171AHW2

Problem 1

Calculate the APY for each of the following investments. Which of the following investments has the highest effective return (APY)? Assume that all CDs are of equal risk.

APY <- function(m, APR) {
  (1+(APR/m))^(m)-1
}
APY(4,0.1);APY(12,.1);APY(1,10.2);APY(2,0.1);APY(365,0.096);APY(9999999,0.096)

Part A

A bank CD which pays 10 percent interest quarterly. \(APY=(1+\frac{APR}{m})^{m}-1=(1+\frac{.10}{4})^{4}-1=0.1038129\)

Part B

A bank CD which pays 10 percent monthly. \(0.1047131\)

Part C

A bank CD which pays 10.2 percent annually. \(0.102\)

Part D

A bank CD which pays 10 percent semiannually. \(0.1025\)

Part E

A bank CD which pays 9.6 percent daily (on a 365-day basis). \(0.1007452\)

Part F

A bank CD which pays 9.6 percent continuously. \(0.1007591\)

Problem 2

Part A

Plot a yield curve based on these data.

library(ggplot2)
Years <- c(0.5,1,2,3,4,5,10,20,30)
Rate <- c(5.1, 5.5, 5.6, 5.7, 5.8, 6, 6.1, 6.5, 6.3)
dat <- data.frame(Years,Rate)
ggplot(data=dat, aes(x=Years, y=Rate, group=1)) +
  geom_line(colour="darkblue", size=0.5) +
  geom_point() +
  labs(x="Years", y="Rate (Percent)") +
  ggtitle("Yield Curve") +
  theme(plot.title=element_text(size=7, face="bold")) +
  theme(axis.title.x=element_text(size=6)) +
  theme(axis.title.y=element_text(size=6)) +
  theme_bw()

Part B

What information does this graph tell you?

Part C

Suppose you need to borrow for 30 years to finance a house. Why not simply borrow short-term money each year for the next 30 years, rather than one a 30 year loan, to save money?

For each year the loan is renewed, interest rates are likely to increase, which thus exposes the borrower to rollover risks. If the lender borrows for 30 years, interest rate is locked at the rate stated in the contract.

Problem 3

You read in the financial pages of your newspaper that 30-day T-bills are currently yielding 5.5 percent. Your brother-in-law, a broker with XYZ Securities, has given you the following estimates of current interest rate premiums:

• Inflation premium (IP) = 3.25%
• Liquidity premium (LP) = 0.6%
• Maturity risk premium (MRP) = 1.8%
• Default risk premium (DRP) = 2.15%

On the basis of these data, what is the real risk-free rate of return?

\(5.5\%=r^{*}+3.25\% \equiv r^{*}=2.25\%\)

Problem 4

The real risk-free rate is 3 percent. Inflation is expected to be 3 percent for the next two years. A 2-year Treasury security yields 6.2 percent. What is the maturity risk premium for the 2-year security?

• \(r^{*}=3\%\)
• \(IP_{2}=3\%\)
• \(r_{T2}=6.2\%\)
• \(T=2\)

Problem 5

What are premium, discount and par bonds?

• Premium bonds are bonds that are sold above par value. \(\rightarrow k>r\)
• Discount bonds are bonds that are sold below par value. \(\rightarrow k<r\)
• Par bonds are bonds that are sold at par value. \(\rightarrow r=k\)
• Note that \(k\) is the coupon rate, and \(r\) is the market rate.

Problem 6

XYZ Inc. has 8 percent coupon bonds on the market that have 14 years left to maturity. If the YTM on these bonds is 9.1 percent, what is the current bond price?

• \(k=0.08\)
• \(r=0.091 \equiv \frac{r}{2}=0.0455\)
• \(m=2\)
• \(N=14\)
• \(PAR=\$1,000\)

Pb <- function(k,r,m,N,PAR) {
((k*PAR)/m)*((1/(r/m))-1/((r/m)*(1+(r/m))^(m*N)))+PAR/(1+(r/m))^(2*N)
}
Pb(0.08,0.091,2,14,1000)

## [1] 913.8968

Problem 7

A bond with maturity of 19.5 years sells for $1,047. If the coupon rate is 6.5%, what is the yield to maturity of the bond?

• \(N=19.5\)
• \(P_{Bond}=\$1,047\)
• \(PAR=\$1,000\)
• \(k=0.065\)
• \(m=2\)
• \(r=?\)

\(\$1,047=\frac{\$1,000*0.065}{2}(\frac{1}{\frac{r}{2}}-\frac{1}{\frac{r}{2}(1+\frac{r}{2})^{2*19.5}})+\frac{\$1,000}{(1+\frac{r}{2})^{2*19.5}} \equiv r = ?\)