1. 2. Basics

2. Future Value The future value (FV) of an amount (PV) is where here
r is the rate of return per period
n is the number of compounding periods
Common compounding periods are: year, half-year, month, day, etc.

3. Present Value The present value (PV) of a future amount (FV) is where
r is the discount rate per period
n is the number of compounding periods
Word “discount” is used because future amount is made smaller.

4. FV of a Fixed Sum
If annual rate of return is 8%, how much invested today will grow to $50,000 after 5 years?

5. Legend has it that Indians sold Manhattan Island to the Dutch for $24 in 1626.

6. Semi-Annual and Monthly Compounding
If the Indians could have invested the money at 6%, how much would they have today?

7. Ordinary Annuity
If same amount is paid at end of each period, with first payment one period from now, series is an ordinary annuity whose PV is given by
where A = amount of each payment
r = discount rate per period
n = total number of periods

8. PV of an Ordinary Annuity
An investor receives $100 on Aug 18 for each of the next 8 years, with the first payment one year from now. Assume 9% is an appropriate discount rate. What is the PV of this annuity?

9. FV of an Ordinary Annuity
You receive $150,000 on Aug 18 each year for the next 5 years. As soon as you receive the payments, you invest them at 7% annually. How much will you have at the end of the 5 years?

10. Bond
Borrower (issuer) promises contractually to make periodic payments (called coupon payments, usually semiannually) to bondholder over a given number of years.
At maturity, bondholder receives last coupon payment and principal (face value or par).

11. Computing Value of a Bond
Compute PV of bond by computing PV of each of the bond’s cash flows and adding up. When first coupon payment is one period from now, formula is:
where
C = amount of each coupon payment
r = appropriate discount rate per period
n = total number of periods
F = principal (par value, face value)
Discount rate is ascertained from yields on similar bonds. The discount usually differs from coupon rate.

12. Value of a Bond
What is the PV of a $1,000 bond that has just made a coupon payment, has 2 years to maturity, pays interest semiannually, and has a coupon rate of 6%? Assume similar bonds yield 7%. Helpful to make a timeline.

13. Zero Coupon Bond Since there is no C, in U.S. the customary formula is
where n is double the number of years. That is, we customarily assume semiannually compounding when pricing a zero coupon bond.
What is price of a $1,000 zero coupon bond that matures in 15 years if it is to yield 9.4%?

14. PV vs. NPV Functions
PV function – cashflows must be evenly spaced and equal. Can be either at beginning or end of period. Can also handle balloon payment at end.
NPV function – cashfows must be evenly spaced but can be variable. Can’t handle time-zero cashflows.

15. Bond Coming Out of Bankruptcy
Consider an 8% (semiannual payments) bond that matures Feb 18, Bondholders are asked to accept:
no coupon payments until Aug 18, 2012, but then at only half the coupon rate
resume full coupon payments on Aug 18, 2014 until bond matures as normal.
As of Aug 18, 2010, what is bond worth if the appropriate discount rate is 10%?

16. Fractional Part of a Period
On Aug 18, 2010 what is the value of a $5,000 zero coupon bond that matures on Jan 18, 2011 assuming that yield to use is 5.8%?

17. Accrued Rent
Clean price vs. Full price. Full price is PV whereas clean price is after subtracting off accrued rent.
Clean Price = Full Price – Accrued Rent
where
On Aug 18, what is accrued rent on a 5% $1000 bond that matures on 10/15/15?

18. Saw-Tooth Pattern
When buy a bond, you pay is Full Price, but Clean Price is what is reported in the media.
Clean Price + Accrued Rent = Full Price
Full Price has a saw-tooth pattern. Clean price smoothes this out.
Full-price saw-tooth pattern of a 3-year 6% bond (semi-annual payments) yielding 6% is: 18

19. Full Price of Purchase
Suppose a 8% bond (next semi-annual coupon payment on Dec 23) is quoted in the media at How much would 2000 of them cost? 19

20. Internal Rate of Return
IRR is the discount rate that causes the NPV of the cashflows to be zero.

21. Multiple Answers with IRR
A problem with IRR is that there can be multiple answers.

22. Beta & Portfolio Beta
Every stock has a beta. Obtained by regressing the stock’s returns onto the returns of a broad stock market index. In U.S. we use the S&P 500.
Portfolio beta: A weighted average of the betas of the securities in the portfolio. The weights are the proportions of the portfolio invested in the given stocks.

23. Yield-to-Maturity
Yield-to-maturity. The annual rate that causes all cash flows to discount back to the bond’s market price. Solved by trial-and-error.
What is the yield-to-maturity on a 12-year, 8% coupon bond (semi-annual payments) whose price is $1,097.37?

24. Discount Rate Risk Discount rate risk. Two components:
Reinvestment risk (chance lender will not be able to reinvest coupon payments for remaining time to maturity at the rate the bond has at the time it was purchased)
Price risk (chance discount rate levels will change and thus affect price of the bond)
Duration is the number of years at which the two components counterbalance one another.

25. Duration
Duration is the number of years at which the two components counterbalance one another.
Duration is year-weighted average of the PV of a bond’s cash flows over price of bond
where r is bond’s current yield-to-maturity.

26. Macaulay Duration
Macaulay duration is first derivative of price of bond with respect to r. Can be used to predict changes in price of bond for small changes in yield (discount rate).

27. % Change in Price of Bond
Consider a 20-year, 5% coupon rate bond (annual payments) yielding 4.5%. By what percentage will price of bond change if yield increases 75 basis points?

28. Managing Discount Rate Risk
Zero-coupon approach: Buy “zero” with maturity equal to desired holding period. Locks in YTM. No reinvestment risk because no coupons payments. No price risk when held to maturity. Best way.
Duration matching: Form a portfolio of bonds whose duration matches desired holding period. Good but not best way.
Maturity matching: Select bonds with terms to maturity equal to desired holding period. Don’t use. Doesn’t work for eliminating discount rate risk.

29. Normal Curve (Mean & Stdev Known)
What is cutoff value when know area wanted lower tail?
What is the area in the lower tail when know cutoff value?
What is height of normal curve at a given x-value?
NORMINV(probability, mean, stdev)
NORMDIST(x-value, mean, stdev, 1)
NORMDIST(x-value, mean, stdev, 0)

30. Value at Risk For instance, “90-day 1% VaR is $3 million”.
Means that in next 90 days there is a 99% chance will not lose more that $3 million.
But what might be lurking in 1%?

31. Matrices Matrix is a rectangular array of elements.
There is matrix algebra: matrix addition, matrix subtraction, matrix multiplication, etc.
In matrix multiplication, two key things:
order is important
#cols of 1st matrix must equal #rows of 2nd

32. Random Weighting Vectors
Create 10 random weighting vectors of length 4 that sum one.
Do by creating 4 random numbers. Then divide each by the total of the four.

33. End