F M213 S ummary notes
Present values
Perpetuity and Annuity
𝐶
𝑃𝑉𝑃 =
𝑟
𝐶 1
𝑃𝑉𝐴 = (1 − )
𝑟 (1 + 𝑟)𝑇
With growth
𝐶
𝑃𝑉𝑃 =
𝑟−𝑔
𝐶 1+𝑔 𝑇
𝑃𝑉𝐴 = (1 − ( ) )
𝑟−𝑔 1+𝑟
Quoted and effective interest rates
• Quoted annual: r%
• Effective annual interest rate under monthly compounding:
𝑟% 12
(1 + ) −1
12
• Effective annual interest rate under quarterly compounding:
𝑟% 4
(1 + ) −1
4
• Quoted annual: r%
• Interest rate on a 3-month loan: r%/4
Real and nominal
1+𝑟
(1 + 𝑖) =
1+𝜋
𝑖 ≈ 𝑟−𝜋
,Value of bonds and stocks
Bonds
Value of bonds = sum of present value of all coupon payments and face value
Face value also known as par value. If coupon rate > YTM, price of the bond will be above par
Bond Yield to maturity solves the following equation. It’s a transform of bond price and can be
used to infer bond price. Negative relationship between YTM and bond price
𝑐 𝑐 𝑐+𝐹 𝑐 1 𝐹
𝑃= + 2 + ⋯+ 𝑘
= (1 − 𝑘
)+
1 + 𝑦 (1 + 𝑦) (1 + 𝑦) 𝑦 (1 + 𝑦) (1 + 𝑦)𝑘
Stocks
Percentage Return of holding a stock = (capital gain + dividend received during
holding)/purchased price
Value of a stock = sum of all dividends (perpetuity if dividends are the same)
𝐸𝑡 (𝐷𝑡+𝑖 )
𝑃=∑
(1 + 𝑟)𝑖
𝐷
𝑃=
𝑟
- Notice that 𝑟 is required rate of return
- Dividends start from next period
Gordon growth formula
Basic:
- All earnings are paid out as dividends
𝐷
𝑃𝑡 =
𝑟−𝑔
Where r is the required rate of return and g is the constant growth rate
With plowback
- Some earnings are kept within the firm, generating growth in the future
- Payout ratio + plowback ratio = 1
- Payout ratio(𝟏 − 𝝆): is the ratio of dividends to earnings.
- Plowback ratio (𝝆): is the proportion of earnings retained by the firm and used
- for investment.
- Return on equity(ROE): a measure of the amount of earnings that a Dollar of equity (book
value) creates.
𝐸𝑃𝑆
𝑅𝑂𝐸 =
𝐵𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑒𝑞𝑢𝑖𝑡𝑦
𝑔 = 𝑅𝑂𝐸 × 𝑝𝑙𝑜𝑤𝑏𝑎𝑐𝑘 𝑟𝑎𝑡𝑖𝑜
, - The difference in value between the firm that plows back earnings and the firm that
doesn’t is called the present value of growth opportunities(PVGO)
o No growth:
𝐸𝑃𝑆1
𝑃′ =
𝑟
o With growth:
𝐸𝑃𝑆1 (1 − 𝜌)
𝑃′′ =
𝑟−𝑔
PVGO = P’’ – P’
𝐸𝑃𝑆
𝑃′′ = + 𝑃𝑉𝐺𝑂
𝑟
Comments:
- The firm that plows back earnings has a greater stock price
𝐸𝑃𝑆1 (1 − 𝜌)
𝑃=
𝑟 − 𝜌 × 𝑅𝑂𝐸
𝐸𝑃𝑆
- If ROE = r, the equation above simplifies to 𝑃 = 𝑟 1, the two firms would be worth the
same amount.
- If ROE > r, the shock price of the firm with growth will be greater than the firm with no
growth.
𝐸𝑃𝑆 𝑃𝑉𝐺𝑂
= 𝑟 (1 − )
⏟𝑃 𝑃
𝐸
𝑃 𝑟𝑎𝑡𝑖𝑜
- E/P ratio is not a good measure of required returns because they will understate required
returns for firms with large PVGO
Present values
Perpetuity and Annuity
𝐶
𝑃𝑉𝑃 =
𝑟
𝐶 1
𝑃𝑉𝐴 = (1 − )
𝑟 (1 + 𝑟)𝑇
With growth
𝐶
𝑃𝑉𝑃 =
𝑟−𝑔
𝐶 1+𝑔 𝑇
𝑃𝑉𝐴 = (1 − ( ) )
𝑟−𝑔 1+𝑟
Quoted and effective interest rates
• Quoted annual: r%
• Effective annual interest rate under monthly compounding:
𝑟% 12
(1 + ) −1
12
• Effective annual interest rate under quarterly compounding:
𝑟% 4
(1 + ) −1
4
• Quoted annual: r%
• Interest rate on a 3-month loan: r%/4
Real and nominal
1+𝑟
(1 + 𝑖) =
1+𝜋
𝑖 ≈ 𝑟−𝜋
,Value of bonds and stocks
Bonds
Value of bonds = sum of present value of all coupon payments and face value
Face value also known as par value. If coupon rate > YTM, price of the bond will be above par
Bond Yield to maturity solves the following equation. It’s a transform of bond price and can be
used to infer bond price. Negative relationship between YTM and bond price
𝑐 𝑐 𝑐+𝐹 𝑐 1 𝐹
𝑃= + 2 + ⋯+ 𝑘
= (1 − 𝑘
)+
1 + 𝑦 (1 + 𝑦) (1 + 𝑦) 𝑦 (1 + 𝑦) (1 + 𝑦)𝑘
Stocks
Percentage Return of holding a stock = (capital gain + dividend received during
holding)/purchased price
Value of a stock = sum of all dividends (perpetuity if dividends are the same)
𝐸𝑡 (𝐷𝑡+𝑖 )
𝑃=∑
(1 + 𝑟)𝑖
𝐷
𝑃=
𝑟
- Notice that 𝑟 is required rate of return
- Dividends start from next period
Gordon growth formula
Basic:
- All earnings are paid out as dividends
𝐷
𝑃𝑡 =
𝑟−𝑔
Where r is the required rate of return and g is the constant growth rate
With plowback
- Some earnings are kept within the firm, generating growth in the future
- Payout ratio + plowback ratio = 1
- Payout ratio(𝟏 − 𝝆): is the ratio of dividends to earnings.
- Plowback ratio (𝝆): is the proportion of earnings retained by the firm and used
- for investment.
- Return on equity(ROE): a measure of the amount of earnings that a Dollar of equity (book
value) creates.
𝐸𝑃𝑆
𝑅𝑂𝐸 =
𝐵𝑜𝑜𝑘 𝑣𝑎𝑙𝑢𝑒 𝑜𝑓 𝑒𝑞𝑢𝑖𝑡𝑦
𝑔 = 𝑅𝑂𝐸 × 𝑝𝑙𝑜𝑤𝑏𝑎𝑐𝑘 𝑟𝑎𝑡𝑖𝑜
, - The difference in value between the firm that plows back earnings and the firm that
doesn’t is called the present value of growth opportunities(PVGO)
o No growth:
𝐸𝑃𝑆1
𝑃′ =
𝑟
o With growth:
𝐸𝑃𝑆1 (1 − 𝜌)
𝑃′′ =
𝑟−𝑔
PVGO = P’’ – P’
𝐸𝑃𝑆
𝑃′′ = + 𝑃𝑉𝐺𝑂
𝑟
Comments:
- The firm that plows back earnings has a greater stock price
𝐸𝑃𝑆1 (1 − 𝜌)
𝑃=
𝑟 − 𝜌 × 𝑅𝑂𝐸
𝐸𝑃𝑆
- If ROE = r, the equation above simplifies to 𝑃 = 𝑟 1, the two firms would be worth the
same amount.
- If ROE > r, the shock price of the firm with growth will be greater than the firm with no
growth.
𝐸𝑃𝑆 𝑃𝑉𝐺𝑂
= 𝑟 (1 − )
⏟𝑃 𝑃
𝐸
𝑃 𝑟𝑎𝑡𝑖𝑜
- E/P ratio is not a good measure of required returns because they will understate required
returns for firms with large PVGO
