = 9.014%
3 Year
8%
25.97%
f2, 1 = 10.03%
f2, 2 = 9.009%
4 Year
8%
36.05%
f3, 1 = 8.00%
f3, 2 = 6.75%
5 Year
7.5%
43.56%
f4, 1 = 5.52%
f4, 2 = 5.03%
6 Year
7%
50.07%
f5, 1 = 4.53%
f5, 2 = 4.04%
7 Year
6.5%
55.40%
f6, 1 = 3.55%
f6, 2 = 3.06%
8 Year
6%
59.38%
f7, 1 = 2.57%
f7, 2 = n. a.
Term
YTM
1 Year
Find all of the one and two year forward rates that can be determined.
To find the forward rates, we want to find the rate that would set the yield
if we invest for the entire period equal to the yield if we invest for a
shorter period and reinvest at that forward rate.
e.g. (1 + r5)5 = (1 + r4)4(1 + f4,1)
f4,1 = (1.4356 / 1.3605) – 1 = 5.523%
e.g.
(1 + r6)6 = (1 + r4)4(1 + f4,2)2
f4,2 = (1.5007 / 1.3605) – 1 = 5.523%
Tutorial 3: fixed income securities
Term Structure and Future/Present Value:
Present and future value calculations become more complicated if the discount rate is not
constant over the time that is being considered. For future values, the FVIF becomes a
multiple of the various (1+k)s that are appropriate for the time periods covered. Present
values are done in a similar manner using the appropriate (1+k)-1 for each period. For
example; if the appropriate discount rates for the next 3 periods are 9%, 7% and 6%, the
FVIF = (1.09)(1.07)(1.06) = 1.2363.
Problem 5:
A trust company is offering an investment certificate that pays 3% in the first year, with
the rate increasing by 1% per year for up to 8 years, with all of the accumulated interest
reinvested at the current rate.
a) If you invest $5,000 how much will you have at the end of 8 years?
b) What rate, locked in for 8 years would be equivalent?
A)
FVIF = (1.03) (1.04) (1.05) (1.06) (1.07) (1.08) (1.09) (1.10)
= 1.6519
FV = $5,000 x FVIF = $8,259.67
B)
Find k in (1 + k)8 = 1.6519
k = (1.6519)1/8 – 1 = 6.4753%