Time Value of Money
Financial Ratios
Equations
Liquidity Ratios
𝑭𝒊𝒔𝒉𝒆𝒓 𝑬𝒒𝒖𝒂𝒕𝒊𝒐𝒏 = 𝑁𝑜𝑚𝑖𝑛𝑎𝑙 (1 + 𝑛𝑜𝑚𝑖𝑛𝑎𝑙) = 𝑅𝑒𝑎𝑙 (1 + 𝑟𝑒𝑎𝑙) + 𝐼𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 (1 + 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛)
𝑨𝑷𝑹 = $1 (𝑥)[(1 +
𝑪𝒖𝒓𝒓𝒆𝒏𝒕 𝑹𝒂𝒕𝒊𝒐 =
𝑸𝒖𝒊𝒄𝒌 𝑹𝒂𝒕𝒊𝒐 =
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐴𝑠𝑠𝑒𝑡𝑠
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐿𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐴𝑠𝑠𝑒𝑡𝑠 − 𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐿𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠
𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠
𝑵𝒆𝒕 𝑾𝒐𝒓𝒌𝒊𝒏𝒈 𝑪𝒂𝒑𝒊𝒕𝒂𝒍 𝑻𝒖𝒓𝒏𝒐𝒗𝒆𝒓 =
𝑁𝑒𝑡 𝑊𝑜𝑟𝑘𝑖𝑛𝑔 𝐶𝑎𝑝𝑖𝑡𝑎𝑙 (𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐴𝑠𝑠𝑒𝑡𝑠 − 𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝐿𝑖𝑎𝑏𝑖𝑙𝑖𝑡𝑖𝑒𝑠)
Asset Management Ratios
𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠
𝑁𝑒𝑡 𝐴𝑐𝑐𝑐𝑜𝑢𝑛𝑡𝑠 𝑅𝑒𝑐𝑒𝑖𝑣𝑎𝑏𝑙𝑒𝑠
𝐶𝑜𝑠𝑡 𝑜𝑓 𝐺𝑜𝑜𝑑𝑠 𝑆𝑜𝑙𝑑
𝑹𝑴𝑨 𝑰𝒏𝒗𝒆𝒏𝒕𝒐𝒓𝒚 𝑻𝒖𝒓𝒏𝒐𝒗𝒆𝒓 =
𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦
𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠
𝑺&𝑷 𝑰𝒏𝒅𝒖𝒔𝒕𝒓𝒚 𝑺𝒖𝒓𝒗𝒆𝒚𝒔 𝑰𝒏𝒗𝒆𝒏𝒕𝒐𝒓𝒚 𝑻𝒖𝒓𝒏𝒐𝒗𝒆𝒓 =
𝐼𝑛𝑣𝑒𝑛𝑡𝑜𝑟𝑦
𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠
𝑭𝒊𝒙𝒆𝒅 𝑨𝒔𝒔𝒆𝒕 𝑻𝒖𝒓𝒏𝒐𝒗𝒆𝒓 =
𝑁𝑒𝑡 𝐹𝑖𝑥𝑒𝑑 𝐴𝑠𝑠𝑒𝑡𝑠
𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠
𝑻𝒐𝒕𝒂𝒍 𝑨𝒔𝒔𝒆𝒕 𝑻𝒖𝒓𝒏𝒐𝒗𝒆𝒓 =
𝑇𝑜𝑡𝑎𝑙 𝐴𝑠𝑠𝑒𝑡𝑠
𝑨𝒄𝒄𝒐𝒖𝒏𝒕𝒔 𝑹𝒆𝒄𝒆𝒊𝒗𝒂𝒃𝒍𝒆𝒔 𝑻𝒖𝒓𝒏𝒐𝒗𝒆𝒓 =
Debt Management (Leverage) Ratios
𝐶𝑜𝑠𝑡 𝑜𝑓 𝐺𝑜𝑜𝑑𝑠 𝑆𝑜𝑙𝑑
𝑹𝑴𝑨 𝑨𝒄𝒄𝒐𝒖𝒏𝒕𝒔 𝑷𝒂𝒚𝒂𝒃𝒍𝒆𝒔 𝑻𝒖𝒓𝒏𝒐𝒗𝒆𝒓 =
𝐴𝑐𝑐𝑜𝑢𝑛𝑡𝑠 𝑃𝑎𝑦𝑎𝑏𝑙𝑒𝑠
𝑺&𝑷 𝑰𝒏𝒅𝒖𝒔𝒕𝒓𝒚 𝑺𝒖𝒓𝒗𝒆𝒚𝒔 𝑨𝒄𝒄𝒐𝒖𝒏𝒕𝒔 𝑷𝒂𝒚𝒂𝒃𝒍𝒆𝒔 𝑻𝒖𝒓𝒏𝒐𝒗𝒆𝒓 =
𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠
𝐴𝑐𝑐𝑜𝑢𝑛𝑡𝑠 𝑃𝑎𝑦𝑎𝑏𝑙𝑒𝑠
𝑇𝑜𝑡𝑎𝑙 𝐷𝑒𝑏𝑡
𝑇𝑜𝑡𝑎𝑙 𝐴𝑠𝑠𝑒𝑡𝑠
𝐸𝐵𝐼𝑇
𝑻𝒊𝒎𝒆𝒔 𝑰𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝑬𝒂𝒓𝒏𝒆𝒅 =
𝐴𝑛𝑛𝑢𝑎𝑙 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝐸𝑥𝑝𝑒𝑛𝑠𝑒
𝑫𝒆𝒃𝒕 𝒕𝒐 𝑻𝒐𝒕𝒂𝒍 𝑨𝒔𝒔𝒆𝒕𝒔 =
𝑳𝒆𝒗𝒆𝒓𝒂𝒈𝒆 𝑴𝒖𝒍𝒕𝒊𝒑𝒍𝒊𝒆𝒓 𝑹𝒂𝒕𝒊𝒐 =
𝑇𝑜𝑡𝑎𝑙 𝐴𝑠𝑠𝑒𝑡𝑠
𝐶𝑜𝑚𝑚𝑜𝑛 𝐸𝑞𝑢𝑖𝑡𝑦
Profitability Ratios
𝑁𝑒𝑡 𝐼𝑛𝑐𝑜𝑚𝑒
𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠
𝐺𝑟𝑜𝑠𝑠 𝑃𝑟𝑜𝑓𝑖𝑡 (𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠 − 𝐶𝑂𝐺𝑆)
𝑵𝒆𝒕 𝑮𝒓𝒐𝒔𝒔 𝑷𝒓𝒐𝒇𝒊𝒕 𝑴𝒂𝒓𝒈𝒊𝒏 =
𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠
𝐸𝐵𝐼𝑇
𝑵𝒆𝒕 𝑶𝒑𝒆𝒓𝒂𝒕𝒊𝒏𝒈 𝑷𝒓𝒐𝒇𝒊𝒕 𝑴𝒂𝒓𝒈𝒊𝒏 =
𝑁𝑒𝑡 𝑆𝑎𝑙𝑒𝑠
𝑁𝑒𝑡 𝐼𝑛𝑐𝑜𝑚𝑒
𝑹𝒆𝒕𝒖𝒓𝒏 𝑶𝒏 𝑨𝒔𝒔𝒆𝒕𝒔 (𝑹𝑶𝑨) =
𝑇𝑜𝑡𝑎𝑙 𝐴𝑠𝑠𝑒𝑡𝑠
𝑁𝑒𝑡 𝐼𝑛𝑐𝑜𝑚𝑒
𝑹𝒆𝒕𝒖𝒓𝒏 𝒐𝒏 𝑬𝒒𝒖𝒊𝒕𝒚 (𝑹𝑶𝑬) =
𝐶𝑜𝑚𝑚𝑜𝑛 𝐸𝑞𝑢𝑖𝑡𝑦
𝑬𝑨𝑹 = [(1 +
𝐴𝑃𝑅
) 𝑚𝑡]
𝑚
𝐴𝑃𝑅
) 𝑚𝑡] = 1 + 𝐸𝐴𝑅
𝑚
𝑭𝑽-𝑡 = 𝑃𝑉0(𝑥)(1 + 𝑟)𝑡
𝐹𝑉𝑡
𝑷𝑽𝟎 =
𝑡
(1 + 𝑟)
1
𝑫𝒊𝒔𝒄𝒐𝒖𝒏𝒕 𝑭𝒂𝒄𝒕𝒐𝒓 =
𝑡
(1 + 𝑟)
Multiple Cash Flows
∞
𝐶𝐹𝑡
𝐶𝐹1
𝐶𝐹2
𝑃𝑉 = ∑
𝑡 = 𝐶𝐹0 + (1 + 𝑟)1 + (1 + 𝑟)2 + ⋯
(1
+
𝑟)
𝑡=0
Ordinary Annuities & Annuities Due
𝑃𝑉 𝑜𝑓 𝑎𝑛 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 (𝑃𝑉𝐴) =
𝐹𝑉 𝑜𝑓 𝑎𝑛 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 (𝐹𝑉𝐴) =
𝑇
𝐶𝐹𝑡
𝐶𝐹
𝐶𝐹
= 𝐶𝐹 +
+
+⋯
∑
(1 + 𝑟)𝑡
(1 + 𝑟)1 (1 + 𝑟)2
𝑡=0
𝑇
∑ 𝐶𝐹𝑡 (1 + 𝑟)𝑡 = 𝐶𝐹1 (1 + 𝑟)1 + 𝐶𝐹1(1 + 𝑟)2 + ⋯
𝑡=0
1
1−
(1 + 𝑟)𝑇
𝑃𝑉 𝐹𝑎𝑐𝑡𝑜𝑟 = [
]
𝑟
1
1−
(1 + 𝑟)𝑇
𝑃𝑉 𝑜𝑓 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝐷𝑢𝑒 = (1 + 𝑟)𝐶𝐹 [
]
𝑟
(1 + 𝑟)𝑇 − 1
𝐹𝑉 𝑜𝑓 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝐷𝑢𝑒 = (1 + 𝑟)𝐶𝐹 [
]
𝑟
Perpetuities
𝑃𝑉 𝑜𝑓 𝑎 𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 = 𝐶𝐹 ⁄𝑟
𝑟 = 𝐶𝐹 ⁄𝑃𝑉
𝐶𝐹 = 𝑃𝑉 (𝑋) 𝑟
𝑵𝒆𝒕 𝑷𝒓𝒐𝒇𝒊𝒕 𝑴𝒂𝒓𝒈𝒊𝒏 =
Cash Flow Ratios
𝐶𝐹 𝑓𝑟𝑜𝑚 𝑂𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 + 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡
𝐶𝐹 𝑓𝑟𝑜𝑚 𝑂𝑝𝑒𝑟𝑎𝑡𝑖𝑛𝑔 − 𝑃𝑟𝑒𝑓𝑒𝑟𝑟𝑒𝑑 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑𝑠
𝑸𝒖𝒂𝒍𝒊𝒕𝒚 𝒐𝒇 𝑬𝒂𝒓𝒏𝒊𝒏𝒈𝒔 =
𝐸𝑎𝑟𝑛𝑖𝑛𝑔𝑠 𝐴𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑓𝑜𝑟 𝐶𝑜𝑚𝑚𝑜𝑛 𝑆ℎ𝑎𝑟𝑒ℎ𝑜𝑙𝑑𝑒𝑟𝑠
𝑪𝑭 𝑪𝒐𝒗𝒆𝒓𝒂𝒈𝒆 𝒐𝒇 𝑰𝒏𝒕𝒆𝒓𝒆𝒔𝒕 𝑬𝒙𝒑𝒆𝒏𝒔𝒆𝒔 =
DuPont Analysis
Firms ROE is split into three parts
Expense Control: Profit Margin
Asset Utilization: Total Asset Turnover
Financial Leverage: Leverage Multiplier
Interest
𝑁𝑜𝑚𝑖𝑛𝑎𝑙 𝐼𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑅𝑎𝑡𝑒 = 𝑟 = 𝑟 ∗ + 𝐼𝑃 + 𝐷𝑅𝑃 + 𝐿𝑃 + 𝑀𝑅𝑃 (𝑜𝑟 𝑅𝑅𝑃)
𝑟 ∗ = 𝑅𝑒𝑎𝑙 (𝑜𝑟 𝑝𝑢𝑟𝑒) 𝑟𝑖𝑠𝑘 𝑓𝑟𝑒𝑒 𝑖𝑛𝑡𝑒𝑟𝑒𝑠𝑡 𝑟𝑎𝑡𝑒
𝐼𝑃 = 𝑇ℎ𝑒 𝑖𝑛𝑓𝑙𝑎𝑡𝑖𝑜𝑛 𝑝𝑟𝑒𝑚𝑖𝑢𝑚
𝐷𝑅𝑃 = 𝑇ℎ𝑒 𝑑𝑒𝑓𝑎𝑢𝑙𝑡 (𝑐𝑟𝑒𝑑𝑖𝑡) 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚
𝐿𝑃 = 𝑇ℎ𝑒 𝑙𝑖𝑞𝑢𝑖𝑑𝑖𝑡𝑦 𝑝𝑟𝑒𝑚𝑖𝑢𝑚
𝑀𝑅𝑃 = 𝑀𝑎𝑡𝑢𝑟𝑖𝑡𝑦 𝑟𝑖𝑠𝑘 𝑝𝑟𝑒𝑚𝑖𝑢𝑚
𝑅𝑅𝑃 = 𝑅𝑒𝑖𝑛𝑣𝑒𝑠𝑡𝑚𝑒𝑛𝑡 𝑅𝑖𝑠𝑘 𝑃𝑟𝑒𝑚𝑖𝑢𝑚
Types of Yield Curves
Yield Curve - Moderate rate levels rising continuously with increasing
maturity (upward sloping)
Rising Yield Curve - Low rate levels rising substantially with increasing
maturity
Inverted Yield Curve - Short-term rates high and declining over entire
maturity range
Flat Yield Curve - Rates are relatively constant (invariant) over entire
maturity range
Bonds
Bond Valuation
Step 1: PV of an Annuity e.g. 20n; 3i; 30PMT; PV = $446.32
Step 2: PV of a Single Sum e.g. 20n; 3i; 1,000FV; PV = $553.68
Use the market interest rate for i
Yield-to-Maturity (YTM) – Discount rate that equates bond price to the PV of
all promised cash flows.
APR = Bond equivalent yield. E.g. semi-annual is 2*r, not EAR which would
be [1+r/2)]2-1
$𝐴𝑛𝑛𝑢𝑎𝑙 𝐶𝑜𝑢𝑝𝑜𝑛 𝑃𝑎𝑦𝑚𝑒𝑛𝑡
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑌𝑖𝑒𝑙𝑑 =
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑃𝑟𝑖𝑐𝑒
Interest Rate Risk (or price risk) – Risk associated with price fluctuations
caused by interest rate changes
Reinvestment Rate Risk – Risk associated with the rate at which coupons are
reinvested
Default Risk (or credit risk) – Risk associated with issuer’s ability to make
payments as specified.
Stocks
𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝐴𝑚𝑜𝑢𝑛𝑡
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑆𝑡𝑜𝑐𝑘 𝑃𝑟𝑖𝑐𝑒
𝑃𝑉0 =
+
+⋯
(1 + 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑅𝑎𝑡𝑒)1 (1 + 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝑅𝑎𝑡𝑒)1
𝐷1
= 𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝐷𝑖𝑣𝑖𝑑𝑒𝑛𝑑 𝑌𝑖𝑒𝑙𝑑
𝑃0
G = Expected growth rate
𝐴𝑚𝑜𝑟𝑡𝑖𝑧𝑒𝑑 𝐿𝑜𝑎𝑛 𝐴𝑚𝑜𝑢𝑛𝑡 = 𝐶 (𝑥)
𝐷1
+𝑔
𝑃0
1 𝑛]
)
1+𝑟
𝑟
[(1 − (
LT bonds are subject to greater interest rate risk.
ST bonds are subject to greater reinvestment risk
Both LT and ST bonds are subject to reinvestment risk on coupons
Present Value & Future Value
𝐹𝑉 = $1 (𝑥)(1 + 𝑟)𝑡
$1
𝑃𝑉 =
(1 + 𝑟)𝑡
1
(1 − [
)
(1 + 𝑟)𝑡
𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝑃𝑉 = 𝑐 (𝑥)
𝑟
𝑃𝑉 𝑜𝑓 𝑎 𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 = 𝐶 ⁄𝑟
(1 + 𝑔)
(1 − [
]𝑡
(1 + 𝑟)
𝐺𝑟𝑜𝑤𝑖𝑛𝑔 𝐴𝑛𝑛𝑢𝑖𝑡𝑦 𝑃𝑉 = 𝐶 (𝑥)
(𝑟 − 𝑔)
1
𝐶
(𝑥)
𝐺𝑟𝑜𝑤𝑖𝑛𝑔 𝑃𝑒𝑟𝑝𝑒𝑡𝑢𝑖𝑡𝑦 𝑃𝑉 = 𝐶
=
(𝑟 − 𝑔) (𝑟 − 𝑔)
𝑄𝑢𝑜𝑡𝑒𝑑 𝑅𝑎𝑡𝑒
𝐸𝐴𝑅 = [1 + (
)] 𝑚 − 1
𝑚
𝐿𝑜𝑎𝑛 𝐴𝑚𝑜𝑢𝑛𝑡
𝑃𝑢𝑟𝑒 𝐷𝑖𝑠𝑐𝑜𝑢𝑛𝑡 𝐿𝑜𝑎𝑛 𝑃𝑉 =
(1 + 𝑖)𝑛
1
[(1 − (
)]
(1 + 𝑖)𝑛
𝐴𝑚𝑜𝑟𝑡𝑖𝑧𝑒𝑑 𝐿𝑜𝑎𝑛 𝐴𝑚𝑜𝑢𝑛𝑡 = 𝐶 (𝑥)
𝑖
𝐵𝑜𝑛𝑑𝑠 𝐴𝑛𝑛𝑢𝑎𝑙 𝐶𝑜𝑢𝑝𝑜𝑛𝑠
𝐶𝑢𝑟𝑟𝑒𝑛𝑡 𝑌𝑖𝑒𝑙𝑑 =
𝑃𝑟𝑖𝑐𝑒
[1 −
𝐵𝑜𝑛𝑑 𝑉𝑎𝑙𝑢𝑒 = 𝐶 (𝑥)
Cash Flows from Financing Activities
Cash flows from a company's activities relating to the receipt and
repayment of funds provided by creditors and investors.
Indirect Method for C.F.
NI Before Dividends
+ non-cash expenses (Depreciation, depletion, amortization)
+ decreases in assets
+ increases in liabilities
- increases in assets
- decreases in liabilities
= C.F. from Operating Activities
Purchase or Sale of Security
± Securities issued or repurchased
± Δ Principal on securities
± Δ in L.T. Asset
= C.F. from Investing Activities
𝐷1 𝐸𝑃𝑆
=
𝑅
𝑅
𝐷1
̂0 =
𝐷𝐷𝑀 𝑤𝑖𝑡ℎ 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐺𝑟𝑜𝑤𝑡ℎ = 𝑃
(𝑅 − 𝑔)
̂0 =
𝐷𝐷𝑀 𝑤𝑖𝑡ℎ 𝑍𝑒𝑟𝑜 𝐺𝑟𝑜𝑤𝑡ℎ = 𝑃
𝐸𝑥𝑝𝑒𝑐𝑡𝑒𝑑 𝑅𝑎𝑡𝑒 𝑜𝑓 𝑅𝑒𝑡𝑢𝑟𝑛 𝑜𝑛 𝐶𝑜𝑛𝑠𝑡𝑎𝑛𝑡 𝐺𝑟𝑜𝑤𝑡ℎ 𝑆𝑡𝑜𝑐𝑘 = 𝑅̂ =
Cash Flows from Investing Activities
Cash flows from a company's activities relating to asset
acquisition and disposal.
1
]
𝐹
(1 + 𝑟)𝑡
+
(1 + 𝑟)𝑡
𝑟
C = Coupon paid each period
r = Rate per period
t = Number of periods
F = Bond’s face value
Cash Flows from Operating Activities
Cash flows from company's primary business activities. E.g.
Production and delivery of goods & services for sale. Reflects cash effects of
transaction, which are included in the determination of N.I.
Δ in notes payable (use vs. source of cash)
± Δ L.T. Debt
± Preferred or Common Stock on B/S
+ Dividend Payments
= C.F. from Financing Activities