Question 1
BEEM119: Problem Set 1—Indicative Answers
1. What is the main function of the financial system?
The financial systems channels funds from those with a surplus to those with a shortage of funds. It promotes economic efficiency by producing an efficient allocation of capi- tal which increases production. It further directly improves consumers’ well-being by allowing them to better time their purchases.
2. Name financial innovations that affected you? Why?
Deposit accounts. Consumer credit. Pension funds.
3. Why is a share of Microsoft’s common stock an asset for its owner and a liability for Microsoft?
Common stock represents a share of ownership in a corporation. A share is a claim on the residual earnings and assets of the corporation so that stocks make periodic payments.
4. What is the difference between a mortgage and a mortgage-backed security?
A mortgage is a secured loan taken out by real estate property owners or purchasers to raise funds to, e.g., finance the property purchase. The loan is secured by the property. If the borrower does not honour the terms of the loan contract, the lender takes possession of the property and can sell it to collect the debt.
A mortgage-backed security (MBS) is an security that is backed by an asset, in this case a (pool of) mortgage(s). Pooling mortgages is a way of securitisation. The payments made on the mortgages are passed to the MBS holder.
Question 2
1. Would a dollar tomorrow be worth more to you today when the interest rate is 20%, or when it is 10%? Why?
The present value calculations imply
1.2>1.1⇔ 1>1, 1.1 1.2
so that the dollar is worth more if the interest rate is 10%.
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2. Write down the formula that is used to calculate the yield to maturity on a ten-year 10% coupon bond with $1000 face value that sells for $1,000.
1000= 100 + 100 + 100 +···+ 100 + 1000 (1+i) (1+i)2 (1+i)3 (1+i)10 (1+i)10
10 100 1000 =+.
k=1 (1+i)k (1+i)10
1. Compute the present value of a coupon bond with a coupon of $1,000, face value of
$20,000, maturity of 10 years, and interest rate of 5%?
The yield to maturity equals the coupon rate. That is, the present value of the bond equals its face value. Let’s check this. Given an interest rate i, the present value of this bond is given by
pv=1000+ 1000 + 1000 +···+ 1000 + 20000 . 1+i (1+i)2 (1+i)3 (1+i)10 (1+i)10
Question 3
This can be written as
pv = 1000 (1+i)9 +(1+i)8 +(1+i)7 +···+1+ 20000 .
(1 + i)10 Using the geometric series
1000 pv = (1+i)10
(1 + i)10
1 − (1 + i)10 20000 1−(1+i) + (1+i)10
= 1000 (1+i)10−1+ 20000 (1+i)10 i (1+i)10
1000 (1+i)10−1 (1) =(1+i)10 20+ i .
With i = 0.05, we get
pv = (1.05)10
1000 = (1.05)10
20 +
1000
(1.05)10−1 0.05
20(1.05)10 −20 20+ 1
= 1000 (1.05)10
20(1.05)10 = 20000.
2. Similarly, compute the present value of a coupon bond with a coupon of $1,000, face
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value of $20,000, maturity of 10 years, and interest rate of 10%?
Clearly, the present value should be smaller than the face value. Using equation (1)
with i = 0.1, we get
(1.1)10−1 0.1
= 10000 + 10000 = 13855.43. (1.1)10
pv = (1.1)10
1000 = (1.1)10
= 1000 (1.1)10
20 +
10(1.1)10 −10 1
1000
20 +
10 + 10(1.1)10
3. What is the amount to be repaid at maturity of a loan of $10,000, 3 years time to maturity, and annual interest rate of 6%?
10000(1 + 0.06)3 = 11910.16.
4. What is the implied annual interest rate of a zero-coupon bond with face value
$100,000, 4 years time to maturity, and a current price of $95,000? 1
95000(1+i)4 =100000⇔(1+i)4 = 100 = 20 ⇔i= 20 4 −1=0.0129. 9519 19
That is, i = 1.29%.
5. A coupon bond has face value of $1,000, market value of $900, maturity of 3 years, and
yield to maturity of 10%. What is the coupon rate?
p=c+c+c+···+f 1+i (1+i)2 (1+i)3 (1+i)n
with f = 1000, p = 900, n = 3, and i = 0.1, so that
900= c + c + c +1000.
1.1 1.12 1.13 1.13
Rearranging yields
1000 1 1 1 1.12+1.1+1
900− 1.13 =c 1.1+1.12 +1.13 =c 1.13 3
or
1.13 900 · 1.13 − 1000 900 · 1.13 − 1000
c = 3.31 1.13 = 3.31 = 59.79.
6. The Bank of England issued a consol (or perpetuity) with a coupon of $100. The market price of this consol is $8,000. What is the yield to maturity?
i= c = $100 = 1 =0.0125 or 1.25%. p $8000 80
Question 4
Interest rates were lower in the mid-1980 than in the late 1970s, yet many economists have commented that real interest rates were actually much higher in the mid-1980s than in the late 1970s. Does this make sense? Do you think these economists are right?
Yes. The real interest rate is approximately given by r = i − πe. The real interest rate in the mid-1980s can be much higher than in the late 1970s despite nominal interest rates being lower, if inflation expectations were a lot higher in the late 1970s. Yes (Mishkin, chapter 4, figure 1).
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