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Lecture Weeks 3 and 4

This lecture discusses and cover the concept of time value of money. The lecture discusses the future and present value of annuities and perpetuities. The lecture further discusses the valuation of bonds and stocks. The lecture will be covered over two weeks. The lecture is divided into five parts.

Lecture Week 3

PART A

Time Value of Money, Future Value and Present Value

Time value of Money

  • The value of £1 today as compared to some other period in the future or past.
  • Almost every commodity loses value or gain value over time.
  • Since all values are recorded in currency (£ in the UK), time value of commodities also implies time value of money.
  • Money over time will lose its value because of inflation. Suppose price of X in 2010 was £1.00 per unit but now in 2020 it is £2.00 per unit. The value of £ has fallen.
  • The trade off between the value of money today and some other period depends upon the relevant rate of interest.

Time Value of Money, Future Value and Present Value

Future Value (FV)

  • The value of money (or commodity) at a certain period in the future at a certain interest rate.
  • For example, the value of a house in 10 years at 10% interest rate per year. The value of a saving accounts in 5 years at 12% interest rate per year.
  • Future value is a compounded value. It is added on.

Future Value Formula

  • Future of £1 invested for t period at a rate of interest r per year is
  • FV = £1 x (1 + r)t
  • (1 + r)t is called the future value interest factor.

Time Value of Money, Future Value and Present Value

Two Factors Affect FV

  • (i) Interest Rate (r)
  • (ii) Time period (t)
  • The relationship between FV and r is positive. As r increases, FV increases and the opposite also holds.
  • The relationship between FV and t is positive. As t increases, FV increases and the opposite also holds.

  • What if interest is paid more than once a year?
  • FV = £1 x (1 + r/n)txn
  • n = number of times interest is paid.
  • For example, n=4, interest is paid quarterly (4 times a year). If n=12, interest is paid monthly.
  • What is the relationship between n and FV?

Time Value of Money, Future Value and Present Value

Present Value (PV)

  • The current value of money (or commodity) discounted at a certain rate of interest.
  • For example, how much investment today will grow to £1 million at 8% per year in 5 years?

Present Value Formula

  • The current value of £1 t period in the future at the interest rate r.
  • PV = £1 x [1/(1+r)t]
  • 1/(1+r)t is called the present value interest factor.

Time Value of Money, Future Value and Present Value

Two Factors Affect the PV

  • (i) Interest Rate (r)
  • (ii) Time period (t)
  • The relationship between PV and r is inverse. As r increases, PV decreases and the opposite also holds.
  • The relationship between PV and t is inverse. As t increases, PV decreases and the opposite also holds.

  • What if interest is paid more than once a year?
  • PV = £1 x [1/(1 + r/n)txn]
  • n = number of times interest is paid.
  • For example, n=4, interest is paid quarterly (4 times a year). If n=12, interest is paid monthly.
  • What is the relationship between n and PV?

Lecture Week 3

  • PART B

Time Value of Money, Future Value and Present Value

Annuities and Perpetuities

Annuity

  • A series of constant cash flow for some fixed number of period.
  • Annuities are very common in the financial market. Examples, car loan, most mortgage loan, etc.

  • PV of annuity when interest is paid once a year
  • PV = c x [1-(1/(1 + r)t)]/r
  • c = constant cash flow per year for time period t.

  • PV of annuity when interest is paid more than once a year
  • PV = c x [1-(1/(1 + r/n)txn)]/(r/n)
  • c = constant cash flow per year for time period t.

Time Value of Money, Future Value and Present Value

  • FV of annuity when interest is paid once a year
  • FV = c x [(1 + r)t – 1]/r
  • FV of annuity when interest is paid more than once a year
  • FV = c x [(1 + r/n)txn – 1]/(r/n)

Time Value of Money, Future Value and Present Value

Perpetuity

  • An annuity in which the cash flow continues forever. For example, Consols bonds.
  • PV = c/r

Growing Perpetuity

  • An annuity in which growing cash flow continues forever.
  • PV = c/(r – g)
  • The rate of interest r has to be higher than the growth rate g. Why?

Lecture Week 3

PART C

Bond Valuation and Stock Valuation

Characteristics of a Bond and Bondholders

  • Bonds are debt securities.
  • A bond is a promise by the borrower (firm or the government) to pay the lender (bondholder) certain amount of money per period for a certain length of time.
  • Bondholders are not owners of the firms. They are debt holders.
  • Since they are not owners, they do not control the firm.
  • Since they are debt holders, bondholders get paid first before anybody.
  • In case the firm is liquidated, the bondholders are first in line to get paid from the bankrupt firm.
  • Life of a bond is limited.
  • The interest payment on a bond is fixed (usually).

Bond Valuation and Stock Valuation

Features of a Feature Bond

Coupon Payment

  • Stated interest payment on the bonds.
  • The coupon payment is (usually) fixed or constant.

Face Value

  • The amount paid during the last period of the bond along with the last coupon payment.
  • A bond does not have to sell for the face value.

Coupon Rate

  • The interest rate on the bond. It is equal to coupon payment divided by the face value. It is fixed (usually).

Maturity

  • The length of life of a bond.
  • The numbers of years until the face value is paid.

Bond Valuation and Stock Valuation

Problem with Bond Value

  • Cash flow from the bonds is (usually) constant or fixed but the interest rates in the market are always changing.
  • As market interest rate changes, the value of the bond will change.
  • Market value of the bond depends on the market interest rate and not on the coupon rate.

Bond Valuation

  • To find the actual value of a bond at a certain time, interest rate known as the yield to maturity have to be used.
  • Yield to maturity is the most accurate interest rate. It is a market interest rate.
  • It equates the present value of a bond’s interest payments and face value repayment with its current price.

  • Bond value = C x [1-(1/(1 + r)t)]/r + F/(1 + r)t

  • C = Coupon payment per period.
  • r = Yield to maturity interest rate
  • F = Face value
  • The first part is the PV of the coupon payment using the annuity formula.
  • The second part is the PV of the face value.

Lecture Week 4

PART D

Bond Valuation and Stock Valuation

Factors affecting the Value of Bonds

  • A bond’s value (yield) is influenced by prevailing market interest rates and characteristics unique to that bond.

  • Four factors are common to most bonds that affect the value of the bond.

Bond Valuation and Stock Valuation

Default Risk

  • Default risk is the important risk associated with bonds.

  • Default risk is the probability that the borrower (firms or government) will default on the coupon payments and face value.

  • Investors must consider the creditworthiness of the bond issuer (the borrower).

Bond Valuation and Stock Valuation

  • Bonds are rated by the level of potential default riskiness of the bonds.

  • The two well know rating systems are the one provided by Moody’s and Standard and Poor’s. Investors tend to use these rating systems to assess the creditworthiness of the borrower (issuer). Fitch is another bond rating company.
  • These ratings are based on a financial assessment of the issuing corporation. The higher the rating, the lower the perceived default risk. Of course, rating can change over time as conditions and situations change.

Bond Valuation and Stock Valuation

  • If all other characteristics besides the default risk were equal, bonds with higher degree of default risk would have to offer higher yield (return). Why?

  • Default risk is especially relevant for longer-term bonds that expose investors to the possibility of default for a longer time. Why?

  • Short-term Treasury bills issued by UK or US governments are default risk free. Why?

Rating Chart

Bond Valuation and Stock Valuation

Liquidity

  • Investors prefer securities that are liquid.

  • Liquidity implies how fast and easily a security can be converted to cash without loss of value. Cash is the most liquid asset and real estate is one of the least liquid assets.

  • If all other characteristics were equal, bonds with lower liquidity would have to offer higher return or yield.

  • Bonds with a short-term maturity or an active secondary market have higher liquidity.

  • For investors who will not need their funds till the bond matures, low liquidity is tolerable.

Bond Valuation and Stock Valuation

Tax Status

  • Investors are more concern with after-tax income than before-tax income earned on securities.

  • If all other characteristics were equal, taxable bonds would have to offer a higher before-tax yield to investors than tax-exempt bonds. Why?

  • Investors in high tax brackets benefit most from tax-exempt securities.

  • When assessing the expected yields of various securities, it is common to convert them into an after-tax form, as follows

  • Yat = Ybt(1 – T)
  • Where Yat = after-tax yield
  • Ybt = before-tax yield
  • T = Tax rate
  • Investors retain only a percentage (1-T) of the before tax yield once taxes are paid.

Bond Valuation and Stock Valuation

Term to Maturity

  • Maturity differs among bonds is another reason that bond yield differ.

  • Bonds with identical risk, liquidity and tax characteristics may have to offer different interest rates because the time remaining to maturity is different. Why?

  • The longer the bond takes to mature probability increases that value of the bond will fall or default.

Lecture Week 4

PART E

Bond Valuation and Stock Valuation

Characteristics of Common Shares and Shareholder

  • Common shareholders are the true owners of the firm.
  • They control the firm.
  • They are last to be paid.
  • In case of bankruptcy shareholders are the last in line for payment from the liquidation of the firm.
  • No promised cash flows. No coupon payment or face value.
  • Life of a share could be essentially forever.
  • No stated interest rate on the share. No easy way to observe the rate of return.
  • Shareholders are paid dividend. Dividends are part of firm’s profit paid to the owners.
  • Dividends may not be paid regularly.

Bond Valuation and Stock Valuation

Common Share Valuation

  • P0 = Price of a share today. This is the present value (PV) of the share.
  • P1 = Price of the share next period.
  • D1 = Dividend paid in the next period.
  • r = The minimum required rate of return in the market on this investment. This could be a market interest rate.
  • P0 = D1/(1 + r) + P1/(1 + r) (Equation 1)
  • D1/(1 + r) = PV of the dividend next period.
  • P1/(1 + r) = PV of the share price next period.
  • What is P1 equal to?
  • P1 = D2/(1 + r) + P2/(1 + r) (Equation 2)
  • Substitute equation 2 into equation 1

Bond Valuation and Stock Valuation

  • P0 = D1/(1 + r) + D2/(1 + r)2 + P2/(1 + r)2 (Equation 3)
  • What is P2 equal to?
  • P2 = D3/(1 + r) + P3/(1 + r) (Equation 4)

  • Substitute equation 4 into equation 3
  • P0 = D1/(1 + r) + D2/(1 + r)2 + D3/(1 + r)3 + P3 /(1 + r)3 (Equation 5)
  • What is P3 equal to?
  • P3 = D4/(1 + r) + P4/(1 + r)

  • Keep on doing these substitutions, until present value of the future share price is almost zero.
  • P0 = D1/(1+ r) + D2/(1+ r)2 + D3/(1+ r)3 + D4/(1+ r)4 + …+ Pn /(1+ r)n (Eq 6)

Bond Valuation and Stock Valuation

  • What does equation 6 show?
  • It shows that the price of a share today is equal to the sum of present value of all of the future dividends.
  • How many future dividends are there?
  • There could be infinite number or very few, if the firm does not survive.
  • What is effect of the share price in the last period (Pn)?
  • Almost nothing or very little. The last element of equation 6, Pn /(1+ r)n is a very small number.

Question

  • All assets of a bankrupt firm belong to the bond holders and other debt holders. Stock holders lose all control over the assets. If that is true, then why do bondholders (sometimes) promise to pay the stockholders part of the money from the sale of the assets?

  • In your opinion what is better to invest in, stocks or bonds?