EC1008 T1
Tutorial 1 – Finance
1. How can the adverse selection problem explain why you are more likely to make a loan to a
family member than to a stranger?
You have a closer relationship with the family member than a stranger. You have more information
on their ability to pay back, and their financial habits. The lack of information you have on the
behaviour of the stranger causes the adverse selection problem. As you have information on the
family member, and have a level of bias towards them as family, you are more inclined to feel like
giving money to the family member is a better choice, even though you cannot know that without
full information on both parties
-
People who are willing to pay me a high amount, I don’t trust. The higher the rate
you can offer, the higher the risk in your idea. Becomes too good to be true.
2. If the interest rate is 15%, what is the present value of a security that pays you $1,100 next
year, $1,250 the year after, and $1,347 the year after that?
(
1100
1250
1347
)+(
)+(
) = £2787.38
1
2
(1 + 0.15)
(1 + 0.15)
(1 + 0.15)3
3. Calculate the present value of a $1,300 discount bond with seven years to maturity if the
yield to maturity is 8%.
1300
(1+0.08)7
= £758.54
4. A lottery claims its grand prize is $15 million, payable over 5 years at $3,000,000 per year. If
the first payment is made immediately, what is this grand prize worth? Use an interest rate
of 7%.
3,000,000 + (
3,000,0000
3,000,000
3,000,000
3,000,000
)+(
)+(
)+(
)
1
2
3
(1 + 0.07)
(1 + 0.07)
(1 + 0.07)
(1 + 0.07)4
= £13,161,633.77
− 𝑖𝑓 𝑝𝑎𝑦𝑚𝑒𝑛𝑡𝑠 𝑤𝑒𝑟𝑒 𝑒𝑣𝑒𝑟𝑦 6 𝑚𝑜𝑛𝑡ℎ, 𝑡ℎ𝑒𝑛 𝑤𝑒 ′ 𝑑𝑔𝑜 𝑢𝑝 𝑖𝑛 0.5
5. What is the yield to maturity on a $10,000-face-value discount bond maturing in one year
that sells for $9,523.81?
1000
(
− 1) 𝑋 100 = 5%
9523.81
EC1008 T1
6. What is the yield to maturity (YTM) on a simple loan for $1,500 that requires a repayment of
$15,000 in five years’ time?
15,000 0.2
) − 1) 𝑋 100 = 58.49%
((
1500
7. Which $10,000 bond has the higher yield to maturity, a twenty-year bond selling for $8,000
with a current yield of 20% or a one-year bond selling for $8,000 with a current yield of 10%?
In regard to the one year bond with 10% yield
8,000 =
8,000𝑥0.1 10,000
+
1+𝑖
1+𝑖
𝑖 = 35%
8,000 =
8,000𝑥0.2 8,000𝑥0.2 8,000𝑥0.2
8,000𝑥0.2 10,000𝑥0.2
+
+
+ ⋯+
+
2
3
(1 + 𝑖)
(1 + 𝑖)
(1 + 𝑖)
(1 + 𝑖)20
(1 + 𝑖)20
𝑖 = 20%
8. Consider a bond with a 6% annual coupon and a face value of $1,000. Complete the
following table. What relationships do you observe between years to maturity, yield to
maturity, and the current price?
1,037.72 =
1,000 =
1,000 =
1,089.04 =
920.15 =
1000𝑥0.06 1000𝑥0.06
1000
+
+
(1 + 0.04) (1 + 0.04)2 (1 + 0.04)2
1000𝑥0.06 1000𝑥0.06
1000
+
+
2
(1 + 0.06) (1 + 0.06)
(1 + 0.06)2
1,000𝑥0.06 1,000𝑥0.06 1,000𝑥0.06
1,000
+
+
+
2
3
(1 + 0.06) (1 + 0.06)
(1 + 0.06)
(1 + 0.06)3
1,000𝑥0.06 1000𝑥0.06 1,000𝑥0.06 1,000𝑥0.06 1,000𝑥0.06
1,000
+
+
+
+
+
(1 + 0.04) (1 + 0.04)2 (1 + 0.04)3 (1 + 0.04)4 (1 + 0.04)5 (1 + 0.06)5
1000𝑥0.06 1,000𝑥0.06 1,000𝑥0.06 1000𝑥0.06 1000𝑥0.06
1,000
+
+
+
+
+
(1 + 0.08) (1 + 0.08)2 (1 + 0.08)3 (1 + 0.08)4 (1 + 0.08)5 (1 + 0.08)5
EC1008 T1
When the yield to maturity is equal to the annual coupon, the present value is equal to the
face value. If the yield to maturity is higher than the annual coupon, then the present value
is lower than the face value, but if the yield to maturity is lower than the annual coupon, the
present value is higher than the face value. The higher the yield to maturity, the lower the
present value of the bond.
9. Consider a coupon bond that has a $900 par value and a coupon rate of 6%. The bond is
currently selling for $860.15 and has two years to maturity. What is the bond’s yield to
maturity (YTM)?
$860.15 =
900𝑥0.06
(1+𝑖)
+
900𝑥0.06
(1+𝑖)2
900
+ (1+𝑖)2
YTM = i = 8.5%
10. What is the price of a perpetuity that has a coupon of $70 per year and a yield to maturity of
1.5%? If the yield to maturity doubles, what will happen to the perpetuity’s price?
Yield to maturity = Coupon payment/price
70/0.15=$466.67
If the yield to maturity doubles, the price will half. 70/0.3 = $233.34
11. Property taxes in a particular district are 2% of the purchase price of a home every year. If
you just purchased a $150,000 home, what is the present value of all the future property tax
payments? Assume that the house remains worth $150,000 forever, property tax rates
never change, and a 4% interest rate is used for discounting.
Assume the house is held forever, it is not sold. By that logic, this acts as a perpetuity.
Yield to maturity = Coupon payment/price
The coupon payment takes on the role of how much you pay yearly in property tax. In this example
0.02 x 150,000 = 3,000. The yield to maturity is the interest rate held at 4%. The present value =
3000
=
0.04
$75,000
12. A $1,100-face-value bond has a 5% coupon rate, its current price is $1,040, and it is expected
to increase to $1070 next year. Calculate the current yield, the expected rate of capital
gains, and the expected rate of return.
Current yield = Coupon payment/purchase price =
1100 𝑥 0.05
1040
= 5.29%
Expected rate of capital gains = (future price – current price)/ current price =
Expected rate of return = current yield + expected rate of return =
1070−1040
=
1040
1100 𝑥 0.05 1070−1040
+ 1040
1040
2.88%
= 8.17%
EC1008 T1
13. Assume you just deposited $1,250 into a bank account. The current real interest rate is 1%,
and inflation is expected to be 5% over the next year.
1. What nominal rate would you require from the bank over the next year?
If real interest rate is 1%, and inflation is 5%. Then if the bank gives you a nominal
rate of 6%, then you receive a real increase in the value of your account by 1%. So,
6%.
2. How much money will you have at the end of one year?
1,250 x 1.06 = $1,325
3. If you are saving to buy a fancy bicycle that currently sells for $1,300, will you have
enough money to buy it?
The bike currently costs 1,300. In one year, with 5% inflation it will cost $1365. In one
year, you will have 1325 as per question 3, so you will not have enough money.
14. Go to the St. Louis Federal Reserve FRED database, and find data on the interest rate on a
four-year auto loan (TERMCBAUTO48NS) and consider February 2014. Assume that you
borrow $20,000 to purchase a new automobile and that you finance it with a four-year loan
at the most recent interest rate given in the database. If you make one payment per year for
four years, what will the yearly payment be? What is the total amount that will be paid out
on the $20,000 loan?
According to the St. Louis Federal Reserve FRED database, the interest rate on a four year auto loan
1
1
was 4.35%. In year 1, the interest increases the value he owes by
in year 2 by (1+0.0453)2 in
year 3
1
by (1+0.0453)3
1+0.0453
and similarly for year 4.
𝑓𝑝
𝑓𝑝
𝑓𝑝
𝑓𝑝
20,000 = (1+0.0453 +(1+0.0453)2 +(1+0.0453)3 +(1+0.0453)4 )
Pull out a factor of fp
Fixed payment = 5535.79
Over 4 years 22,143.18
15. Changing interest rates. If the interest rate is 5% in the first year, and 10% in the second
year, what is the present value of a security that pays you $100 next year, and $200 the year
after?
$268.40 =
100
200
+
1 + 0.05 (1 + 0.05)(1 + 0.1)
EC1008 T1
16. Negative payments. If the interest rate is 5%, what is the present value of a security that
pays you £100 next year, -£50 the year after, and £35 the third year?
$80.12 =
100
50
35
−
+
(1 + 0.05) (1 + 0.05)2 (1 + 0.05)3
17. Negative interest rates. If the interest rate is -0.01, what is the present value of a security
that pays you £100 next year? How does the present values change if the interest rate is
0.01, instead?
$101.01 =
100
(1 + (−0.01))
$99.01 =
100
(1 + 0.01)
If the interest rate is positive rather than negative, the present value is lower than the payment. If
the interest rate is negative, the present value is greater than the payment. This is because if the
interest rate is negative, you would have lost money through interest, so the payment is a better
outcome.